A Systematic Approach to Control Structure Design

A Systematic Approach to Control Structure Design

Zur Erlangung des akademischen Grades eines Doktor-Ingenieurs vom Fachbereich Chemietechnik der Universität Dortmund genehmigte Dissertation

von M.Sc. Jorge Otávio Trierweiler aus Porto Alegre / Brasilien

Tag der mündlichen Prüfung: 27. September 1996 1. Gutachter: Prof. Dr.-Ing. S. Engell 2. Gutachter: Prof. Dr.-Ing. W. Marquardt Dortmund 1997

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Dedicated to Vovó

Gedruckt mit Unterstützung des Deutschen Akademischen Austauschdienstes (DAAD).

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Abstract The control literature has only paid limited attention to control structure design (CSD), although an appropriate selection and pairing of measured and manipulated variables are as important as controller design. A wrong choice of the control structure may put fundamental limitations on the system's performance, which cannot be overcome by advanced controller design methods. Moreover, the complexity of a control system is largely determined by the underlying control structure. For these reasons, CSD is a very important issue in modern control design. This thesis introduces two new indices: the Robust Performance Number (RPN) and the Robust Performance Number for a plant set (RPPN). RPN and RPPN are the key elements of a seven step systematic approach to CSD that permits to analyze all different aspects that must be considered in CSD (e.g., model uncertainties, nonlinearities of the process, input saturations, interactions between the control loops and process units, failure sensitivity, sensor noise, etc.). The first step is the determination of the number of process degrees of freedom and the understanding of the main process dynamics. In the second step, heuristics or "feelings" are used to make a preselection. The third, fourth and fifth steps are the determination of the IO-scalings, the nominal performance, and nonlinearities and robustness aspects, which can successfully be analyzed using the RPN and RPPN. In steps 1 to 5, the number of possible control structures is reduced to a few possible structures that are then analyzed for failure sensitivity and controller structure and order in step 6. After performing these six steps, one may obtain more than one good solution, so that additional criteria must be considered in the final step. This work also discusses the role of the minimized condition number γ*(G) and RGA in the determination of the system controllability. It is shown that a conjectured upper bound for γ*(G) is wrong in general. A strategy to scale the system which is based on RPN is presented. This allows a correct evaluation of the sensitivity of the feedback system. The desired performance plays an important role for determination of the optimal scaling. Although the emphasis of this thesis is on CSD, contributions are also made to the controller design step. We present different approaches to performance weight selection which can be applied in H∞-controller design methods. As the H∞- and µ-synthesis usually yield high order controllers, an efficient closed-loop model reduction method is applied, which can also be used in controller design. Here the contribution is to show that the application of this procedure to the scaled system after the scaling procedure based on RPN gives very good results. A criterion based on RPN is given to identify when this procedure will produce good results in controller design.

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Acknowledgments

The distance makes no difference, only the first step is difficult. Mme. du Deffand

I would like to thank my supervisor, Prof. Dr.-Ing. S. Engell, for his NOs that have helped me to find other ways of thinking, his efforts to improve the presentation of the results in this thesis, and to give me the possibility to live and to work in Germany for five years. This time in Germany helped me to broaden my horizons such that I can say that, at this time, I concluded my Ph.D. in life too. Of course, this experience would not have happened, without DAAD (Deutscher Akademischer Austauschdienst) scholarship. Here, a special thanks to Frau H. Wahre from DAAD for her attention, friendship, and helpfulness. I would like to express my acknowledgment to Prof. Dr.-Ing. W. Marquardt for assuming the role of the external examiner of this thesis. The friendship and assistance of Ralf Müller should be noted. He was for me like my older brother in Germany and helped me to overcome my first and most difficult year in the foreign country. I want to express my gratitude to all persons that have made Studienarbeiten and Diplomarbeiten with me (in the chronological sequence: Dietmar Saecker, Dirk Schütz, Kordula Mawick, Tim Schubert, Volker Roßmann, and Bernhard Schulte). In our work together I also learned very much and made many friends. I must, of course, thank all my other friends and colleagues in the Lehrstuhl für Anlagensteuerungstechnik for the nice and positive atmosphere. In particular, I would like to mention my room sharing friends Wang Wei and Volker Roßmann. Finally, I would like to thank my girlfriend CiCi for her love and understanding and to acknowledge all the people who did not get a mention here, who added to make my life as a student in Dortmund interesting and pleasant.

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Notation and Symbols for details see

C R

field of complex number

L∞ H∞

field of real number square integrable function subspace of L2 with functions analytic in Re(s) > 0 function bounded on Re(s) = 0 including at ∞ the set of L∞ functions analytic in Re(s) > 0

G(s) P

transfer function matrix plant model (Polytopic, LFT, Affine Parameter)

|α|

absolute value of α ∈ C

Re(α)

real part of α ∈ C

MT MH M† Mij

transpose of M complex conjugate transpose of M pseudo inverse of M denotes the matrix M with row i and column j deleted

RGA λij(M)

relative gain array ij-element of RGA

_ σ(M), σ(M) σi(M) γ(M) γ*(M) L,R γ*O, γ*I(M) γ#(P) σ#(P)

maximum and minimum singular values of M ith singular value of M euclidean condition number minimized ( euclidean ) condition number of M left (output) and right (input) scaling diagonal matrices output and input minimized condition numbers minimized condition number of a plant set P maximal singular value ratio of P

Γ(G) Γ#(P) ΓLR (G) Γ#LR(P)

RPN-plot RPPN-plot RPNLR-plot with constant scalings RPPNLR-plot with constant scalings

L2 H2

appendix A.2 appendix A.2 appendix A.2 appendix A.2 section 3.1 section 3.1

section 2.3.1

appendix A.1 appendix A.1 appendix A.1 appendix A.1 section 3.3.6 section 3.3.6 section 3.4 section 3.4 section 3.4.3 section 3.4.3

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z uz, yz p up, yp BO,z BI,p

transmission zero input and output zero directions pole input and output pole directions output Blaschke factorization of zeros input Blaschke factorization of poles

=∆ ! "

defined as end of definition, theorem, or proof end of example

appendix A.4 appendix A.4 appendix A.4 appendix A.4 appendix A.5 appendix A.5

List of Acronyms for details see section 1.2

CSD

control structure design

DOF 2 FDOF 2 RDOF 3 CDOF PDOF DDOF ODOF

degrees of freedom feedforward and feedback control prefilter and feedback control feedforward, prefilter, and feedback control process degrees of freedom design degrees of freedom operation degrees of freedom

section 4.1 section 4.1 section 4.1 section 2.1 section 2.1 section 2.1

iff LFT ULFT LLFT

if and only if linear fractional transformation upper linear fractional transformation lower linear fractional transformation

section 3.1 section 3.1 section 3.1

µ (SSV) LTI MIMO NP RHP RP RPN RPPN RS SISO SVD

structured singular value linear time invariant multi-input multi-output nominal performance right-half plane Re(s) > 0 robust performance robust performance number robust performance number for a plant set robust stability single-input single-output singular value decomposition

section 3.2.4 section 3.1 section 3.2 section 3.2 section 3.4 section 3.4 section 3.2 appendix A.1

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Contents Contents ...................................................................................................................vii Chapter 1 - Introduction ............................................................................................1 1.1 Contributions of Control Theory to Process Control ......................................................3 Robust Control..........................................................................................................4 Model Predictive Control..........................................................................................5 The gap between the current control theory and the process control applications ...6 1.2 Control and Controller Structure Design ........................................................................6 1.3 Overview of the Contents ...............................................................................................9 Chapter 2 - A Systematic Approach to Control Structure Design .......................11 2.1 Process Degrees of Freedom and Process Dynamics....................................................13 2.1.1 Process degrees of freedom..................................................................................13 2.1.2 Process dynamics .................................................................................................16 Internal and external recycle streams ......................................................................16 Ill-conditioned systems and the second law of thermodynamics ............................17 Process Nonlinearity ...............................................................................................18 2.2 Heuristic Rules..............................................................................................................18 2.3 Failure Sensitivity and Controller Structure & Order ...................................................20 2.3.1 Interaction measure ..............................................................................................20 Relative Gain Array (RGA) ....................................................................................20 Some useful properties of the RGA for control purposes.......................................21 2.3.2 Failure sensitivity.................................................................................................22 RGA pairing rules:..................................................................................................23 2.3.3 RHP-zeros in the matrix elements of a transfer function.....................................24 2.3.4 Controller structure and order..............................................................................24 2.4 Additional Considerations ............................................................................................25 Number of additional measurements ......................................................................25 Operator aceptance..................................................................................................25 Input and output equipment sensitivity...................................................................25 Chapter 3 - System Description and Analysis Tools ............................................27 3.1 Representations of Systems ..........................................................................................27 Linear Time-Invariant System ................................................................................27 Polytopic Models (Multi-Model representation) ....................................................28 Affine Parameter-Dependent Models .....................................................................28 Linear Fractional Transformation (LFT) Models....................................................29 3.2 Robust Stability Analysis..............................................................................................30 3.2.1 Standard Multivariable Feedback Loop...............................................................30 3.2.2 Representations of Uncertainty............................................................................33 Unstructured Uncertainty........................................................................................33 Structured Uncertainty ............................................................................................34 3.2.3 Time invariant and time-varying perturbations....................................................35 3.2.4 µ or SSV (Structured Singular Value) .................................................................35

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3.2.5 The small µ theorem ............................................................................................38 Robust Performance (RP) .......................................................................................39 Robust performance of inverse-based controllers...................................................41 3.3 System Directionality and Scaling ................................................................................42 3.3.1 The scaling problem and the minimized condition number.................................43 3.3.2 The minimized condition number for 2x2 matrices.............................................45 Optimal scaling matrices L and R for 2 x 2 matrices..............................................45 Minimized condition number for 2 x 2 matrices ....................................................46 3.3.3 Relations between RGA and γ* for n x n matrices...............................................46 Example 3.3.6 Counter-example for the upper bound of γ*....................................47 3.3.4 Numerical solution for the minimized condition number....................................48 1. Using the convex upper bound for µ ..................................................................48 2. The LMI approach...............................................................................................49 3.3.5 Robustness interpretation of the minimized condition number ...........................49 3.3.6 Minimized condition number of P and maximal singular value ratio of P..........51 Numerical computation of the minimized condition number of P ........................51 Continuation of example 3.3.5: Calculating γ# and σ # for LV and DV structures. 52 3.4 RP- and RPP-numbers ..................................................................................................53 3.4.1Definitions of RPN and RPPN..............................................................................54 An expression for the RP-number based on RGA ..................................................55 3.4.2 RPN-scaling procedure ........................................................................................55 3.4.3 RPN and RPPN with constant scalings................................................................56 Chapter 4 - Attainable Performance and Frequency Domain Approximation ....57 4.1 Performance Requirements and Limitations.................................................................57 4.1.1 General control configuration ..............................................................................57 4.1.2 Disturbance attenuation and feedforward action .................................................59 Ratio control structures ...........................................................................................61 4.1.3 Tracking problem and prefilter compensation .....................................................61 4.1.4 Limitations imposed by sensor noise ...................................................................61 4.1.5 Limitations imposed by input constraints ............................................................62 4.1.6 Review of the design requirements......................................................................62 4.2 Performance Limitations due to Internal Stability ........................................................63 4.2.1 RHP-zero and RHP-pole constraints ...................................................................63 RHP-zero constraints ..............................................................................................63 RHP-pole constraints ..............................................................................................63 4.2.2 Performance limitations by pure time delays.......................................................64 4.3 Determination of the Attainable Performance ..............................................................65 4.3.1 Specification of the desired performance.............................................................65 4.3.2 Attainable performance considering the internal stability constraints .................66 RHP-zeros ...............................................................................................................66 RHP-Poles...............................................................................................................67 Pure time delays ......................................................................................................67 4.4 Controller Design Methods...........................................................................................68 4.4.1 H∞ controller synthesis .........................................................................................68 Lumping and moving uncertainty ...........................................................................68

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The mixed NP-RS approach: ([RbTB92],[SP96])..................................................69 The RP-approach: ...................................................................................................69 µ controller design via DK-iteration .......................................................................70 Performance and uncertainty weights .....................................................................71 Procedure for performance weight determination...................................................73 4.4.2 Controller design by frequency response approximation.....................................73 Column-by-column optimization (the least squares problem)................................74 The overall optimization ( the non-convex problem ) ............................................74 4.4.3 Controller design using the attainable closed-loop function T ............................75 4.4.4 Sequential Design Procedure ...............................................................................77 4.4.5 Controller design of a heat integrated distillation column...................................78 Chapter 5 - Control of a CSTR with the Van de Vusse Reaction Scheme ..........83 5.1 Process Description.......................................................................................................83 5.1.1 Nonlinear model...................................................................................................84 5.1.2 Controlled and Manipulated Variables ................................................................85 5.1.3 Process disturbance and parametric sensitivity....................................................86 5.1.4 Linearized Model .................................................................................................87 5.2 Analysis of Different Operating Points.........................................................................87 5.2.1 Transmission zero ................................................................................................88 5.2.2 Input saturation ....................................................................................................91 5.2.3 Operating points...................................................................................................92 5.3 µ-Optimal Controller for the First Operating Point ......................................................94 5.3.1 Uncertainty description ........................................................................................94 5.3.2 Performance weighting function..........................................................................96 5.3.3 Controller performance and order reduction........................................................96 Controller reduction by frequency response approximation ...................................97 5.4 Analysis and Controller Design using RPN..................................................................99 5.4.1 Analysis of the OPs 1, 2, and 7 using RPN and RPPN........................................99 5.4.2 Controller design for the 1st and 7th OPs using RPN .........................................103 5.4.3 A new control structure for the 2nd OP ..............................................................105 Appendix A5.....................................................................................................................112 Chapter 6 - Control of a Pilot Plant Distillation Column.....................................115 6.1 Process Description.....................................................................................................115 6.1.1 Purpose of the experimental pilot plant distillation column ..............................115 6.1.2 Description of the pilot plant .............................................................................115 6.2 Heating System ...........................................................................................................121 6.2.1 Nonlinear grey-box model .................................................................................122 ODE-System based on energy balances................................................................122 Heat transfer in the vertical thermosiphon -reboiler .............................................123 Total flow rate and the influence of bypass ..........................................................124 The dynamics of the control valve RV3 ...............................................................125 Final model structure ............................................................................................126 Parameter optimization with open-loop data ........................................................127 Parameter optimization with closed-loop data......................................................129

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6.2.2 Control structure design.....................................................................................129 Possible control structures ....................................................................................130 Understanding the ∆T - structure ..........................................................................130 Linearized Model ..................................................................................................131 Controllability analysis (RPN- and RPPN-plots)..................................................132 System dynamics...................................................................................................132 Instability with integral control action ..................................................................136 Experimental verification of the instability ..........................................................137 Solution of the instability problem .......................................................................140 6.2.3 The final control concept and the reboiler startup procedure ............................142 Final control concept.............................................................................................142 Startup procedure for the Heating System ............................................................145 6.3 Composition Control...................................................................................................147 6.3.1 Process model ....................................................................................................147 Equilibrium vs. nonequilibrium stage models ......................................................147 Discrete vs. continuous models.............................................................................148 Model equations....................................................................................................148 Determination of HETP using steady-state experimental results .........................149 6.3.2 Open-loop verification of the mathematical model ...........................................150 Nonlinear simulation (comparison in the time domain) .......................................150 Linearized model vs. black-box model (comparison in the frequency domain)...154 6.3.3 Closed-loop validation of the mathematical model ...........................................156 Controller Performance.........................................................................................156 Improving the startup with a controller.................................................................157 Chapter 7 - Conclusions and Directions for Future Works................................159 7.1 Conclusions.................................................................................................................159 7.2 Directions for Future Works .......................................................................................161 Studies of plant wide control ................................................................................161 Integrated process and control design ...................................................................162 Future theoretical analysis of RPN and RPPN......................................................162 Controller design for the 3 CDOF control configuration......................................162 Experimental verification of the "multi-effect batch distillation system" (MEBAD)...................162 Appendix A - Preliminaries ...................................................................................163 A.1 SVD (Singular Value Decomposition) ......................................................................163 A.2 Function Spaces for Systems and Signals ..................................................................164 A.3 Linearized dynamic model representation .................................................................166 A.4 Zeros and Poles of Multivariable Systems.................................................................167 Remarks about zeros:............................................................................................168 A.5 Input and output Blaschke factorization ....................................................................169 Input (Blaschke) factorization...............................................................................169 Output (Blaschke) factorization............................................................................169 References .............................................................................................................171

Chapter 1 Introduction There exist many reasons for the automation of industrial processes. In the past, the most significant ones were productivity and quality, but society has now realized that working conditions and the environment are also extremely important. An additional reason is, of course, competition. Here the economic benefits of improving control tend to be significantly underestimated. A benchmark study presented by Brisk [Br93] indicated that "There is an increasing body of evidence that effective use of control technology achieves real benefits for the process industries, contributing dramatically to increased profitability. Yet these industries have been slow to take advantage of process control." The slow process of automation is due to the fact that planning, design, construction, and automation of a modern industrial system are comprehensive tasks. The result can easily be that the process engineer underestimates or misunderstands the control problem and prescribes a control system structure that cannot perform satisfactorily. Few modern industrial plants are exact copies of plants that were build earlier, even though individual processes may be quite similar. Nowadays, control is not only related to the process units, since the processes are becoming more and more heat integrated and unit integrated, i.e., two or more different unit operations are performed together in the same equipment (e.g., reactive distillation), what often (but not always) reduces the number of manipulated variables and increases process interaction and nonlinearities. Moreover, for processes with recycles the control strategy must often be developed considering associated unit operations in a plant wide fashion. Therefore, one cannot expect the control schemes of such processes to be directly transferable, and the design of the control scheme must be evaluated for each possible complex process as a new challenge, at least to some extent. To solve the complex automation problem, a modern process control system is structured in different layers as illustrated in Figure 1.1. The traditional focus of process control has been on the lower layers ( the regulatory control layer), concerned with the unit operations, and the regulation of equipment operating conditions. Control at this layer ensures safety, rejects disturbances and regulates the operation in response to supervisory actions that determine the overall performance. Historically, the supervisory control layer has been the domain of human

1. INTRODUCTION

2

operators. Being only human, the operator tends to keep the process in a "comfort zone", far enough from the constraints to reduce the occurrence of alarms caused by controls failing to reject disturbances adequately. If this does not succeed, the controllers are put on manual until a stable operation is reestablished. This is unacceptable in a competitive environment where the operations are pushed to the constraints. The regulatory control systems must reject disturbances adequately, coping intrinsically with the constraints without operator intervention. They must also respond rapidly and stably to setpoint changes, which today should originate less frequently from the operators, but should rather result from the plant wide optimization layer by supervising and optimizing process operation. There is a vital need for effective control on this layer too, which in turn is driven by changing operational, market and other business factors communicated from the management layer (the world wide optimization layer). Many information feedback and control action paths exist, and permeate throughout the structure.

Figure 1.1 : Typical control system hierarchy in the process industry The regulatory control layer is usually concerned with the inventory control of the process, whereas the supervisory control layer is strongly related to quality (composition) control. For complex units, an additional supervisory control layer can be inserted. Observe that the layers of the hierarchy illustrated in Figure 1.1 present a time scale distribution that allows a separate analysis and a sequential design of the different layers. This means that the setpoints for a

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1.1 CONTRIBUTIONS OF CONTROL THEORY TO PROCESS CONTROL

given layer in the hierarchy, are updated only periodically. The time scale hierarchy is not only important for both control layers, but also for the plant wide optimization layer, since here usually steady-state simulators are applied. For multipurpose plants, because of the smaller difference in the time scale, it can be in some cases advantageous to apply a dynamic simulator on this layer. For the control layers, restrictions on the feedback compensator structure are often encountered in chemical plants, when the control stations are provided only with local measurements. Such decentralized information structures result in block-diagonal compensator matrices. Decentralized controllers are also attractive because the feedback is concentrated in the diagonal blocks. Therefore, they are easier to understand and to put into and out of operation and more easily made failure tolerant than general multivariable control systems. Skogestad and Postlethwaite [SP96, chapter 10] point out that: "One important reason for decomposing the control system into a specific control configuration is that it may allow for tuning of the subcontrollers without the need for a detailed plant model describing the dynamics and interactions in the process. Multivariable centralized controllers may always outperform decomposed (decentralized) controllers, but this performance gain must be traded off against the cost of obtaining and maintaining a sufficiently detailed plant model." It is advantageous to decentralize the control structure as much as possible and to include, if necessary to achieve the desired performance, additional paths for the information flow between the controllers and the units (e.g., feedforward control, one-way decoupling, etc.). At this point the question arises: What does the current control theory offer to solve the demanding automatization problems of the process industry? As an answer to this question we give a short review of the state of controller design theory and methods in the next section.

1.1 Contributions of Control Theory to Process Control From 1930 to 1960 the two-dimensional complex plane was the workhorse of control theory. Design in this space allows placement of poles and zeros. Various other two-dimensional plots are used in such designs, including the Bode plot, the Nichols plot, the Nyquist plot and the Root Locus. The essential tool here is complex analysis. These methods are known as the classical control theory and are very successful to solve single-input-single-output (SISO) problems [FPE91]. In the 1960s, the state space became the popular design space. Because the n-dimensional state space does not lend itself to plotting, the graphical methods made popular for twodimensional complex plane played a lesser role in this period [KwSi72]. The tools used were optimization, the calculus of variations, ordinary differential equations and Hilbert spaces. These methods became known as "modern" control theory. This theory had little impact in the process industry and was not able to handle the realities of constrained, nonlinear processes for which the assumed "perfect" models were not available. However, it led to a much deeper understanding of dynamical systems and provided the basis for nonlinear control system design.

1. INTRODUCTION

4

During the decade 1970-1980, the well established frequency domain approach was extended to deal with MIMO systems, where significant interaction is present. Rosenbrock [Ro74] created the Inverse (and Direct) Nyquist Array design method and MacFarlane [Ma80] proposed the Characteristic Locus design method. Current research is now focused on the establishment of robust versions of these techniques for systems with uncertain parameters [Lev96, chapter 45]. Nowadays, we see the development of many different approaches. Many "new" methods are a combination of two or three of the following branches of control theory: robust control, model predictive control, nonlinear control, and adaptive control. The general area of methods for feedback design is covered in The Control Handbook [Lev96] by several specialists in the different areas. Here we concentrate on the robust and model predictive control branches. Robust Control To cope with the robustness problem of the "modern" methods, in the 1980s the classical notions of stability margins were extended to multi-input-multi-output (MIMO) systems in a systematic way (see, e.g., [Fr87], [Mac89]). Several important developments originated from the seminal work of Zames [Za81]. By the small gain theorem and the more general small µ theorem, the stability margin of a system can be quantified by the "size" of certain transfer functions of the closed loop system. For SISO systems, the size can be measured by the peak of the Bode magnitude plot. The structured singular value (SSV or µ) (or the multivariable stability margin) was proposed to extend the concept of SISO stability margin for MIMO transfer matrices. Exact computation of these quantities is a nonconvex problem and therefore approximate computation of the scaled H∞-norm, which is a convex optimization problem, has been used in practice. In the late 1980s, a state space interpretation of H∞-theory was provided [DGKF89] and a synthesis algorithm based on two Riccati equations became available (e.g., [RbTB92], [MuTB93], [LMITB95]). Guaranteeing an upper bound on the scaled H∞ norm remains a major concern for control theorists. Since the late 1980s, the application of convex programming techniques to systems analysis and synthesis has been gaining attention. By formulating control problems as convex optimization problems, one can compute, in principle, global solutions that indicate "limits of performance". In the early stage of this method, the Youla parametrization played the central role, leading to computationally expensive infinite-dimensional convex problems (see, e.g., [BoBa91], [WE96]). In the 1990s, Linear Matrix Inequalities (LMI) were recognized to be a suitable tool for control system design based on convex programming (e.g., [BGFB94], [LMI95]). LMIs are not as universal as the Youla parameterization in the sense that the class of control specifications that can be considered by LMIs is not as large as that for Youla parametrization. However, LMIs can be solved by finite-dimmensional convex programming, and several efficient algorithms are now available (e.g., [LMITB95], [LMITL95]). Recent successes using the LMI approach include the calculation of upper bounds to µ, full order output feedback H∞ controller synthesis, full-state-feedback and dynamical outputfeedback H∞ controller synthesis with constant diagonal scalings, full-order gain-scheduled controller synthesis for plants with rapidly-varying parameters, full-order simultaneous

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1.1 CONTRIBUTIONS OF CONTROL THEORY TO PROCESS CONTROL

stabilization of multiple plants, and the computation of optimal fixed-order dynamical diagonal scalings and generalized Popov multipliers for µ analysis (for details, see [SGL94], [LMI95] and the references therein).

Model Predictive Control Another branch that has developed since the end of the 1970's within both the research community and industry is Model Predictive Control (MPC). Richalet was a pioneer of the application of MPC techniques through of the Identification and Command algorithm (IDCOM) and subsequently the Model Algorithmic Control (MAC) [RRTP78] controllers. IDCOM algorithm was improved by the development of the Dynamic Matrix Control (DMC) algorithm by Cutler and Ramaker [CR80]. Perhaps, the most popular MPC method at the moment is the Generalized Predictive Control (GPC) developed by Clarke et al. [CMT87]. The acceptance in the industry of MPC can be attributed to the fact that MPC is perhaps the most general way of posing the process control problem in the time domain. The MPC utilizes the available system model to incorporate the predicted future behavior of the process into the controller design procedure. This method of control design and implementation usually comprises: (1) a process model (e.g., linear model, neural networks, etc.); (2) a predictor equation (this is a forward simulation for a fixed number of time steps to predict the process behavior); (3) a known future reference trajectory, and (4) a cost function (this is usually a quadratic function of future process output errors and controls). Camacho and Bordons [CB95] pointed out a number of advantages of MPC over other methods. Some of them are: •

It introduces feedforward control in a natural way to compensate for measurable disturbances.

•

Its extension to the treatment of constraints and nonlinear processes is conceptually simple, because of the finite control horizon used in the online optimization procedure.

•

It is very useful when future process actions are known, since some feedforward action to a future event can easily be integrated in a control strategy using MPC.

•

MPC formulation can integrate optimal control, stochastic control, control of processes with dead time (due to its intrinsically compensation for dead times capability), multivariable control, etc.

While robust control theory has been developed in a rigorous mathematical framework by academics, MPC has not had a solid fundamental basis until recently (see, e.g., [CB95] for a historical perspective and detailed discussion). However, several successful implementations of different variants of MPC have shown the strength of this control concept. Due to its versatility, MPC is often used on the supervisory control layer. However, the robust control theory seems to be more suitable for process analysis and therefore will be used in this work as the basis of our theoretical developments. Moreover, the information obtained by the process analysis can be later applied when tuning MPC algorithms (e.g., [Lun94, chapter 5]).

1. INTRODUCTION

6

The gap between the current control theory and the process control applications In spite of the improvements in robust controller design methods, the problem is not yet completely solved. Safonov et al. [SGL94] wrote: "It remains true however that much difficulty remains in the robust synthesis of practical, non-conservative controllers, Three very important classes of robust control design issues have not been found to be readily transformable into the LMI framework. These are (1) µ-synthesis via dynamical scalings/multipliers, (2) fixed-order control synthesis and (3) decentralized controller design (i.e., synthesis of controllers with "block diagonal" or other specified structure)." Note that the points 2 and 3 are specially important for the process control, since the standard controllers in process control systems are decentralized PID controllers. About 90% to 95% of all control problems in the process industry are (or can be ) solved by PID controllers. A factor that contributes to increase the gap between the available control techniques and standard industrial practice is due not only to the fact that practicing engineers are frequently unaware of the available tools, but also the pressure to use simple and general solutions. Design, maintenance, operation, understanding, and many other related conditions, beyond basic principles, and theoretical possibilities, must all be considered when one is choosing a control system. Therefore one prefers to use simple PID controllers as much as possible, and a high-order centralized controller, a MPC, or a nonlinear controller are only used when the performance improvements are really considerable. Observe that often in the literature the PID controller is used as basis of comparison for a new algorithm. However, many of these comparisons are not fair. For example, many MPC controllers can be interpreted as two degree-of-freedom controllers, so a two degree-of-freedom control structure should be the correct basis for a comparison with a PID-like controller. The PID controller usually gives satisfactory performance. It can often be used on processes that are difficult to control provided that extreme performance is not required. There are, however, situations when it is possible to obtain better performance by other types of controllers. Typical examples are processes with relatively long pure time delays and oscillatory systems. There are also cases where PID controllers are clearly inadequate. If we consider the fact that a PI controller always has a phase lag and that a PID controller can provide a phase lead of at most 90°, it is clear that neither will work for systems that require more phase advance. A typical example is stabilization of unstable systems with time delays. However by a correct choice of the control structure and/or process modifications, the number of the cases where a PID controller will not work can be largely reduced.

1.2 Control and Controller Structure Design If the reader looks, for example, in The Control Handbook [Lev96] to get a general idea about the different methods currently used in control system design, he or she will recognize that all methods presented assume that a control structure is given at the outset. Many basic questions are not answered by any design method. These questions can be grouped into three different classes as follows:

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1.2 CONTROL AND CONTROLLER STRUCTURE DESIGN

1. Which variables should be controlled, which variables should be measured, which inputs should be manipulated, which connections should be made between them? 2. Which control configurations (e.g., cascade control, feedforward control etc.) should be used, how must the different controls be organized hierarchically, how many degrees of freedom must the controller have to achieve the desired performance? 3. Which control type (e.g., linear, nonlinear) must be applied, which controller structure (e.g., decentralized, block-diagonal) should be used, which is the necessary controller order to achieve a given performance under the given constraints? We call the investigation of these questions Control Structure Design (CSD).

Figure 1.2 : Schematic representation of Control Structure Design CSD can be defined more formally as the phase of the control system design which precedes the actual controller design, as illustrated in Figure 1.2, consisting of three main stages in which decisions are made on the number, place and type of actuators (inputs) and sensors (outputs) to be used (IO-selection phase); on the hierarchical interconnections between measured and manipulated variables ( CC-selection phase); and finally on the structure, order, and type of controller that must be used to solve the control problem (CSO-selection).

1. INTRODUCTION

8

The IO-selection phase is the first step in CSD and has a strong relation to process design. To decide where the sensors and manipulators must be placed, it is necessary to understand the process dynamics. Therefore, the process understanding is a key activity for successful control system design. For example, if recycle streams occur in the plant, the procedure for designing an effective "plant wide" control system becomes much more complicated and is much less understood. Processes with recycle streams are quite common, but their dynamics are often complex. As pointed out by Luyben [Luy93]: " This is one of the most important areas in process control that cries out for some engineering research. The typical approach in the past for plants with recycle streams has been to install large surge tanks. This isolates sequences of units and permits the use of conventional process design procedures. However, this practice can be very expensive in tankage capital costs and in working capital investment. In addition and increasingly more important, the large inventories of chemicals can greatly increase safety and environmental hazards if dangerous or environmentally unfriendly chemicals are involved." Often, changing the control structure is more fruitful than trying to solve the problem with a complex control algorithm. The conventional approach to synthesizing the control system is to analyze the process after it was completely designed. The process design step assumes steady-state conditions, yet the control system must cope with the dynamic situation. This may lead to a poor match between the process characteristics and the control system characteristics. In order to design a better controlled plant, the process design and the process control design should be considered simultaneously or sequential-interactively (i.e., process design, CS-analysis, process redesign, CS-analysis, etc.). Doing so, additional sensors and actuators can be installed in the process or equipment dimensions and connections can be modified before the plant is built and changes become very costly. Real control problems are seldom solved using a single controller. Many control systems are designed using a "bottom up " approach where usually PID controllers are combined with other elements, such as filters, selectors and others. The choice and the combination of these different elements are called Control Configuration. The choice of cascade, feedforward, 2 degree-of-freedom, and hierarchical control belongs to the control configuration task. •

Cascade control. Cascade control is used, for example, when there are several measured signals and only one control variable. It is particularly useful when there are significant dynamics (e.g., long dead times or large time constants) between the controlling variable and the controlled variable. Tighter control can then be achieved by using an intermediate measured signal that responds faster to the control signal. Cascade control is built by nesting the control loops. The inner loop is called the secondary loop; the outer loop is called the primary loop. The reason for this terminology is that the outer loop controls the signal we are primarily interested in. It is also possible to have cascade control with more nested loops.

•

Feedforward Control. Disturbances can be eliminated by feedback. With a feedback system it is, however, necessary that there is an error before the controller can take actions to eliminate disturbances. In some situations it is possible to measure disturbances before they have influenced the processes or to react immediately to

9

1.3 OVERVIEW OF THE CONTENTS

setpoint changes. It is then natural to try to eliminate the effects of the disturbances before they have created control errors. This control paradigm is called feedforward. The determination of the necessary Controller Structure and Order is strongly related to controller design, but, as we will see in this work, much can be said about the necessary type of the final controller (e.g., linear vs. nonlinear, centralized vs. decentralized, high order vs. low order) before designing the controller. The current literature has only paid limited attention to CSD, although appropriate selection and pairing of measured and manipulated variables are as important as controller design by itself. A wrong choice of the controller structure may put fundamental limitations on the system performance, which cannot be overcome by advanced controller design. Moreover, the complexity of a control system is largely determined by the underlying control structure. Many different points must be considered and analyzed by the CSD, for example, model uncertainty, nonlinearities of the process, input saturations, interactions between the control loops and process units, failure sensitivity, sensor noise, disturbance sensitivity, etc. A recent survey on the literature on control structure design is given by Van de Wal and de Jager [WJ95] and the authors concluded that an intensive further research has to be done in the area of CSD for both nonlinear and linear control systems, since none of the currently available CSD methods seems to be completely satisfactory. Here we intent to contribute to the development of systematic procedures to solve this problem. The next section presents the outline of this thesis.

1.3 Overview of the Contents The main goal of this thesis is to present a systematic approach to CSD. The results and procedures are presented in such a way that our approach can also be understood by a reader who is not an expert in robust control theory, which provides the theoretical basis of this work. A systematic approach to CSD is presented in Chapter 2. The method consists of seven steps. The first step is the determination of the number of degrees of freedom of the process and understanding the main process dynamics. In the second step, we use some heuristic or "feelings" to make a preselection. The IO-scaling and the determination of nominal performance represent the third and the fourth step in our procedure and are performed using the Robust Performance Number (RPN) concept. Since the system in general is nonlinear and the robustness properties depend on the control structure, we need to consider this factor in CSD. This is done in step 5 using the Robust Performance Number of a plant set (RPPN). In steps 1 to 5, the number of possible control structures is reduced to a few possible structures that are then analyzed for failure sensitivity and controller structure and order in step 6. After performing these 6 steps, usually we obtain more than one good solution, so that additional criteria must be considered, e.g., the number of necessary measurements, operator acceptance, etc. This is the seventh step of the proposed procedure. Chapter 3 introduces the general framework of linear controller analysis and design. Robustness issues and uncertainty descriptions are also discussed. Finally, the Robust

1. INTRODUCTION

10

Performance Number (RPN) and the Robust Performance Number of a plant set (RPPN) are defined. Both RPN and RPPN are the key concepts in our approach. In Chapter 4, we discuss the properties of a feedback system. In particular, we consider the benefit of the feedback structure and the design tradeoffs for feedback controller design. A procedure to determine the attainable performance is presented taking into account the limits imposed by plant model mismatches, sensor errors and actuator limits. This chapter ends with the presentation of a controller design method that allows the design of low-order decentralized controllers in a systematic way. In Chapter 5 we treat a continuous stirred tank reactor (CSTR) with the Van de Vusse side and consecutive reaction scheme. Two operating points of the CSTR were considered in the literature as benchmark problems for nonlinear process control. Here an analytical expression for the linearized model is used and a new operating point is suggested. It is also shown how the change of the controlled variable can improve the controllability of the system. Finally, three representative operating points are analyzed in detail via the methodology presented in Chapters 2 and 3. At the University of Dortmund, a Multi-Purpose Packed Distillation Column (MPPDC) was built in cooperation by the Thermal Separation Processes Group and the Process Control Group to analyze the static and dynamic behavior and the control of packed distillation columns. The MPPDC is described in chapter 6. The idea of hierarchical control is applied to the MPPDC. Special attention is given to the heating system (HS), since it constitutes an experimental plant for a process with recycle streams. Another issue, which is also analyzed in this chapter are the special dynamic characteristics presented by Packed Distillation Columns. Here, experimental results of the pilot plant are modelled by a simplified dynamic model and special attention is given to the properties which are important for feedback control. Chapter 6 also shows how the process dynamics can be modified very advantageously by applying a feedback controller. A new controlled variable is proposed to take the temperature nonequibrium dynamics into account. Finally, conclusions and recommendations for future research are given in Chapter 7. Appendix A presents the necessary background to understand this work. All results are presented in the differential algebraic equations (DAE) setting. Often, chemical processes are modeled naturally by DAE systems, where explicit differential equations arise from the dynamic balances of mass and energy, while algebraic equations are obtained from empirical correlations, thermodynamical equilibrium relations, pseudo steady-state assumptions, etc.

Chapter 2 A Systematic Approach to Control Structure Design In this chapter we present a systematic approach to control structure design. The method consists of seven steps as illustrated in Figure 2.1. The first step is the determination of the number of degrees of freedom of the process and to understand the dominant process dynamics. In the second step we use some heuristics or "feelings" to make a preselection. For many processes one can already reduce the number of possible control structures considerably. Moreover, this allows us to group all possible structures in different sets with essentially equivalent properties. The third step is IO-scaling, it can be applied to the general process, i.e., to all manipulated and controlled variables, or to each set of possible control structures separately. In Chapters 3 and 4, we will see that the IO-scaling plays an important role in the system analysis and design. In Chapter 3, an efficient IO-scaling procedure based on RPN will be presented. The fourth step of our procedure, the choice of the desired nominal performance, will be the main theme of Chapter 4. Here we analyze the effects of input saturation, sensor noise, RHP-zeros, etc. on the nominal performance. In other words, we determine the attainable performance of a particular control structure using the procedure that will be presented in Chapter 4 and the structures that cannot achieve the desired nominal performance are discarded. As the degree of nonlinearity and robustness requirements depend on the control structure, we also have to consider this in CSD and it is done in step 5 of our procedure. Here the basis of the analysis will be the RP- and RPP-numbers developed in Chapter 3. In steps 1 to 5, the number of possible control structures is reduced to a few possible structures that are then analyzed for failure sensitivity and necessary controller structure and order in step 6. After performing of these 6 steps, usually we find more than one good solution, so that additional criterions must be considered, e.g., the number of necessary measurements, operator acceptance, etc. This step that is more or less subjective and strongly dependent on the particular problem at hand constitutes the seventh step of our procedure. Sometimes we can bypass some step of the procedure or make two or three steps together. Moreover, the sequence in which the different steps are applied is not rigid. For example,

2. A SYSTEMATIC APPROACH TO CONTROL STRUCTURE DESIGN

12

some of the considerations in step 7 can (or must) already be applied in step 2, e.g., if the set of measurements is fixed, the number of possible control structures is automatically reduced.

Figure 2.1 : Seven steps of Control Structure Design: 1. Process dynamics and degrees of freedom, 2. Heuristic preselection, 3. IO-Scaling, 4. Nominal performance determination, 5. Degree of nonlinearity (DN) and degree of uncertainty (DU), 6. Failure sensitivity and controller structure and order, 7. Additional criterions like: number of necessary measurements, operator acceptance, etc.

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2.1 PROCESS DEGREES OF FREEDOM AND PROCESS DYNAMICS

In Chapters 3 and 4, the steps 3, 4, and 5 will be discussed in detail. Therefore, in this chapter we concentrate on the discussion of the remaining steps. In section 2.1 the process dynamics and degrees of freedom are considered. The heuristic preselection is discussed in section 2.2. Failure sensitivity and controller structure and order is the theme of section 2.3. Finally, section 2.4 presents a discussion of the additional criteria.

2.1 Process Degrees of Freedom and Process Dynamics Before we start to develop a CS, we need to know the number of process degrees of freedom (PDOF) and, of course, the control objectives. Without knowing the number of PDOFs is not possible to determine the set of feasible CS. Another important point strongly related to the first screening step is the understanding of the process dynamics. Often qualitative informations are sufficient to reduce the number of feasible control structures considerably. Internal and external recycle streams, the directionality of the process, and the process nonlinearities have a strong influence on the final process dynamics and properties.

2.1.1 Process degrees of freedom The first step in the development of control strategies is the determination of the degrees of freedom of a process. Only when this was done the following questions can be answered: • How many operating conditions or desired outputs can be specified? • Is the process self-regulatory and stable, or does it need feedback control? How many control loops are required? • Can we optimize the process by changing some of the operating conditions? • How much information is necessary to solve the problem? What are "Degrees of Freedom"? The number of additional pieces of information needed to uniquely specify a problem is termed as the number of "degrees of freedom" of the problem. In general, the following holds:   Number of  Number of   Number of  Degrees   unknowns   independent         = −  of Freedom  of the  equations relating   .  (DOF)   problem   unknowns

(2.1)

It would be extremely tedious if every time an engineer is faced with a process, he or she had to count up all the equations (e.g., material balances, heat balances, composition restrictions, relations of enthalpy to temperature, equilibrium relations, etc.) and all the unknowns (e.g., compositions, flow rates, pressures, temperatures, heat flows, numbers of stages, etc.) and take the difference, as in (2.1), to find the degrees of freedom of the problem. Of course, if we develop a process model, equation (2.1) must be applied to verify the consistency of the


14

model, before we can solve it. Usually, simulation languages (e.g., [SpeedUp], [gPROMS]) perform this automatically. For process analysis, we can simplify this analysis. As for different goals we have different variables for which we can specify a value, it is useful to divide the DOFs into: design DOF, operation DOF, dynamic DOF, and process DOF. Definition 2.1.1 Design degrees of freedom (DDOF). The design DOFs are the number of variables needed to specify the steady-state system design procedure uniquely. Definition 2.1.2 Operation degrees of freedom (ODOF). The operation DOF are a subset of the DDOF which can be modified during the normal operation of the system. Definition 2.1.3 Dynamic degrees of freedom. The dynamic DOFs are the number of the variables needed to specify the dynamic behavior of the system uniquely. It includes the DDOF and additional variables usually related to inventories of the system. Definition 2.1.4 Process degrees of freedom (PDOF). The process DOF are a subset of the dynamic DOF which can be modified during the normal operation of the system. In other words, the DDOF correspond to the number of variables which have to be specified when we work with a steady-state simulator. For example, for a simple distillation column, it is necessary to specify the number of trays, the feed tray, column pressure, reflux ratio (or distillate composition xD ) and distillate (or bottom ) flow (see Figure 2.2(a)). Usually, the DDOF consists of all the possible choices we make at the design stage, such as: wall tickness, column internals, etc. Observe that DDOF includes many variables (e.g., number of trays) that cannot be changed during the normal operation of the unit. Therefore, it is useful to define the subset of the design variables that can be manipulated during the normal operation. We give the name ODOF to this subset of the variables. For process control, we need to know the number of PDOF, so that we do not attempt to overor undercontrol the process. The PDOF is strongly related to the manipulated variables of a system. Since the manipulated variables in the process industry are mainly valves and sometimes electrical heatings, we only have to determine the number of the rationally placed control valves and eletrical heatings to determine the number of PDOF. The "rationally placed" qualification is to emphasize that we have avoided poorly conceived design such as the placement of two control valves in series in a liquid filled system. In Figure 2.2(b) there are five control valves, one on each of the following streams: distillate (D), reflux (L), coolant, bottoms (B), and heating medium (Q or V). We are assuming for the moment that the feed stream is set by the upstream unit. So this simple column has 5 PDOF. The PDOF can be used to control variables related to inventory or quality (composition) controls. The inventories in a process must always be controlled. Inventory loops involve usually liquid levels for liquid systems, pressures for gas systems, and weights for solid systems. In our simple distillation column example, this means that the liquid level in the reflux drum (MD), the liquid level in the base of the column (MB), and the column pressure must be controlled. In addition, there are two variables that can be controlled, independent of the number of the components, i.e., a simple, ideal, binary system has 2 composition (quality)

15


DOF; a complex, multicomponent, nonideal distillation system also has 2 composition (quality) DOF.

(a) Design DOF

(b) Process DOF

Figure 2.2 : Different types of DOF for a distillation column: design DOF (a), process DOF (b)

Often primary variables, i.e., variables that we are interested to control, are substituted by secondary variables that are in some way related to the primary variables, but are more easily measured. Here, a typical example is the composition in a distillation column (primary variable) that is usually substituted by the control of a temperature. In this case, the temperatures are only secondary variables that allow us to estimate the composition via phase equilibrium relations. The relation between temperature and compositions is a bijective function for ideal binary systems under a given pressure. For a multicomponent system there is usually more than one possible composition for a given temperature and pressure. In this case, the temperature control is usually applied in a cascade control structure where the setpoint of the temperature control is determined by the composition control loop. Observe that by closing secondary control loop is equivalent to replace the original manipulated variables (typically, flows and valve positions) by some new manipulated variable, i.e., the setpoints for the secondary control loop, but the number of PDOF remain the same (see, e.g., [WS96], [Luy92, chapters 8 and 9]). As the ODOF are related to the operating conditions, the ODOF are the natural variable set for the steady-state plant optimization usually performed at the upper layer of the hierarchical control structure which is responsible to determine the setpoints for the lower layers.


16

2.1.2 Process dynamics Process simulation is a valuable tool for understanding of the process dynamics and for the design of process control strategies. The scope and complexity of process simulation will depend on the problem being solved and must normally be determined on a case-by-case basis. In defining the size of the simulation, for example, which unit operations to simulate, the engineer looks for logical cuts in the process. Large interprocess inventories are good boundaries for simulation models because most disturbances are significantly attenuated at those points. Also, portions of a process that are not connected by recycles can be segregated. To illustrate this point, consider distillation columns which operate somewhat independently of other unit operations and therefore can be studied one at a time. This occurs, for example, for a process where all unit operations are in series with no recycle streams. In this case, the control strategy can be developed for each unit operation separately. Additionally, tanks for column feed or for product streams isolate columns, allowing column control strategies to be developed independently. However, for processes with recycles or processes with coupled distillation columns, a distillation column control strategy must often be developed taking associated unit operations into account. Studying the control and operations of distillation columns and associated equipment for processes with large recycles usually requires a plant wide process model and simulation. The primary benefit of process simulation is improved process insight and understanding. The thinking process that the user goes through in developing and running a simulation is what leads to improved process insight and understanding. Better process understanding results in the formulation of better control strategy designs, as well as better flowsheet designs and better safety systems. A process simulation does not explicitly provide such solutions, but as a thinking tool it assists in the development of solutions. A simulation reveals which process parameters are important and which ones are not. Sometimes the dynamic behavior is easily explained, but counterintuitive. The mind does not easily predict dynamic responses by superimposing all the simultaneous dynamic effects in a chemical process, particularly, if the process contains one or more recycles. The explicit results from process control simulations tend to be more qualitative than quantitative. Application of process control simulation often results in a relative decision such as "control structure A is better than B" or "structures B and C are almost equivalent", whether the inclusion of a continuous analyzer will improve product quality, selection of appropriated inference variable for a product quality, if a surge tank is necessary, etc. In contrast, process design simulations often provide absolute numbers, for example, reboiler heat duty, distillation column diameter, etc. For some successful applications of plantwide dynamic simulation the reader is referred to [Luy92, chapter 6 and 20]. We will discuss here three important aspects that influence the process dynamics. Internal and external recycle streams Recycle streams occur when two systems are connected with at least a forward and a backward stream. We speak of internal recycle streams when the two systems in consideration are at the same equipment or unit operation. Typical examples of this class are all countercurrent unit operations like, for example, distillation columns. Here the interconnected

17


systems are the column trays placed one after another, in such a way that each tray is influenced only by its neighbors' trays. Process with internal streams are sometimes in the literature called cascade processes [Mor95]. External recycle streams (or simply recycle streams) represent the case where the two different pieces of equipments or unit operations are involved. A typical example of this class is a reactor/separation/recycle process. Although most subsystems of a system with recycle streams are rather simple (such as the trays of a column) their overall behavior can be complex, and very different from the behavior of the individual elements. The complex dynamics come from the positive feedback effect between the systems. This positive feedback characteristic of systems with recycle streams was often used in the literature to explain the usually large time constant of process with recycle streams. For example, Kapoor et al. [KMcM86] used a simple positive feedback transfer function model to show why high-purity distillation column time constants are so large. Similar results are presented by Morud [Mor95, chapter 4 and 5]. Luyben in a series of papers [Luy93/1/2/3] discussed the dynamic of process with external recycle streams showing with some examples that the positive feedback is also responsible for the modification of the system dynamics. In section 6.2 we present an experimental system with an external recycle stream. Ill-conditioned systems and the second law of thermodynamics Output constraints always exist on a process. Most of them are only a consequence of the input constraints that limit the achievable output values and can be easily removed with a change in the input range (e.g., a new pump or valve) or a process modification (e.g., increase the number of column trays, increase the heat transfer area in heat exchanger, etc.). Another type of output constraints is due to the second law of thermodynamics which determines the direction that some process will take. For example, the energy flow in a heat exchanger is from the hot to the cold temperature and it will be not possible to transfer energy from one medium to the other if the temperature difference is zero. Therefore, the second law of thermodynamics is responsible for an unremovable output constraint. Moreover, the difficulty to go to the direction of the thermodynamic attainable value increases with the closeness to this point, whereas it will be more and more easy to go to the opposite direction. Another illustrative example is a separation process where the necessary energy to improve the purity of a product (i.e., to improve the system organization) increases with product purity. The different facility to go in one direction in comparison to another is responsible for the illconditionedness of a system and occurs when the inputs are aligned to the "second law of thermodynamics direction". For example, the temperature of the hot stream will be always above the temperature of the cold stream at the same point in a countercurrent heat exchanger. When the heat transfer is very effective such that the two outlet temperatures are almost the same, it will be difficult to make one outlet stream hotter and the other colder (this is the weak or difficult or output low-gain direction of the plant), whereas we may easily make them both hotter or colder ( this is the strong or easy or output high-gain direction of the plant). Another example is high-purity distillation columns operating with reflux L and boilup V as independent variables (i.e., LV-structure). In this case the low-gain direction is related to manipulations of L and V in the same direction (i.e., ∆L=∆V) corresponding to a simultaneous


18

variation of both product purities. The high-gain direction consists of manipulations of L and V in opposite directions (i.e., ∆L = -∆V) which increases only one product purity by decreasing the other one. This asymmetrical behavior is also responsible for the over proportional energy increase necessary to improve the product quality in comparison to the energy saving produced by reducing the product quality. This fact is one of the motivations for dual-composition control for simple distillation columns. Another motivation is the reduction of the product variability (see, e.g., [Luy92, chapter 16]). Process Nonlinearity Most chemical processes are inherently nonlinear in nature. Nevertheless, they are often treated using linear analysis and design techniques in order to simplify the development, implementation, and operation of a control strategy. However, the extent of nonlinearity in many chemical processes is such that controller design and analysis methods based on linear process models may no longer be satisfactory. Nonlinearities may arise from chemical process characteristics (e.g., the Arrhenius rate expression), input saturation (e.g., valve limits), and output thermodynamical limitations (physical limits on output variables as a consequence of the second law of thermodynamics). The impact of the nonlinearities on controller stability and performance depends on the degree of nonlinearity of the process and the intended range and direction of operation. For example, low purity distillation columns exhibit moderate nonlinearity caused mainly by the nonlinearities related to the thermodynamical equilibrium model (or correlation), whereas a high purity columns with a large number of trays will present strong nonlinearity and directionality caused by the second law of thermodynamics (output limitations). As we will see in Chapter 3, we can assess the degree of process nonlinearity using the RPP-number. Note that the nonlinearity of the process depends on the control structure. For example, the plant nonlinearity of a high purity distillation column can often be compensated using "logarithmic compositions" instead of "absolute compositions" ( see, e.g., [MZ89, chapter 16], [Luy92, section 7.3.6]). If large variations in the operating point of the column are expected, one may prefer to use the weighted average of several tray temperatures instead of only one temperature. This will avoid the problem of an insensitive measurement if the temperature profile becomes flat at the selected tray location. The setpoint of the average temperature can easily be reset to its correct value by an outer cascade controller based on composition measurements. Wolf and Skogestad [WS96] interpret the weighted average temperature as a static composition estimator.

2.2 Heuristic Rules A large system will typically have a large number of inputs and outputs. With a top-down approach, a system can be divided into small subsystems. It is then desirable to group different inputs and outputs together, so that a collection of smaller systems is obtained. If possible, the grouping should be done such that (i) there are only weak couplings between the subsystems; and (ii) each subsystem has the same dynamic scale; i.e., time constants, time delays, and RHP zeros are of the same magnitude. There are no general rules for the grouping. Nevertheless, there are some useful and intuitive rules that can be used as guidelines. For

19

2.2 HEURISTIC RULES

example, Seborg et al. [SEM89, chapter 28] present several guidelines for the selection of controlled, manipulated, and measured variables. Some of them are listed below with some modifications.

Guidelines for the Selection of Controlled Variables: 1. Select variables that are not self-regulating. E.g., the column pressure of an atmospheric column is a self-regulating variable. Therefore, it need not be controlled. 2. Choose output variables that may exceed equipment and operating constraints (e.g., temperatures, pressure, and compositions). 3. Select output variables that are a direct measure of product quality and have favorable dynamic and static characteristics. Guidelines for the Selection of Manipulated Variables: 1. Select inputs that have large effects on the controlled variables and rapidly affect the controlled variables. The above guidelines are usually applied to each unit operation. The next level of complexity is to look at an entire plant which is made up of many unit operations connected in series and parallel, with recycles of material and energy among the various parts of the plant. As pointed out by Luyben [Luy90], Buckley was one of the pioneers in this aspect of control. He developed a procedure that is still widely used today. Since this procedure is very useful, we state the Buckley Procedure following Luyben [Luy90], which consists of the following steps: 1. Lay out a logical control scheme to handle all the liquid levels and pressure loops throughout the plant so that the flows from one unit to the next are as smooth as possible. These loops are called the material-balance or inventory loops. If the feed rate is set in front of the process, the material-balance loops should be set up in the direction of flow, i.e., the flow out of each unit is set by liquid level or pressure in the unit. If the product flow rate out of the plant is set, the material-balance loops should be in the direction opposite of flow; i.e., the flow into each unit is set by a liquid level or pressure in the unit. Doing so, all control loops will be simple low order systems and there will be no stability problems. With the wrong choice, there may be instabilities due to the feedback around all units. Moreover, it can also be shown that choosing the right control flow direction can result in smaller inventories (e.g., smaller storage tanks). 2. Then design the composition (quality) control loops for each unit operation. Determine the closed-loop time constants of these product-quality loops. Size the holdup volumes so that the closed-loop time constants of the inventory loops are a factor of ten smaller than the closed-loop time constants of the product-quality loops. This breaks the interaction between the two types of loops. Luyben [Luy94] reported the "snowball" phenomenon, i.e., a small change in a load variable causes a very large change in the recycle flow rates around the system. It is important to note that snowballing has nothing to do with dynamics. It is a purely steady state phenomenon and depends on the applied control structure. In [Luy94] it is shown that the snowball problem can


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be prevented by using a control structure that fixes the flow rate of one stream somewhere in a liquid recycle loop. For example, in processes with one recycle stream, the flow rate of the reactor effluent can be set, whereas for processes with two or more recycle streams, the flow rate of each recycle can be fixed. Heuristic rules can also be used during the process design. For example, controllability of heat exchanger networks depends on the control configuration or bypass locations. As heat exchanger network synthesis has in general flat optima, a large number of designs with near minimum cost will be possible. So the application of some heuristics for controllability should be used as additional criterion. Doing so we can achieve a design close to the optimal value with very good operability and controllability characteristics. Mathisen [Mat94] formulates heuristics that describe common features of heat exchanger networks that are easy to control and operate and explains how these operability heuristics may be taken into account during synthesis with the pinch design method, which is the most common synthesis method for heat exchanger networks (see, e.g., [Dog88], [Sm95]). In [Mat94, chapter 8] it is shown how the operability heuristics may be formulated as logic statements and linear constraints by including a new set of binary variables for synthesis approach using mathematical programming. The rules of thumb presented here should be regarded only as guidelines. The advantage of these heuristic rules is that with qualitative models of the process we can suggest some possible structure that will work. But for a final decision it is usually necessary to consider a more exact and mathematical criterion like the one presented in the next chapters. The reader will find many illustrative examples and additional discussions and references in [SEM89, chapter 28], [Luy90, section 8.8], [NL89, appendix D], and [PL93].

2.3 Failure Sensitivity and Controller Structure & Order 2.3.1 Interaction measure Interaction is an important system property and constitutes to the difference between MIMO and SISO systems. A MIMO system with little interaction can be considered as a multi-SISO system. It means that each channel can be considered independent of the others making the controller design easier and the on-line tuning straightforward. Interaction is not bad in principle, in many cases even quite beneficial, since the actuator constraints problem can be reduced, the effect of not pinned RHP-zeros can be moved from one to other channel, the same applies to some extent for deadtimes, etc. The main problem with interacting systems is that we have more freedom, and usually more freedom needs more knowledge and understanding to use it correctly. Therefore, to know how much we need to know about a system, it is necessary to measure its degree and type of interaction. The 'classical' interaction measure is the relative gain array (RGA).

Relative Gain Array (RGA) The RGA was originally introduced by Bristol [Bri66] to measure steady-state interactions caused by decentralized diagonal control, but the RGA also has a number of important

21

2.3 FAILURE SENSITIVITY AND CONTROLLER STRUCTURE & ORDER

properties as a frequency-dependent function. Most authors have confined themselves to use the RGA at steady state. A complete review of the use and interpretation of the steady state RGA is given by Grosdidier et al. [GMH85]. Some additional properties are given by Hovd and Skogestad [HS92].

Definition 2.3.1 RGA The RGA of a complex non-singular n x n matrix M, denoted by RGA(M), is a complex n x n matrix defined by RGA( M ) = M × (M −1 ) , where the operation × denotes element by element ∆

T

multiplication (often called the Hadamard or Schur product). ! The RGA-elements λij can also be calculated by

λij = (− 1)i + j

( )

mij det M ij det (M )

where Mij denotes the matrix M with row i and column j deleted. For 2 x 2 matrices, we can write the following special equation for the RGA: 1 − λ11   λ11 1 RGA( M ) =  and λ11 = .  m m λ11  1 − λ11 1 − 12 21 m11m22

(2.2)

Some useful properties of the RGA for control purposes Here we list some useful properties for control purposes. For more details the reader is referred to [GMH85], [HS92], and [SP96]. In particular, Appendix A.4 in [SP96] gives a summary of the algebraic properties of the RGA. 1. The RGA is an interaction measure. The RGA is a measure of diagonal dominance. For decentralized control we prefer pairings for which the diagonal elements of RGA at the crossover frequency are close to 1. In addition, Skogestad and Postlethwaite [SP96] point out that for multivariable design methods, like H∞ loop-shaping ([MG90]), it is simpler to choose the weights and shape the plant if we first rearrange the inputs and outputs to make the plant diagonally dominant. 2. The RGA is scaling invariant. More precisely, RGA( LMR ) = RGA(M) where L and R are diagonal matrices. 3. The RGA is a measure of sensitivity to relative element-by-element uncertainty in the matrix. The matrix M becomes singular if we make a relative change -1/λij in its ij'th element, that is, if a single element in M is perturbed from mij to mij'= mij (1-1/λij). 4. The norm of the RGA is closely related to the minimized condition number γ*(M). (See Section 3.3.3 for details). 5. RGA and DIC. RGA can be used as criterion for Decentralized Integral Controllability (DIC) (see the next section for details).


22

2.3.2 Failure sensitivity A desirable property of a control system is that it has integrity, i.e., the closed-loop system should remain stable when subsystem controllers are brought into and out of service. This property is especially important for decentralized control systems where usually the different controllers are put into (or out) of operation one after the other. Mathematically, the system possesses integrity if it remains stable when the decentralized controller Kd is replaced by diag{εi} Kd where εi can take the values of 0 or 1. A stronger requirement is that the system remains stable as the gains in various loops are reduced (detuned) by an arbitrary factor, i.e., 0 ≤ εi€ ≤ 1. This stronger CS property for a controller with integral action is called decentralized integral controllability (DIC) (see, e.g., [MZ89, chapter 14], [SP96, chapter 10]). Note that DIC is very desirable for both integrity and on-line tuning. Definition 2.3.2 Decentralized Integral Controllability (DIC). A system G (corresponding to a given input-output pairing) is DIC if there exists a decentralized controller K such that the standard closed loop system of Figure 2.3 with MK = I and A= I is stable and such that each individual loop may be detuned independently by a factor εi , i.e., 0 ≤ εi€ ≤ 1, without introducing instability.

Figure 2.3 : Standard Feedback Configuration As the DIC considers only the existence of a controller with integral action, it depends on the plant G and the chosen pairings only. The steady-state RGA provides a very useful tool to test whether a given plant G and pairing is DIC. The following result which was first proved by Grosdidier et al. [GMH85] illustrates this relation between RGA and integrity: Theorem 2.3.3 Implications of negative RGA. Consider a stable square plant G and a diagonal controller K with integral action in all elements, and assume that the loop transfer function GK is strictly proper. If a pairing of outputs and manipulated inputs corresponding to a negative steady state relative gain is used, then the closed loop system has at least one of the following properties: 1. The overall closed loop system is unstable. 2. The loop with the negative relative gain is unstable by itself. 3. The closed loop system is unstable if the loop with the negative relative gain is opened. ! DIC means that the controller gains of the individual loops can be reduced independently without introducing instability what is equivalent to require that all the eigenvalues of GK(0) stay in the open RHP for all positive scalings. Yu and Fan [YF90] point out that the

23

2.3 FAILURE SENSITIVITY AND CONTROLLER STRUCTURE & ORDER

mathematical formulation of DIC of square systems is related to the mathematical D-stability1 problem. As discussed by Campo and Morari [CM94], D-stability provides necessary and sufficient conditions except in a few special cases, such as when the determinant of one or more of the submatrices (obtained by deleting rows and corresponding columns in G(0)) is zero. RGA pairing rules: Bristol's intention by the introduction of RGA was to develop a steady-state measure of interactions for the input-output pairing for efficient design of a decentralized controller. RGA-elements λij close to 1 mean that the gain between the input variable uj and the output variable yi is unaffected by closing the other loops. In other words, λij close to 1 indicates small system interaction. In this sense, the RGA is also used as a frequency dependent pairing selection tool. The common rule is to pair the variables for which the RGA-matrix in the crossover region is close to identity (see, e.g., [SP96]). The other common pairing rule is related to the DIC property of the CS. As it is desired to pair input and output variables such that the system is DIC, we will now state necessary and sufficient conditions for DIC. For 2 × 2 and 3 × 3 plants, we have a tighter condition for DIC based on RGA(0). The plant G is DIC if and only if: • 2 × 2 systems: λ11(0) > 0. Note that for 2 × 2 plants there is always a pairing such that the system is DIC. • 3 × 3 systems:([YF90]) (i) NI{G(0)} = det(G(0))/Πi gii(0) > 0, (ii) λii > 0, i=1,2,3, and (iii) λ11+ λ22+ λ33 > 1 , where λii is the diagonal element i of the RGA(G(0)). For n x n systems, the following condition is necessary for DIC (see [SP96, chapter 10] for details): • n × n systems. Assume without loss of generality that the signs of the rows or columns of G have been adjusted such that all diagonal elements of G are positive. Then one may compute the determinant of G(0) and all its principal submatrices (obtained by deleting rows and corresponding columns in G(0)), which must all have the same sign to have DIC. Observe that the DIC-pairing rule is related to the low frequency range whereas the interaction degree is related to the RGA elements at the crossover frequency.

When the controller K in Figure 2.3 is a multivariable controller, both matrices A and MK must be used for a correct description of actuactor and sensor failure, respectively. However, it is not clear for a MIMO controller what a failure may lead to. For example, if an actuator fails, in order to maintain an offset free subsystem, which controlled variable should be left uncontrolled? Will the subsystem remain stable and/or retain integral controllability? For a decentralized controller we have already answered all these questions. Note that for the decentralized controller, the pairing automatically defines which variable must be left uncontrolled or set manually. In other words, the matrices A and MK in Figure 2.3 have the 1

The real matrix M is D-stable iff the matrix MD has its eigenvalues in RHP for all positive definite diagonal matrices D.


24

same role for a decentralized controller. Chang and Yu [CY91] studied this problem for an inverse-based controller. In [CY91] failure tolerance is addressed on the basis of the stability of the subsystems, e.g., to avoid instability as the result of positive feedback. Based on that work we recommend to use the same "pairing" for the MIMO controllers that would give a DIC control structure. Doing so, we make the controller design easier, since it is easy to choose the weighting function necessary for the design procedure, and we have a good indication about which variable should be put out of operation when a subsystem failure occurs. Of course, many other solutions could also be applied, e.g., in model predictive control the failure can be handled as an additional constraint, another possibility is to switch to another MIMO controller developed for the specific failure case, etc.

2.3.3 RHP-zeros in the matrix elements of a transfer function Zeros of a system may arise when competing effects in the system are such that the output is zero even while the inputs (and the states) are not themselves identically zero. The notion of zeros of a multivariable system usually adopted is the concept of transmission zeros, i.e., z is a transmission zero or simply zero of G(s) if the rank of G(z) is less than the normal rank of the transfer function matrix G(s). Appendix A.4 presents a detailed discussion about transmission zeros. A RHP-zero of G(s) limits the achievable bandwidth of the plant. This holds irrespective of the type of controller used. In the multivariable case, a RHP transmission zero of G(s) does not imply that the matrix elements gij(s) have RHP-zeros. Vice versa, the presence of RHP zeros in the elements does not necessarily imply a RHP transmission zero of G(s). If we use a multivariable controller, then RHP zeros in the elements do not imply any particular problem. However, Zafiriou and Chiou [ZC94] showed that when we move from the unconstrained to the constrained input problem, the zeros of individual elements of a MIMO process may become very important and that RHP-zeros of individual elements of G(s) can cause instability in the presence of constraints. Certainly, this result can be generalized to the case where instead of RHP-zeros of individual elements of G(s) we have RHP-zeros in subsystems Gij(s) of G(s). Particular attention must be paid for decentralized controllers, since in this case we should generally avoid pairing of elements with significant RHP zeros (i.e., RHP zeros close to the origin in relation to the system time scale), because otherwise this loop may become unstable if left by itself (i.e., when the other loops are open or the other input variables are saturated). Therefore, it is also recommended to reject control structures with significant RHP-zeros in its matrix elements gii(s) or in its subsystems, even when the transfer function G(s) does not have significant (or critical) RHP-zeros.

2.3.4 Controller structure and order The controller structure and order must be also considered in the CSD. Of course, decentralized or block-decentralized controller structures with low order are preferable, since they are easy to understand and to put into operation. The problem with a decentralized controller structure is that the performance can be deteriorated in comparison to a more centralized structure (i.e., full feedback controller). The controller structure and order selection is the CSD step that is strongly related to the controller design and depends therefore

25

2.4 ADDITIONAL CONSIDERATIONS

both on the plant model and on the desired closed loop performance. A full controller for a simple decoupled desired closed loop behavior is based on the inverse of the process model. Here, to make possible a correct comparison between the different channels it is necessary that the system must be optimally scaled. A procedure for this is presented in Section 3.4.2. To get an idea which is the necessary order and structure of the controller we plot the magnitude for each channel of the inverse of the scaled system. The structure is given by looking for elements close to 0 and the order by the slope of the curves. An alternative is to apply the controller design algorithms presented in Chapter 4.

2.4 Additional Considerations Often more than one structure can be used to control a given process. Therefore, additional considerations must be taken into account for the final decision. Here we discuss some of these additional points. Number of additional measurements For process control, as few pieces of control hardware and measurements as possible should be used. Every additional gadget that is included in the system is one more item that can fail or drift. More complex structures usually need additional measurements. Therefore, it must be analized if the higher complexity is justifiable. For example, for high-purity columns, the (D,V) and (L/D,V/B) control structures usually give good results. Nevertheless, the (L/D,V/B) structure has some drawbacks like the necessity of more measurements and the interdependence between level and composition controls. This can make the on-line tuning of the level control problematic. Remember that to reduce the interaction between the inventory and composition control, the level control must be made about 10 times faster than the composition control loop. On the other side, flow disturbances are very well handled by the (L/D,V/B) control structure. The reason for this is that the internal flows are adjusted automatically to the external flows when the level control responds to flow disturbances. Operator aceptance The control system must be kept as simple as possible. Everyone involved in the process, specially the operators, should be able to understand the system. If the operators do not understand the control strategy, the probability that they will use it is not high. When a more elaborated control structure must be applied, we have to explain in simple terms which advantages the special control structure has. Usually some analogies and simple models can be used to explain how the control structure works. Input and output equipment sensitivity Input and output equipment sensitivity are important considerations in selection of the valves and sensor types. Although such issues are fundamental, they are frequently overlooked in the design of control strategies. Of course, this issue is more related to the equipment selection (e.g., valve type, range, resolution, linearity, dynamic, etc.). Failing to understand the importance of the equipment sensitivity issues can lead to serious operational and control problems. The sensor problem occurs when the control sensor has a narrow span compared to


26

the range over which the process variable can be manipulated. In such cases, even small changes in the manipulated variable will cause the sensor to saturate. When disturbances occur, most of the time the sensor will send the maximum or minimum value of its possible outputs. Valves and other actuators all have a minimum resolution with respect to positioning. These limitations restrict the fine adjustments often necessary for steady state operation. If this fine adjustment is less than the resolution of the valve, sustained oscillation will usually occur. This kind of problem is typical for oversized control valves, for example, a linear valve where 100% of the maximum flow is achieved when the signal to the valve is only 10% of its range. Here, if the flow for the linear valve can be positioned with an error of ±1.0% of its maximum value, the corresponding flow error will be ±10.0% in this case. Low equipment sensitivity is also undesirable. Here, the most obvious problem caused by the low sensitivity is the saturated input. The control system is severely limited with respect to the magnitude of errors that can be corrected by the valve. The magnitude of the change necessary to correct an error may well exceed the capability of the control valve. The second problem caused by the low process sensitivity concerns the effect which sensor errors have on loop performance. In order to compensate for the low process sensitivity, the feedback controller gains are typically set high. Such high controller gains are a source of problems if the control system is connected to a sensor with significant noise in its signal.

Chapter 3 System Description and Analysis Tools This chapter introduces the general framework of linear controller analysis and design. Robustness issues and uncertainty descriptions are also discussed. Finally, the Robust Performance Number (RPN) and the Robust Performance Number for a set of plants (RPPN) are defined. RPN and RPPN are the key concepts of our approach.

3.1 Representations of Systems Linear Time-Invariant System A finite dimensional Linear Time-Invariant (LTI) System can be described by the following linear constant coefficient differential-algebraic equations (DAE): dx = A x (t ) + Bu(t ) dt y(t ) = C x(t ) + Du(t )

E

(3.1)

where A, B, C, D and E are real matrices. The corresponding transfer matrix from u to y is defined as y(s) = G(s )u(s) (3.2) where u(s) and y(s) are the Laplace transforms of u(t) and y(t) with zero initial conditions. If (sE - A)-1 is invertible, G(s) for the system (3.1) is given by (see Appendix A.3 for details) G(s) = C(sE − A) B + D . −1

(3.3)

Equation (3.3) will be abreviated as G=(A,B,C,D,E) or just (A,B,C,D) when E=I. By the DAE form we can represent arbitrary rational matrices, i.e., nonproper transfer functions can also be represented in this form. The following example illustrates this fact:

3. SYSTEM DESCRIPTION AND ANALYSIS TOOLS

0 1  d  x1  1 0   x1   0  0 0   x  = 0 1   x  + −1u   dt  2    2      y = 1 0  x1  ]  x2   [   

28

⇔ G(s) = s

(3.4)

The letter G in this work is always used to represent a transfer matrix, which can be obtained by linearizing a nonlinear model at one operating point or by process identification. More generally, G can include irrational elements as time delays. Physical models of a system often lead to a state-space description of its dynamical behavior. The resulting state-space equations typically involve physical parameters whose values are only approximations of complex and possibly nonlinear phenomena. The system is then described by an uncertain state-space model G=(A,B,C,D,E), where the state-space matrices A,B,C,D and E can vary in some bounded subsets of the space of matrices. The set P of linear models can be represented in different forms. In this work, polytopic models, affine parameter dependent models and linear fractional transformation models are used. Polytopic Models (Multi-Model representation) A polytopic model P is defined by P ∈ Co

{ G ,...,G } = ∑ α G ∆

1

k

k

 i=1

i

i

k  : α i ≥ 0 ,∑ α i = 1  , i =1 

(3.5)

where G1,...,Gk are given transfer matrices. In words, P is a convex combination of transfer matrices. The nonnegative numbers α1,..., αk are called the polytopic coordinates of P. Such models arise in many practical situations, including: • multi-model representation of a system, where each model is derived for particular operating conditions • nonlinear systems and time-varying system of the form dx E (v ) (3.6) = A(ν ) x(t ) + B(ν ) u(t ) , y(t ) = C(ν ) x(t ) + D (ν ) u(t ) dt where ν = t or ν = x for time-varying systems and nonlinear systems, respectively. • state-space models depending affinely on time-varying parameters (see [LMITB95] for some examples) Affine Parameter-Dependent Models Physical equations often involve uncertain or time-varying coefficients. When the system is linear, this naturally leads to parameter-dependent models of the form E( p) x! = A( p) x + B( p) u y = C( p) x + D( p)u

where

(3.7)

29

3.1 REPRESENTATIONS OF SYSTEMS A( p) = A0 + p1 A1 + ...+ pk Ak , B( p) = B0 + p1 B1 + ...+ pk Bk , C( p) = C0 + p1C1 + ...+ pk Ck , D( p) = D0 + p1 D1 + ...+ pk Dk , and E( p) = E0 + p1 E1 + ...+ pk Ek

(3.8)

are known functions of some parameter vector p=(p1,...,pn). Such models are useful to represent bilinear systems and/or systems where a given parameter p has a strong influence on the system's dynamics. Linear Fractional Transformation (LFT) Models The Linear Fractional Transformation (LFT) provides a standardized framework for a variety of feedback arrangements and model descriptions. Definition 3.1.1 LFT (a) A (lower) linear fractional transformation (LLFT) is denoted by (see figure 3.1 )   P11 FL     P21

P12   −1 ,K  = P11 + P12 K ( I − P22 K ) P21  P22  

(3.9)

whenever det( I - P22 K) ≠ 0. An alternative expression for a LLFT, if P21-1 exists, is the following:  U11 U12   −1 FB   , K  = ( U11 K + U12 )( U21 K + U22 )   U21 U22   (3.10) −1 −1 U11 U12   P12 − P11 P21 P22 P11 P21  where U =   = P21−1  − P21 P22 U21 U22   with det(U21K + U22) ≠ 0. This alternative expression is useful because the LLFT in this form is written as a stable factorization, and can be linked with results on coprime factorizations [MG90].

Figure 3.1: Lower Form of LFT

Figure 3.2: Upper Form of LFT

(b) An (upper) linear fractional transformation (ULFT) is denoted by (see figure 3.2)   N11 N12   −1 FU   ,∆ = N 22 + N21∆( I − N11∆ ) N12    N21 N22   whenever det( I - P11 ∆) ≠ 0.

(3.11)


30

The LLFT provides a convenient framework for posing many H∞ problems, while the ULFT provides a very general way to describe uncertain systems. The generalized plant P consists of everything that is fixed at the start of the control design: the plant, actuators which generate inputs to the plant, sensors measuring certain signals, analog-to-digital and digital-to-analog converters, etc. The controller K consists of the free part: it may be an electric circuit, a programmable logic controller, a vector-valued function of time. The components of w, z, y, and u are, in general, vector-valued functions of time. The components of w are the exogenous inputs: references, disturbances, sensor noises, etc. The components of z are the signals which we wish to control: tracking errors between reference signals and plant outputs, actuator signals whose values must be kept between certain limits, etc. The vector y contains the outputs of all sensors. Finally, u contains all controlled inputs to the generalized plant P.

3.2 Robust Stability Analysis Whether the nominal model is obtained from theoretical considerations or identification methods, it is often necessary to neglect part of the dynamics to obtain a model which is suitable for mathematical manipulations. Differences between the actual plant and the nominal model are generally called modeling errors or system uncertainty. Modeling errors can arise in many ways. For example, in design it is in general necessary to consider a linear nominal model, while most physical systems are nonlinear in some way. As a linear model cannot truly represent nonlinear behavior, a modeling error is necessarily introduced. Also, even when a physical system is predominantly linear, it is common to include only the most dominant modes in the nominal model, and again this introduces a modeling error. A further source of modeling error arises from inaccurate determination of model parameters, which is particularly common when the nominal model is determined using a system identification method where parameters are estimated from input/output information.

3.2.1 Standard Multivariable Feedback Loop Consider the standard feedback system shown in Figure 3.3. Since matrix multiplication is in general noncommutative, breaking the loop at the plant input ( point I in Figure 3.3) generally yields a different open loop than breaking the loop at the plant output ( point O).

Figure 3.3 : Standard Feedback Configuration

31

3.2 ROBUST STABILITY ANALYSIS

Hence, in the multivariable case, two sets of transfer functions must be defined. Let the open loop transfer function (LO), sensitivity function (SO), and complementary sensitivity function (TO) at the plant output be denoted by ∆

LO (s ) = G(s )K (s ) ∆

SO (s) = [ I + LO (s)] ∆

(3.12)

−1

(3.13)

TO (s) = LO [ I + LO ] = [ I + LO ] LO −1

−1

(3.14)

and denote the corresponding transfer functions at the plant input by ∆

LI (s) = K (s) G(s) ∆

SI (s ) = [ I + LI (s )] ∆

(3.15)

−1

(3.16)

TI (s) = LI [ I + LI ] = [ I + LI ] LI . −1

−1

(3.17)

The input and output sensitivity and complementary sensitivity functions characterize stability robustness of the feedback system against different classes of plant uncertainty ( see, e.g., [ZDG96, table 9.1]). Uncertainty at high frequencies due to unmodelled dynamics is frequently described by a multiplicative perturbation. Unlike the scalar case, however, one must distinguish between multiplicative uncertainties at the plant input ∆

P (s ) = G(s)[ I + EI (s)] = G(s )[ I + W2 (s)∆ I W1 (s)] ⇒ EI = G −1 ( P − G)

(3.18)

and multiplicative uncertainties at the plant output ∆

P (s ) =

[ I + EO (s)]G(s) = [I + W4 (s)∆ OW3 (s)] G(s)

⇒ EO = ( P − G)G −1 .

(3.19)

Both uncertainties are illustrated in Figure 3.4. For completeness an additive uncertainty given by ∆

P (s ) = G(s) + EA (s) = G(s) + W4 (s)∆ AW1 (s ) ⇒ E A = P − G

(3.20)

is also shown in Figure 3.4. In (3.18), (3.19), and (3.20), ∆I represents an unknown input multiplicative perturbation, and ∆O an output multiplicative perturbation, and ∆A an additive perturbation and the fixed minimum-phase transfer_ functions W1, W2, W3, and W4 are used to normalized the 'size' of ∆, which is measured by σ(∆) ( the maximal singular value of ∆ ) at each frequency ( for definitions see Appendix A.1 ).


32

Figure 3.4 : Standard feedback configuration for plant P represented by a nominal model G and input, output, and additive uncertainties corresponding to the ∆I, ∆O, and ∆A blocks, rescpectively. The fact that robustness against uncertainty in different loop-breaking locations is governed by different transfer functions is responsible for several phenomena unique to multivariable systems. One of these is that robustness against uncertainty at one loop location can be quite large while, at the same frequency, the stability margin against uncertainty in the other location can be extremely small. It turns out that this can occur only when both plant and compensator transfer functions have a large condition number γ. To show this, assume for simplicity that the plant is square. It is easy to show that TI (s) = G −1 (s ) TO (s) G(s ) .

(3.21)

_ The plant condition number is defined by γ(G) = σ(G)/σ(G) (see Appendix A.1). Using wellknown properties of the singular value decomposition (e.g., [SP96, Appendix A] ), it follows that

and

σ (TO ) ≤ σ (TI ) ≤ σ (TO )γ (G) γ (G)

(3.22)

σ (TI ) ≤ σ (TO ) ≤ σ (TI )γ (G) . γ (G)

(3.23)

These bounds show that, if γ(G) ≈ 1, then the two complementary sensitivity functions have approximately the same size. Hence the associated stability margins are also almost equal. Bounds similar to (3.22) and (3.23), with γ(G) replaced by γ(K), show that the use of a wellconditioned compensator also guarantees that stability margins will be approximately the same at both loop-breaking points. If both plant and compensator have high condition

33


numbers, it follows that knowing the stability margin at one point does not imply the same stability margin at the other point. Similar remarks apply to the two sensitivity functions, as may be seen by using the identities S+T=I in conjunction with (3.22) and (3.23). It may be shown (e.g., [ZDG96],[SP96]) that the stability margins against individual perturbations at the plant input and output can yield misleading estimates of robustness against simultaneous perturbations only if the plant and/or compensator are ill-conditioned. More precise bounds can be obtained via substitution of γ(G) by the minimized condition number γ*(G) (see Appendix A.1) in (3.22) and (3.23), since the plant G can be, as we will see later, only illscaled, but not ill-conditioned. To clarify the effect of input and output uncertainty on the closed-loop transfer functions we will present some relations between the perturbed output sensitivity function SO′ = ( I + PK )

−1

(3.24)

and the nominal closed-loop transfer functions TO , TI , and SO. The relations for EO : for EI :

SO′ = SO ( I + EOTO )

−1

SO′ = SO (I + GEI G −1TO ) = SO G( I + EI TI ) G −1 −1

−1

(3.25)

can be easily derived using the fact that for square plants E0 = G EI G-1 and T0 = G TI G-1 . Note that S'_O is a function of SO, EO (or EI ), and TO. At the crossover frequency, where both _ σ(SO) and σ(TO) are large, the magnitude of uncertainties has a strong influence on the closed loop performance.

3.2.2 Representations of Uncertainty In H∞- and µ-theory, it is necessary to model the plant uncertainty as a separate transfer function. Common approaches as illustrated in Figure 3.4 are to model uncertainty in a multiplicative or additive way with respect to the nominal plant or as stable additive perturbations of the factors in a coprime factorization of the nominal plant (e.g., [MG90], [ZDG96]). It is useful to make the distinction between unstructured and structured forms of uncertainty. Unstructured Uncertainty The uncertainty is called unstructured, if no information is available about its effects on a process, except that an upper bound on its magnitude as a function of frequency can be estimated. Uncertainty due to unmodeled actuator dynamics can be described appropriately by plant input uncertainty while uncertainty associated with the sensors would be better described by plant output uncertainty. If the only available information about the system uncertainty is that it is multiplicative and occurs either at the plant input or at the plant output, then the perturbation ∆ is termed an unstructured uncertainty. The three commonly used models are given by equations (3.18), (3.19), and (3.20). Unstructured uncertainty usually represents frequency-dependent elements such as unmodeled structural modes in the high frequency range or plant disturbances in the low frequency range. If more than one ∆-block in Figure 3.4 is used to describe the uncertainty, the perturbation ∆ will have a block-diagonal structure and then ∆ is called structured uncertainty, although each block may be unstructured.


34

Structured Uncertainty Structured Uncertainty is uncertainty about which 'structural' information is available, which will typically restrict the uncertainty to a part of a process model. Uncertainty in input channels: Consider a distillation column that operates with two flowrates L and V (see Chapter 2). Assume that the flow-rates are in the region of 100 kg/h. Typically, changes of up to 10 kg/h are required. These changes are realized by servocontrolled valves which rely on measurements of the flow, but the flow measurement error is 1%. So when a change of flow-rate from 100 to 110 kg/h is required, an actual change to 111 kg/h may occur. The error in the required change is therefore about 10%. Thus our plant model, which describes changes about some operating point, is subject to errors of up to 10% on each input channel. Since the error on each input channel is independent of the other ones, a suitable representation of this model uncertainty is given by δ 1 0  ∆I =  1 k = 1,2 .  , δ k ≤ 0.,  0 δ2

(3.26)

In the above discussion of flow control, we used information about the structure of the input uncertainty: we know that the uncertainty for each valve is independent of the uncertainty of the others. However, when we write ||10∆I||∞ ≤ 1 instead, we lose this information, since this description also allows perturbations such as 0 0.1 ∆I =   0 0 

and

0.05 0.05 ∆I =   0.05 0.05

(3.27)

which do not correspond to the real perturbations. The use of an unstructured description generally leads to compensator designs which are unnecessarily conservative, because they perform satisfactorily (in some sense) even in the presence of perturbations which never occur. The controller may have to be 'detuned' to guard against non-existent perturbations. We recommend applying the following criterion to identify when a full block representation (i.e., wI (jω) ∆n×n) can be used instead of a diagonal input uncertainty description (i.e., wI (jω) diag {δ1,..., δn }) without increasing considerably the set of possible plants:   1 ≈ 2 max RGA 1 ,RGA ∞  , γ (G( jω )) + γ ( G ( jω ) )   then w I ( jω ) ∆ n× n can be used instead of w I ( jω ) diag {δ 1 ," ,δ n }

{

if

}

(3.28)

where ||RGA||1 and ||RGA||∞ are respectively the induced 1 and ∞ norm of RGA = RGA(G(jω)) defined by ∆   M 1 = max ∑ mij  and M j  i 

and wI(s) is a scalar weighting function.

∆  = max ∑ mij ∞ i  j

   

(3.29)

35


In words, it means that when the condition in (3.28) is satisfied at a given frequency ω, we can substitute a diagonal input uncertainty description by a full block representation without becoming more conservative. For a 2×2 system it is always possible to scale the inputs and outputs at a given frequency such that the condition (3.28) is satisfied. For n×n systems, equation (3.28) can be used as an indicator whether a more structured input uncertainty description is necessary. When the system is optimally scaled and the condition in (3.28) is approximately satisfied, a full block description is sufficient to represent the system's behavior. Working with simple uncertainty descriptions should always be preferred as will be discussed later.

3.2.3 Time invariant and time-varying perturbations It is important to distinguish between time-invariant and time-varying/nonlinear parameters. Time-invariant uncertain parameters have a fixed value that is known only approximately. In this case, the underlying dynamical system is time invariant. Time-varying parameters vi(t) are parameters that their values vary during operation. The resulting dynamical system is then time-varying. In general, robust stability against time-varying parameters is more demanding than_ robust stability against fixed but uncertain parameters for the same uncertainty range [vi , vi] (see, e.g., [Sch94], [PT95]). To analyze a time-varying system we . . _. need to specify the rate of variation vi of vi that will be usually in a given range [ v i ,v i ]. This description . encompasses time-invariant parameters as the special case vi = 0. In principle, a very slowly time-varying parameter (e.g., very slowly deactivation of a catalyst ) can be analyzed for control purposes as a time-invariant parameter. We will here define three classes of parameter . variations by considering how large vi is in comparison to the closed-loop and process dynamics: . Definition 3.2.1 Time invariant parameter (TIP). If the rate of variation vi is much slower than the main process time constant, then the parameter vi is called a time invariant parameter. ! . Definition 3.2.2 Slowly time-varying parameter (STVP). If the rate of variation vi is slower than the dominant closed-loop time constant (i.e. the dominant time constant of the output complementary sensitivity function TO), but faster than or in the same range as the dominant process time constant, then the parameter vi is called a slowly time-varying parameter. ! . Definition 3.2.3 Fast time-varying parameter (FTVP). If the rate of variation vi is in the same range or faster than the dominant closed-loop time constant (i.e., the dominant time constant of TO), then the parameter vi is called a fast time-varying parameter. !

3.2.4 µ or SSV (Structured Singular Value) Conceptually, the structured singular value is a straightforward generalization of the maximum singular value for a matrix M considering a special block-diagonal structure. It was introduced to allow robust stability analysis for systems with stable time-invariant structured uncertainies. Here we will give only the formal definition and characterization of the structured singular value of a complex matrix M. In many recent books (e.g., [SP96], [ZDG96], [Mac89], [MZ89], among others) the reader will find a complete treatment.


36

Definition 3.2.4 Block-Diagonal Perturbation (BD). The BD denotes the set of normbounded block-diagonal perturbations with a particular structure given by ∆

BD

=

[

]

 ∆ = diag δ 1r Ir1 ,...,δ rR IrR , δ 1c Ic1 ,...,δ C c IcC , ∆1 ,...,∆ F :  r r no x ni c c −1 < δ i < 1,δ i ∈ R ; δ i < 1 , σ ∆ j < 1, δ i ∈ C , ∆ j ∈ C

( )

 . 

(3.30)

! There are two types of blocks: repeated scalar (δi) and full blocks (∆j). The scalar blocks (δi) can be restricted to be real blocks. The nonnegative integers R, C, and F represent the number of repeated real scalar blocks, complex scalar blocks, and the number of full blocks, respectively. In [HBC95] a generalization of the block structure is discussed permitting the treatment of nondiagonal real uncertain blocks. Definition 3.2.5 Structured Singular Value (µ). For a complex matrix M, the real nonnegative function µ(M), is defined by ∆ 1 (3.31) µ(M )= min{k m :∆ ∈ BD,det (I − k m M ∆ )= 0 } with µ(M) =∆ 0 if no km> 0 and ∆ ∈BD makes ( I − km M ∆ ) singular (i.e., det (I − k m M ∆ ) ≠ 0 for all km ). ! The smallest value of km in (3.31) is sometimes called the multivariable stability radius or multivariable stability margin and represents the robust stability margin for time invariant perturbations, that is, the largest amount of uncertainty ∆ that can be tolerated without losing stability. µ and km were defined in 1982 independently by Doyle and Safonov, respectively (see [SP96], [ZDG96] for the historical details). It is important to note that the value of µ(M) depends on the structure of ∆. This is sometimes shown explicitly by _ using the notation µ∆(M). A value of µ = 1 means that there exists a perturbation with σ(∆) = 1 which is just large enough to make (I-M∆) singular. A large value of µ is "bad" as it means that a smaller perturbation makes (I-M∆) singular, whereas a small value of µ is "good". Although (3.31) provides an exact expression for µ, it involves an optimization problem which is, in general, not convex. It has been established that the computational complexity of computing µ may have a non-polynomial growth with the number of parameters even for purely complex perturbations [BYDM94]. So to yield tractable computation schemes, attention has focused on upper and lower bounds for µ. In [YD96] the relation of these bounds to each other and to µ is examined. The most important properties that allow us to derive a lower and upper bound for complex µ are:

µ (αM ) = α µ ( M ) , µ ( I ) = 1 , µ ( AM ) ≤ σ ( A)µ ( M ) .

(3.32)

Let D be any real, diagonal, positive matrix with the structure

{

}

D = diag D1 ,..., DC , d1 I j1 ,..., d F −1 I jF −1 , I jF and d j > 0, d j ∈ R , Di ∈ C noxni , Di = Di H > 0,

37


then for complex µ

µ ( DMD −1 ) = µ ( M ).

(3.33)

Let U be any unitary matrix (i.e., U UH = I) with the same block-diagonal structure as the set BD, then the following inequality holds [PD93]

(

max ρ (UM ) ≤ µ ( M ) ≤ min σ DMD −1 U

D

)

(3.34)

where ρ represents the spectral radius of a complex matrix M (i.e., ρ(M) = max |λi(M)|, where λi are the eigenvalues of M ). The first inequality in (3.34) is a non-convex maximization problem. This does not imply, however, that practical algorithms are not possible (e.g., [MuTB93], [RbTB92]). The second inequality leads to a convex problem and therefore is more useful for estimating the µ-value. When all blocks in BD are complex, the lower and upper bounds of µ given in (3.34) may easily be computed. _ For complex perturbations the scaled singular value σ(DMD-1) is usually a tight upper bound on µ(M) and it is used in the DK-iteration controller synthesis procedure where the upper bound is minimized (i.e., min ║DMD-1║∞) (see Section 4.4.1). However, ║DMD-1║∞ is of interest also in its own right. The reason is that when all uncertainty blocks are full and complex, then this upper bound provides a necessary and sufficient condition for robustness to arbitrary-slow time-varying linear uncertainty [PT95]. On the other hand, the use of µ assumes the uncertain perturbations to be time-invariant. It can be argued that slowly timevarying uncertainty is more useful for practical problems than a constant perturbation, so that it is better to minimize ║DMD-1║∞ instead of µ(M). In addition, by considering how D(jω) varies with frequency, one can find bounds on the allowed variations of the perturbations [PT95]. Another interesting fact is that use of constant D-scales provides a necessary and sufficient condition for robustness to arbitrary-fast time varying linear uncertainty [Sch94]. It may be argued that such perturbations are unlikely in a practical situation. Nevertheless, we see that if we can get an acceptable controller design using constant D-scales, then we know that this controller will work very well even for rapid changes in the plant model. For _ real-1or mixed real/complex perturbations there exists a generalization of the upper bound σ(DMD ) by the inclusion_ of a scaling matrix related to the real perturbation structure. In general, the upper bound σ(DMD-1) for complex perturbations is tight, whereas the present upper bounds for mixed perturbations may be arbitrarily conservative [YD96]. An implementation of the upper bound for mixed µ can be found in [MuTB93] and [LMITB95]. Another approach to developing bounds for mixed µ is to specialize absolute stability criteria for sector-bounded nonlinearities to the case of linear uncertainty. An implementation of this approach can be found in [RbTB92] and [LMITB95].


38

3.2.5 The small µ theorem The best-known use of µ is as a robustness analysis tool in the frequency domain. In fact, µ was proposed to permit the generalization of the results of the small gain theorem for robust stability [Zam81] to a structured block diagonal perturbation. Definition 3.2.6 Permissible perturbations (D). A permissible perturbation, ∆, is one such that ∆ ∈ D where D = DS ∪ DU and D S = {∆ : ∆ ∈ RH ∞ ; ∆

∞

< 1}

D U = {∆ : ∆ ∈ RL∞ ;η( FU ( M ,0)) = η( FU ( M ,∆ )) ; ∆

∞

< 1}

(3.35)

where M is the generalized plant model ( see Figure 3.5 ), η(...) denotes the number of closed RHP poles of a transfer function, and the function spaces RH∞ and RL∞ are defined in Appendix A.2. ! Remark 3.2.7 The set Ds describes a set of stable bounded perturbations, and Du , describes a set of bounded perturbations such that the nominal plant and the perturbed plant have the same number of unstable poles. In the case of coprime factor uncertainty, all possible perturbations are stable and, therefore, D is equal to Ds (see, e.g., [Kw93], [MG90]).

Figure 3.5 : Generalized regulator with robustness interpretation

39


Theorem 3.2.8 The Small µ Theorem. A controller K stabilizes P = FU ( M, ∆ ) ( see Figure 3.5 ) for all ∆ ∈ BD with each block ∆i ∈ D, if and only if (a) the generalized plant model M is stabilizable and detectable, (b) K stabilizes FU ( M, 0 ), i.e., the nominal model G, and (c) µ ∆ ( FL ( M ,K )) ≤ 1. ! Proof : The proof of this theorem can be found in many textbooks (e.g., [ZDG96], [Mac89]) It uses the same construction as in the proof of the small gain theorem. In fact, this theorem is exactly the generalization of the small gain theorem to a structured perturbation ∆. The formulation of the small µ theorem presented here follows the recommendations in [TF95]. ! Remark 3.2.9. The allowed perturbation for the case where only complex blocks exist in the uncertainty description can be changed by the modification of the condition that the uncertainty belongs to RH∞ and RL∞ in Ds , Du , and M by belonging to H∞ and L∞ ( the function spaces RH∞ , RL∞ , H∞ ,and L∞ are defined in Appendix A.2). For the mixed real/complex µ problem some additional conditions must be satisfied and are stated in [Ti95].

µ analysis investigates the robust stability or performance of systems with linear time invariant block-diagonal uncertainty in the linear fractional transformation (LFT) form. It is also applicable to time-invariant parameter-dependent systems by first deriving an equivalent LFT representation. For structured uncertainties with four or more complex full block perturbations, there is a gap between µ and the upper convex bound for µ (e.g., [ZDG96], [PD93]). As already mentioned, Polla and Tikku [PT95] showed that this conservatism is in fact desirable because it allows for some, at least modest, time variation. Another interesting fact is that use of constant D-scales, provides a necessary and sufficient condition for robustness to a fast time-varying linear uncertainty [Sch94]. So for time invariant, slowly time-varying, and fast time-varying systems µ , µ upper bound, and µ upper bound with constant D-scales must be used in the analysis procedure. It is important to mention that there are several different tests for robust stability and/or performance of uncertain dynamical systems. These include quadratic stability, Karitonov's theorem, techniques based on affine parameter-dependent Lyapunov functions and on the Popov criterion. The most appropriate tool for each problem should be selected depending on the nature of the uncertainty (e.g., linear vs. nonlinear, time invariant vs. time-varying, dynamical (complex blocks) vs. parametric (real blocks), etc.). The interested reader is referred to [LMI95] (and references therein) and [Bar94] for details. Robust Performance (RP) In practice, we are usually interested in more than stability. We would like to maintain an acceptable level of performance, as well as stability, in the presence of a prespecified class of perturbations, i.e., we would like to achieve performance robustness as well as stability robustness. The expressions nominal performance ( NP ), robust stability ( RS ), and robust performance ( RP ) are often used to characterize a closed-loop system. They mean:


40

Definition 3.2.10 Nominal Performance ( NP ). The closed-loop system achieves nominal performance if the performance objective is satisfied for the nominal plant model G. The performance objectives are measured by some set of norms of transfer functions ( e.g., H2, H∞, etc.) and consider a given set of disturbances/inputs ( e.g., steps, sinusoids, etc.). ! Definition 3.2.11 Robust Stability ( RS ). The closed-loop system achieves robust stability if the closed loop system is internally stable for all possible plant models G ∈ P. ! Definition 3.2.12 Robust Performance ( RP ). The closed-loop system achieves robust performance if the closed-loop system is RS and, in addition to that, the performance objectives are satisfied for all G ∈ P. ! For the case where the performance objectives are measured in terms of the H∞-norm, the RP problem can be transformed to a RS problem with the inclusion of a full perturbation block related to the performance requirements ([Mac89], [SP96]). Figure 3.6 illustrates the case where the performance requirement is stated as WP ( I + GK ) Gd −1

∞

≤ 1.

Figure 3.6 : RP problem converted to a RS problem (compare with Figure 3.5)

(3.36)

41


Robust performance of inverse-based controllers Stein and Doyle [SD91] developed an analytical expression to calculate µ of the RP problem when an inverse-based controller is used. They considered a complex full block input uncertainty description with weighting function wt and performance requirement ws. It is equivalent to make the following assumptions for the weighting functions in Figure 3.6: ∆ A = 0 ,∆ O = 0 ,W1 = wt I ,W2 = I , WP = ws I ,and Gd = I .

(3.37)

Putting the system in Figure 3.6 into the upper linear fractional transformation (ULFT) form yields1 wt KSO   wt TI N = FL ( M ,K ) =  . ws SO G ws SO 

(3.38)

We assume that G is stable with stable inverse (i.e., G is minimum-phase)2. Furthermore, we will assume that the controller has the form K (s) = G −1 (s) k (s )

(3.39)

where k(s) is a desired scalar loop transfer function which makes K(s) proper and stabilizes the closed-loop. This compensator produces diagonal sensitivity and complementary sensitivity functions with identical diagonal elements, namely SO = SI =

1 k (s ) I , TO = TI = I. 1 + k (s ) 1 + k (s) #$ % % & #$ % % & ζ ( s)

(3.40)

τ ( s)

Substituting these expressions into (3.38), we obtain  w τ I wtτ G −1  N= t . ζ ζ w G w I s s  

(3.41)

The structured singular value for N at a frequency ω can be computed as ( for details, see [ZDG96, section 11.3.3] )

µ ∆ ( N ( jω ) ) =

wsζ

∆ full where ∆ =  I  0 1 2

2

+ wtτ

0  . ∆ full P 

2

+ wsζ

 1  wtτ γ (G) + γ (G)  

(3.42)

-I 0

For simplicitiy, we have eliminated the negative signs in N , since µ(N) = µ(UN), with U=[ 0 I ] . G can be strictly proper.


42

Note that if | ws ζ | and | wt τ | are small, which are guaranteed if the nominal performance and robust stability conditions are satisfied, then (3.42) can be approximated by

µ ∆ ( N ( jω ) ) ≈

wsζ

 1  wtτ γ (G) + . γ (G)  

(3.43)

Equation (3.42) was derived by considering a full block input uncertainty with a scalar weighting function wt. This description can be very conservative. To reduce this conservativeness is necessary to scale the system before the determination of the weighting function wt. This is to some extent similar to substituting γ(G) by γ*(G) in (3.42). The exploration of this idea is the basis of the Robust Performance Number (RPN) that will be defined at the end of this chapter.

3.3 System Directionality and Scaling Directionality analysis is usually based upon the singular value decomposition (SVD) of a matrix ( see Appendix A.1 ). The analysis of a multivariable system with SVD has an intuitive appeal. The reason for this is that the concept of SISO gain in this way can be extended to MIMO systems. The gain of a MIMO system is then sandwiched between the largest and the smallest singular value of the transfer function matrix of the plant model. The degree of directionality for a given system G is measured by how much the gain between one set of outputs and one set of inputs may vary. Different sets of inputs may affect the outputs in different ways. Some systems have the same gain for all possible input directions, whereas the gains of other systems vary considerably. Such systems are called well-conditioned and illconditioned, respectively. The common definition of an ill-conditioned process is based on the condition number. However, for MIMO systems with different types of outputs (e.g., temperatures, compositions, pressures, etc.) and inputs (e.g., volumetric flows, energetic flows, etc.) the choice of units for the input and output variables has significant effects. Although the choice of the units (or scalings) does not affect the system characteristics, it has an obvious effect on the singular values and thus on the condition number. Mathematically, the scalings are represented by real, diagonal, and nonsingular matrices like L and R, which ∆ can be used to minimize the condition number, i.e., γ*(G(jω)) = min γ (LG(jω)R) . L,R

Due to the scaling dependence of the condition number, the minimized condition number γ* should be used for the correct defintion of ill-conditionedness instead of the condition number γ, since γ can be large whereas γ* is small. In this case the process is only ill-scaled but wellconditioned. Moreover, how ill-conditioned a system is depends on the frequency range. It means that a process can be ill-conditioned only in some frequency range and at other frequencies it can be well-conditioned. These concepts are defined below.

Definition 3.3.1 Ill-Conditioned System. A system G is called ill-conditioned at a frequency ω if the minimized condition number γ*( G(jω) ) is large. !

43

3.3 SYSTEM DIRECTIONALITY AND SCALING

Definition 3.3.2 Well-Conditioned System. A system G is called well-conditioned at a frequency ω if the minimized condition number γ*( G(jω) ) is small. ! Definition 3.3.3 Ill-Scaled System. A system is called ill-scaled at a frequency ω if the condition number γ ( G(jω) ) is large, but the minimized condition number γ*( G(jω) ) is small. ! Here, large and small are fuzzy concepts. We mean by small values smaller than 5 and by large values larger than 50. In between, there is some grey zone. There is a common belief that plants with a large condition number (plants with large directionality) are difficult to control (e.g., [Fr93], [Ch92]), however, no conclusive proof of this connection has yet been presented. The refinement proposed in this section provides further indications of the connection between ill-conditionedness and control difficulties.

3.3.1 The scaling problem and the minimized condition number Theorem 3.3.4 Equal Magnitude Components (EMC) (Golub and Varah [GV74]). Let the singular value decomposition of a complex square matrix G( jω ) be UΣVH. If σ1 ≠ σ2 and σn-1 ≠ σn , then G( jω ) is optimally scaled in sense of the l2-norm (euclidean norm), if and only if the first and last columns of U and V have elements of the equal magnitude, that is, ui = u i and vi = v i

∀ i = 1," ,n .

(3.44)

! The EMC-theorem has both practical and theoretical significance. The practical application of EMC-theorem is to determine how well scaled a given matrix is without calculating the scaling input and output matrices. Moreover, by the analysis of the input and output vectors v and u, we can see, if the system is ill-scaled at the input and/or at the output. This analysis will be illustrated by an example where two common configurations for the composition control of a high purity distillation column are considered. The models used in the example (i.e., (3.45) and (3.46)) are very crude representations of a real distillation system, but they provide an excellent example of an ill-conditioned and ill-scaled process. The original control problem was formulated by Skogestad et al. [SMD88]. Limebeer [Lim91] included time and frequency domain performance specification and the uncertainty description (3.47). Example 3.3.5 Motivation example (High Purity Distillation Column ). The following idealized and simplified models of a distillation column for two commonly used control structures are given by 1 LV(s) = 75s +1

−τ s 0.878 -0.864 k1e 1.082 -1.096  0   1

0   k2 e −τ s  2

(3.45)


1 DV(s) = 75s +1

44

−τ s -0.878 0.014  k1e -1.082 -0.014  0   1

0   k2 e − τ s 

(3.46)

2

ki ∈[0.8 ,1.2] ;τ i ∈[0.0 ,1.0]

(3.47)

where ki and τi represent the system input uncertainties. In physical terms this means 20% gain uncertainty and up to 1 min delay in each input channel. The corresponding SVD structures are  k1e −τ s  75s + 1   0 

  −τ s  k2 e  75s + 1 

1

0.6246 -0.7809 1.972 0   0.7066 −0.7077 LV =  0.7809 0.6246   0 0.01391 −0.7077 −0.7066   

H

0

2

(3.48)

#%%%$%%% & #%% %$%%% & #%%%$%%%& U V Σ #%% %$%%% & DI  k1e − τ s  75s + 1   0 

  −τ s  k2 e  75s + 1 

1

-0.6301 0.7765  1.393 0  1 −0.00147 DV =  -0.7765 -0.6301  0    0.01969 0 .00147 1   

H

0

2

(3.49)

#%%%$%%% & #%% %$%%% & #%%%$%%%& U V Σ #%% %$%%% & DI

Since both structures have σ1 ≠ σ2 and σn-1 ≠ σn , we can apply the EMC theorem to verify whether LV and DV are ill-scaled or not. For the LV-structure both matrices U and V have elements with almost the same magnitude. In particular, the matrix V has its elements quite close to the value 1/ 2 for optimally scaled systems. Thus γ(LV) ≈ γ*(LV). The values γ(LV) = 141.73 and γ*(LV) = 138.2680 confirm our prediction. For the DV-structure γO*(DV) ≈ γ(DV), γI*(DV) ≈ γ*(DV), and γ(DV) » γ*(DV) are expected. Comparison with the correct values γI*(DV) = 1.11, γ *(DV) = 1 , γO*(DV) =69.62, and γ(DV) = 70.76 shows that our conclusions are correct. The corresponding optimally scaled systems are given by  k1e −τ s  75s + 1   0  1

0.7071 -0.7071 138.27 0   0.7071 −0.7071 LV* =   1  −0 .7071 −0.7071 0.7071 0.7071   0

H

  −τ s  k2 e  75s + 1  0

2

(3.50)

#%% %$%%% & #% %$%% & #%%%$%%%& U V Σ #%% %$%%% & DI  k1e −τ s  75s + 1   0  1

-0.6693 -0.7430 1 0  1 0  DV* =     -0.7430 0.6693  0 1  0 −1

H

  −τ s  k2 e  . 75s + 1  0

2

#%%%$%%%& #$& # %$% & U V Σ #%% %$%%% & DI

(3.51)

45


Observe that now the EMC theorem is no more applicable to DV*, since in this case σ1 = σ2 = 1. Finally, we can say that the LV-model is an ill-conditioned systems while the DV system is only ill-scaled. "

γ*(G) and RGA are scaling-independent measures of the system properties that take the relative distribution of the elements of G into account. Thus input and output diagonal matrices, as the matrix DI in example 3.3.5, will have no influence on γ*(G) and RGA, since they can be interpreted as scaling matrices. Of course, DI plays a very important role for the control of the system. In section 3.4, we show how the effect of DI can be taken into account by the definition of the RPP-number, but now we first discuss the relation between RGA and γ*(G).

3.3.2 The minimized condition number for 2x2 matrices Here we present some analytical results for 2 x 2 systems. These results help us to understand different aspects of scaling matrices that cannot be visualized by numerical solution. Moreover, 2 x 2 systems often occur in process control, so there is a direct application of the results of this section. Optimal scaling matrices L and R for 2 x 2 matrices When the diagonal and transversal diagonal elements of the scaled 2 x 2 matrix G*(jω) have the same magnitude, i.e., g11 ,s = g22 ,s and g12 ,s = g21,s , the input direction dependence (i.e., system directionality) will be minimal. We can use this fact to determine the optimal scaling matrices. A possibility, for the case where gij ≠ 0 with i,j = 1,2 , is  g  21  g11 L=  0  

  g  12 0    g11  and R =  g12   0  g22  

 0    g21  g22 

(3.52)

which yields the following scaled matrix  g11 g12 g21  g 11 G *( jω ) = LG( jω ) R =  g21 g12  g11g22 

  g11g22  . g22 g12 g21  g22  g12 g21

(3.53)

Since L'=α L and R'=β R, where α and β are nonzero numbers, will also give a scaled matrix with g11,s = g22 ,s and g12 ,s = g21,s , we can conclude that the matrices L and R give only information about the relative influence of the different outputs and inputs. With the inclusion of α and β in the scaling matrices L and R we will obtain the scaled matrix


46

G**(jω)=α β G*(jω). It means that we can always find a G**(jω) such that σ(G**(jω) ) = 13. So all criteria which use σ(G*(jω)) or σ(G(jω)) as screening tools for CSD (e.g., σ(G(jω))/γ*(G(jω)), σ(G(jω)), etc.) cannot be useful. The only case where σ(G(jω)) gives a clear conclusion is for σ(G(jω)) = 0, since in this case γ*(G(jω)) = ∞ and the matrix G is rank deficient (singular). Minimized condition number for 2 x 2 matrices Grosdidier et al. [GMH85] present the following analytical solution for γ* based on RGA for 2 x 2 matrices

γ *(G( jω )) = RGA 1 +

RGA 1 − 1 = RGA 2

∞

+

RGA

2 ∞

−1

(3.54)

where ||RGA||1 and ||RGA||∞ are the induced 1 and ∞ norm of RGA= RGA(G(jω)) defined by (3.29). For 2 x 2 matrices, it is easy to see that ||RGA||1 = ||RGA||∞ = |λ11| + |1-λ11|. Equation (3.54) can be written as

γ *(G( jω )) +

1 = 2 RGA 1 = 2 RGA ∞ . γ (G( jω ))

(3.55)

*

It is interesting to see that for real 2 x 2 matrices with an odd number of negative elements in the transfer function matrix always γ∗ = 1. It can be understood if we consider that for this kind of matrices λ11≤ 1 ⇒ ||RGA||1 = |λ11| + |1-λ11| = 1 ⇒ γ∗ = 1 (equation (3.54)). Roughly speaking, the interaction paths for this matrix type have different sign what compensates in some sense the system directionality, but not the interaction. The DV-structure of the example 3.3.5 belong to this class of matrices. Note also that system directionality and interaction are different properties. As we have already seen, a system can present strong interaction, but no directionality. However, the converse is true, i.e., a system with strong directionality will always have strong interaction.

3.3.3 Relations between RGA and γ* for n x n matrices Equation (3.54) shows that there exists a strong relation between RGA and minimized condition number for 2 x 2 systems. For n x n systems, we have the following lower bound [NM87] ∆

{

RGA = max RGA 1, RGA ∞ 1 γ * (G( jω )) + * ≥ 2 RGA γ (G( jω )) 3

}

⇔ γ * (G( jω )) = RGA +

RGA 2 − 1

Since σ(G** (jω)) = αβσ(G*(jω) ), then for αβ€ = 1/σ(G*) ⇒ σ(G**) = 1 .

(3.56)

47


and a conjectured upper bound of γ*(G(jω)) ([SM87], [NM87])

γ *(G( jω )) ≤

(∑

ij

)

λ ij + k (n)

(3.57)

where k(n) is a constant (conjectured to be n-2) and ||⋅||1 and ||⋅||∞ are respectively the maximum column and maximum row sums defined by (3.29). The upper bound was proven for 2 x 2 matrices with k(2) = 0 in [GMH85], but it was only conjectured for the general case in [SM87] and [NM87]. Here we will show by an example that this bound is in general wrong. Example 3.3.6 Counter-example for the upper bound of γ* The upper bound works quite well for 3x3 systems, but for systems of higher order it will depend on relations between all singular values and especially on the unitary matrices U and V of the optimally scaled matrix. Using the EMC theorem and the fact that U and V are unitary matrices (i.e., UUH=UHU=I) we can easily construct the following optimally scaled matrices: -0.5 -0.5 0.5 -0.5 0.5 0.5  0.5   0.5  0.5 0.5 0.5 0.5 0.5 0.5 -0.5 0.5 U1 =  0.5 0.5 -0.5 -0.5  V1 =  0.5 0.5 0.5 -0.5   0.5 -0.5 0.5 -0.5   -0.5 0.5 0.5 0.5 

 0.3305 0.5601 U2 =  -0.2776  -0.7071

-0.8395 0.2771 0.3305  0.5369 0.2904 0.5601 0.0837 0.9159 -0.2776  0 0 0.7071 

 -0.3507 0.4272 V2 =  -0.614  0.5635

-0.6135 -0.5639 0.3505 0.4275

-0.6144 0.5631 0.351 -0.4268

-0.3507  -0.4272 -0.614  -0.5635 

The first set of matrices corresponds to the generalization of equation (3.53). Therefore, all systems with the matrices U1 and V1 will have ||RGA||1 = ||RGA||∞ as for the 2 x 2 systems. The other set, i.e., U2 and V2 , represents an usually distribution. To analyze the influence of the singular value distribution we chose

Σ1=diag(20, 19, 2, 1) Σ2=diag(20, 15, 5, 1)

Σ3=diag(200, 199, 2, 1) Σ4=diag(200, 150, 50, 1).

Table 3.1 shows that the conjectured upper bound is wrong (bold numbers in table). Moreover, the influence of the matrices U and V is larger than that of the singular value distribution. Numerical results indicate that when Test = RGA + RGA − 1 + min{ RGA 1 ,RGA #%%%$%%%& Lower bound

∞

} −∑

ij

λ ij

(3.58)

is positive the conjectured upper bound will be verified and for negative values it will give wrong results. But it is only a 'conjecture' for the conjectured upper bound (3.57). Moreover, Table 3.1 shows that when Test < 0 the lower bound (3.56) gives a bad estimation of γ∗.


48

Therefore, (3.58) give us a criterion that allows us to determine how conservative the lower bound for the estimation of the minimized condition number is.

M

Table 3.1: Analysis of lower and upper bounds of γ∗ ( M ). For matrices with Test < 0 the upper bound (3.57) will give wrong results. lower upper * γ (M) ||RGA||1 ||RGA||∞ Σij |λij| bound bound Test (σ2 /σ3) (3.29) (3.29) (3.56) (σ1 /σ4) (3.57) (3.58)

U1 Σ1 V1T

2.99

2.99

11.95

5.80

20.00

13.95

-3.16

9.50

U2 Σ1 V2T

8.07

10.02

24.92

20.00

20.00

26.92

3.15

9.50

U1 Σ2 V1T

4.33

4.33

17.33

8.55

20.00

19.33

-4.45

3.00

U2 Σ2 V2T

7.80

10.02

22.22

20.00

20.00

24.22

5.58

3.00

U1 Σ3 V1T

25.50

25.50

101.99

50.98

200.00

103.99

-25.52

99.50

U2 Σ3 V2T

82.15

100.00

257.31

200.00

200.00

259.31

24.83

99.50

U1 Σ4 V1T

49.30

49.30

197.18

98.58

200.00

199.18

-49.31

3.00

U2 Σ4 V2T

75.64

100.00

222.02

200.00

200.00

224.02

53.62

3.00

Observe that if we apply equation (3.28) to this example, we will conclude that for the system with U1 and V1 matrices the substitution of a diagonal input uncertainty description by a full block representation increases the set of possible plants considerably, whereas for systems with U2 and V2 it does not. "

3.3.4 Numerical solution for the minimized condition number Minimizing of the condition number is a convex problem. There are straightforward and numerical efficient alternatives to do it: 1. Using the convex upper bound for µ To compute the minimized condition numbers we define  0 G −1  H= . G 0 

(3.59)

Then as proven by Braatz and Morari [BM94]  γ *(G ) = min σ DHD −1  R ,L

(

γ

 * G = ( ) min σ I  R

2

  , D = diag ( R ,L) 

)

2

  , D = diag ( R ,I ) 

(DHD ) −1

(3.60)

(3.61)

49


 γ *O (G ) = min σ  L

2

  , D = diag ( I ,L) . 

(DHD ) −1

(3.62)

These convex optimization problems may be solved using available software for the upper bound on µ(H). In calculating µ(H), we use the structure ∆€= diag (∆diag , ∆diag ) for γ*(G), the structure ∆ =€diag (∆diag , ∆full ) for γΙ*(G), and the structure ∆ = diag (∆full , ∆diag ) for γO*(G). 2. The LMI approach Boyd et al. [BGFB94] present a LMI approach to solve the minimized condition number problem. This method consists of the solution of the following generalized eigenvalue minimization problem (GEVP) under LMI constraints: minimize γ 2 subject to P∈R nox no and diagonal, P > 0 Q∈R ni x ni and diagonal, Q > 0

(3.63)

Q ≤ M H P M ≤ γ 2Q The scaling matrices L and R and the minimized condition number γ* are respectively given by the matrix square root of P, Q, and γ 2, i.e., L = sqrtm(P), R = sqrtm(Q), and γ∗(M) = γ 2. The LMI approach has advantages over the µ-upper-bound, because here we can insert additional constraints and conditions. This allows to consider additional system properties as, for example, diagonal dominance.

3.3.5 Robustness interpretation of the minimized condition number Here we will continue the discussion started in section 3.2 about the role of the minimized condition number in the robustness analysis. Our aim is to gain insights into how open-loop properties and thus potential closed-loop properties may vary in the presence of uncertainty. Particularly, we will consider here the variation in the open-loop transfer function LO as a consequence of an input uncertainty given by (3.18), i.e., P=G [ I+EI ]. To proceed with this study, it is useful to analyze the relative deviation of the open-loop transfer function, which is defined by ∆

RL = ( LO′ − LO ) L−O1 .

(3.64)

From (3.64),

[

]

LO′ = PK = G[ I + EI ]K = I + GEI G −1 LO = [ I + RL ] LO .

(3.65)


50

This equation shows us that the worst case magnitude of RL corresponds to the biggest difference between the nominal and actual open-loop transfer function. The problem with (3.65) is its output scaling dependency. We can solve this problem by inserting a real positive diagonal matrix DO which is minimized over all possible scalings. Doing so, (3.65) is transformed to

[

]

[

]

DO LO′ = I + DO RL DO−1 DO LO ⇔ DO LO′ = I + DO GEI G −1 DO−1 DO LO .

(3.66)

The size of the worst case scaled magnitude will be termed the worst case deviation and is defined as follows: Definition 3.3.7 Scaling-independent worst case deviation (η). The scaling-independent worst case deviation is defined as max

σ (DO RL DO−1 ) =

max

σ (DO GW2 ∆ IW1G −1 DO−1 ) ⇔ η = µ ∆O (GW2 ∆ IW1G −1 )

∆

η= η=

EI ∈∆ I , DO∈∆ O EI ∈∆ I , DO∈∆ O

max

EI ∈∆ I , DO∈∆ O

σ (DO GE I G −1 DO−1 )⇔

(3.67)

where ∆Ο has a diagonal structure to represent the scaling matrix DO. ! η may be calculated using the structured singular value (µ). To do so, we apply the following property of µ given by Skogestad and Morari [SM88]. Property of µ. Let ∆€= diag(∆Ο, ∆I). Then 0 µ ∆ O ( A∆ I B) ≤ σ (∆ I )µ 2∆  B

A . 0 

(3.68)

Now, applying (3.68) to calculate η yields  0 W1G −1  η = µ ∆ O (GW2 ∆ I W1G ) ≤ µ   0  GW2 −1

2 ∆

(3.69)

where ∆ = diag(∆O, ∆I). When W1 = W2 =I and ∆Ι = ∆diag the upper bound reduces to the minimized condition number γ*(G), i.e., η ≤ γ*(G), since (3.69) is equal to (3.60) which is one of the possible numerical methods to calculate γ*(G) (see section 3.3.4 ). Therefore, the minimized condition number is a good indicator of the possible deviation of the open-loop transfer function LO. An analogous derivation can be done for uncertainty at the plant output and the end result is given by an expression equivalent to (3.69). Chen et al. [CFN94] present similar conclusions. In addition, they include some additional derivations considering ||RGA||. Their results can be obtained from ours, if we consider the relations between ||RGA|| and γ*(G) given by the lower bound (3.56).

51


A large value of η (or to some extent of γ*(G)) at the frequency of interest indicates that the open-loop transfer function may deviate far from its nominal value and thus will potentially lead to undesirable closed-loop properties. The sensitivity of LO to plant uncertainty however is not of importance, but its effect on the closed-loop system. A simple computation for the perturbed output sensitivity function S'O = (I+L'O) -1 yields

(SO′ − SO )SO−1 = RLTO ( I + RL TO )−1 .

(3.70)

Thus a large η will yield a large RL hence also a big relative deviation of the output sensitivity function when TO ≈ 1 (or when RLTO is also large).

3.3.6 Minimized condition number of P and maximal singular value ratio of P In the last section, we showed that the effect of an input plant uncertainty on SO can be estimated by γ*(G). Although this class of uncertainty is very important, there are many other uncertainties. Moreover, if γ*(G) is large and there are small input uncertainties, we have to calculate η from (3.69) to draw a conclusion. For this, the weighting functions W1 and W2 must be determined. By introducing the concept of the minimized condition number of P, we circumvent this problem. Definition 3.3.8 Minimized Condition Number of P (γ#). Consider the polytopic system representation P k k  P ∈ Co { G1 ,...,Gk } = ∑ α i Gi : α i ≥ 0 ,∑ α i = 1  . i =1  i =1 

(3.71)

Then, the minimized condition number of P is defined as

γ # (P )= min sup γ (L# PR # ) =min sup{γ 1 (L#G1 R # ), " , γ k (L# Gk R # )} # # # # ∆

L ,R

(3.72)

L ,R

where L# and R# are real diagonal nonsingular scaling matrices.

!

Numerical computation of the minimized condition number of P To calculate the scaling matrices L# and R# used to determine γ#(P), we need to solve the following GEVP problem minimize κ 2 subject to P∈R no xno and diagonal, P > 0 Q∈R ni xni and diagonal, Q > 0

(3.73)

Q ≤ M i P M i ≤ κ 2Q ∀ i =1,", k . H

The scaling matrices and γ# are given by L# = sqrtm(P), R# = sqrtm(Q), and γ#(M) = max γ(L#MiR#), respectively. The minimized condition numbers of P and G can only detect changing in the system directionality. The multiplication of all channels by a scalar produces no modification of γ#(P)


52

and γ*(G). Of course, this kind of perturbation is also important for the correct caracterization of the system. Therefore we introduce the maximal singular value ratio of a plant set P to take this effect into account. Definition 3.3.9 Maximal singular value ratio of a plant set P ( σ #(P) ). For the polytopic model P, given by (3.71), the maximal singular value ratio of P is defined by ∆

σ ( P( jω )) = #

{ } min{σ ( L G R ),",σ ( L G R )}

max σ ( L# G1 R # ),",σ ( L# Gk R # ) #

#

#

1

(3.74)

#

k

where L# and R# are the real, diagonal, and nonsingular scaling matrices used to determine ! γ#(P). The maximal singular value ratio of a plant set P is the ratio between the largest and the smallest maximal singular values of the optimally scaled polytopic model P. If the polytope P has almost a ball shape, the scaling matrices L# and R# will be the same as the scaling matrices L and R for the model G in the center of the polytope. The next example illustrates this fact. Continuation of example 3.3.5: Calculating γ# and σ # for LV and DV structures. To calculate γ #(P) and σ #(P) we need to construct a representative polytopic set. Of course, for the system's input uncertainty the largest possible deviation is achieved when a variation like k1=1.2 and k2=0.8 is chosen. This kind of variation is illustrated by the models 1, 2, 5, and 6 in Table 3.2. For control purposes, it is also_interesting to consider the lowest possible value of σ (GK) and the largest possible value of σ(GK) for a given controller K. These will be the plant model that gives the most sluggish and the fastest response, the corresponding models are the models 3 and 4 in Table 3.2. Table 3.2: Minimized condition number of P considering six extreme plants _ σ # # # # k τ τ γ (L LVR ) γ (L DVR )

N

k1

0

1.0

1.0

0.5

0.5

138.27

1

0.8

1.2

0.0

1.0

2

1.2

0.8

1.0

3

0.8

0.8

4

1.2

5 6

2

1

2

_ σ

(L#LVR#)

(L#DVR# )

1.0

172.84

1.25

149.79

1.5

176.26

1.50

0.0

149.79

1.5

176.26

1.50

0.0

0.0

138.27

1.0

138.26

1.00

1.2

1.0

1.0

138.27

1.0

207.40

1.50

0.8

1.2

1.0

0.0

149.79

1.5

176.26

1.50

1.2

0.8

0.0

1.0

149.79

1.5

176.26

1.50

γ#(P)

149.79

1.5

σ#(LV) = 1.50

σ#(DV) = 1.50

0.8 ≤ k1, k2 ≤ 1.2, 0 ≤ τ1,τ2 ≤ 1

53

3.4 RP- AND RPP-NUMBERS

The scaling matrices L# and R# were determined using the numerical solution presented in this section and are the same as L and R that we obtained for the nominal model, since this model is in the center of the ball shaped polytope given in Table 3.2. Observe that the minimized condition number of P, i.e., γ#(P), solves in part the problem that γ*(G) and RGA are only functions of the relative sizes of the elements of G. It means that input and output diagonal matrices have no influence on γ∗(G) and RGA. γ#(P) reflects all variations that modify the relative magnitude of the inputs as in models 1, 2, 5, and 6 but it is not sensitive to simultaneous modifications (models 3 and 4) and to all-pass functions as pure delays, i.e., γ#(P) only represents magnitude changes and no phase changes. On the other hand, σ€#(P) can detect absolut variations like those which occurred in the models 3 and 4. Therefore, σ€#(P) complements the information obtained by γ#(P). In the next section, the problem with the time delays will be solved by the introduction of two new indices which take the feedback control limitations imposed by nonminimum-phase behavior into account automatically. Based on Table 3.2, we can conclude that for the DV-structure it will be simple to design a controller that gives a good performance for all possible models, whereas for the LVstructure, we can expect problems. "

3.4 RP- and RPP-numbers The definitions of the Robust Performance Number (RPN) and the Robust Performance Number of a plant set (RPPN) are based on equation (3.42), i.e., the analytical solution for the RP-problem for the case where an inverse-based controller is used to control a minimum phase system with an unstructured complex input uncertainty. Particularly, when the nominal performance and robust stability conditions are satisfied, | ws ζ | and | wt τ | in (3.42) will be smaller than 1, so (3.42) essentially is determined by (3.43), which is rewritten below to facilitate our discussion

µ RP ≈

wsζ

 1  wtτ  γ ( G) + . γ ( G)  

(3.43)

Now, if we assume ws and wt equal to 1, equation (3.43) essentially states that the condition number of the plant is only critical in the frequency region where neither S nor T are small, i.e., in the crossover frequency range. Using (3.42) resp. (3.43) as controllability measure has two drawbacks: First, (3.42) results from assuming a full block input uncertainty with a scalar weighting function wt . This description can be very conservative. To reduce this conservativeness, the system should be scaled before the determination of the weighting function wt . This is to some extent equivalent to substituting γ€(G) by γ *(G), the minimized condition number, in (3.42). Secondly, the desired performance is indirectly represented in (3.42) by the weighting function ws .


54

Instead of using weighting functions and the free parameter k(s) of (3.39), it is more intuitive and useful to relate the controllability measure to the desired closed-loop performance directly. These ideas led to the definition of the Robust Performance Number (RPN).

3.4.1Definitions of RPN and RPPN Definition 3.4.1 Robust Performance Number ( RPN, Γ€). The RP-number is defined as   ∆ 1 Γ (G ,T ,ω ) = σ [ I − T ( jω )] T ( jω ) γ *(G( jω )) + *  γ (G( jω ))  

(

)

(3.75a)

∆

RPN = Γsup (G, T , ω ) =sup{Γ(G, T , ω )} ω ∈R

(3.75b)

_ where γ*(G(jω)) is the minimized condition number of G(jω); σ( [I-T] T ) is the maximal singular value of the transfer function [I-T] T ; and T is the (attainable) desired output complementary sensitivity function which was determined for the nominal model G. ! The_ RP-number consists of two factors: 1. σ( [I-T] T ). This term acts as a weighting function and is a generalization of the weighting function |1-T||T| proposed by Engell [En88] for SISO system controller design to emphasize the region which affects the damping and speed of the closed loop system. It is well-known that the system uncertainties close to the crossover frequency are more important for robust stability and robust performance than the uncertainties at low and high frequencies (see, e.g., [MZ89],[SP96] ). For example, a system can have large uncertainty at low frequencies, but nevertheless present no stability and performance problems. This _ fact is automatically considered by the function σ( [I-T] T ), which has its peak value in the crossover frequency range. The choice of T depends on the desired closed loop bandwidth, sensor noise, input constraints, and in particular on the nonminimum-phase part of G, i.e., RHP-zeros, RHP-poles, and pure time delays. In Chapter 4, we discuss how T can be determined. 2. γ∗(G)+1/γ∗(G) . The origin of this term and consequently the name RP-number is equation (3.42). In (3.42) we see that the µ for RP is a function of γ(G)+1/γ(G) multiplied by weighting functions that represent _the performance and uncertainty terms. In our definition of RPN this term is substituted by σ( [I-T]T ). The RP-number is a measure of how potentially difficult it is for a given system to achieve the desired performance robustly. The easiest way to design a controller is to use the process inverse. An inverse-based controller will have potentially good performance robustness only when the RP-number is small. As inverse-based controllers are simple and effective, it can be concluded that a good selection of the control structure is one with a small RP-number. Next, we introduce the definition of RPPN.

55

3.4 RP- AND RPP-NUMBERS

Definition 3.4.2 Robust Performance Number of P ( RPPN, Γ# ). The RP-number of P is defined by ∆

[

]

Γ #(P ,T ,ω ) = σ #( P( jω )) + γ #( P( jω )) − γ *( G( jω )) Γ(G ,T ,ω ) ∆

{

}

# RPPN = Γsup (P, T , ω ) =sup Γ # (P, T , ω )

ω ∈R

(3.76a)

(3.76b)

where γ#(P), σ#(P), and Γ(G,T,ω) are defined by (3.72), (3.74), and (3.75a), respectively. ! With the RPPN, the expected plant uncertainties can be reflected in more detail. The potential increase of the directionality of the system is considered by the term ( γ#(P) - γ*(G) ) and the modification of the maximal singular value is captured by σ#(P). By the definition of RPPN we have RPPN ≥ RPN. An expression for the RP-number based on RGA From the lower bound (3.56) and the definition of the RP-number (3.75) we get

(

)

Γ (G, T ,ω ) ≥ σ [ I − T ( jω )] T ( jω ) 2 RGA .

(3.77)

This equation is an equality for 2 x 2 systems and gives a very good estimation for systems for which (3.58) is positive.

3.4.2 RPN-scaling procedure We have already seen how important the scaling of a matrix is for the correct analysis of directionality. Although we can calculate a scaling matrix L and R that makes the γ*(G(jω)) minimal for each frequency ω, we cannot do it in general for the system G. Here, we have the possibility to chose only one set of matrices L and R to be used for all frequencies. Therefore, it is necessary to develop a strategy to system scaling that permits a correct interpretation of the feedback system properties. The crossover frequency range (or equivalently the desired performance) plays a very important role for the determination of the optimal scaling. Now, we state a scaling procedure that takes the performance and robustness factors into account automatically. RPN-scaling procedure: 1. Determine the frequency ωsup where Γ(G,T,ω) achieves its maximal value. 2. Calculate the scaling matrices Ls and Rs , such that γ(Ls G(jωsup)Rs) achieves its minimal value γ*(G(jω)). 3. Scale the system with the matrices Ls and Rs , i.e., GS(s) = Ls G(s) Rs.


56

The analysis and controller design should be performed with the scaled system GS (for details see Chapter 4).

3.4.3 RPN and RPPN with constant scalings The RPPN as defined above is influenced both by the plant nonlinearity (which gives rise to different models at different operating points) and the plant uncertainties. Small differences of RPPN and RPN indicate that the system has small sensitivity to time invariant uncertainties. For changing operating conditions, the nonlinearity can be represented as a time-varying perturbation with some rate of variation. The extreme case of such a variation is to consider the system as arbitrarily fast time-varying. In this case, there is a relation between the D-scales used to calculate the upper µ bound and the scaling matrices L and R defined above [SP96]. Therefore, the calculation of RPN and RPPN using the constant matrices Ls and Rs obtained from the RPN-scaling procedure in section 3.4.2, i.e., ∆

γ jω = γ ( Ls G( jω ) Rs ) ∆

(

ΓLR (G) = σ Ls [ I − T ( jω )]

T ( jω ) L−s 1

)

 1   γ jω +    γ  jω 

(3.78a)

∆

RPN LR =sup{ΓLR (G )}

(3.78b)

ω ∈R

∆

[ (

) (

)

]

# (P )= sup σ # (P ) + γ L#s P( jω )Rs# − γ jω ΓLR (G ) ΓLR

∆

{

}

# RPPN LR = sup ΓLR (P )

ω ∈R

(3.79a)

(3.79b)

provides a criterion to identify whether the system will present problems for arbitrarily fast time-varying parameters. Small RPPNLR values indicate that the use of a nonlinear controller will yield only small improvements over a linear controller. Large RPPNLR values for polytopic models that consider only model uncertainty related to the nonlinear model indicate that a nonlinear controller or a nonlinear transformation might be effective to improve the system's performance.

Chapter 4 Attainable Performance and Frequency Domain Approximation The stabilization of a system is only in exceptional cases the major reason for the introduction of feedback control. Indeed, in the case of stable plants, the feedback can have a detrimental effect on the stability robustness of the system, because the inclusion of feedback system can transform stable systems to unstable ones. The most common reason for the introduction of feedback control is the improvement of performance in the presence of uncertainty. In this context, control goals include disturbance attenuation, sensitivity reduction, the reduction of nonlinear effects, and command tracking improvement. In this chapter, we discuss the benefit of feedback control and the design tradeoffs.

4.1 Performance Requirements and Limitations It is well known that the benefits of feedback control come from high gain and it is also known that high gain increases the danger of loop instability, actuator saturation and sensor noise amplification. This conflict between the high and low gain requirements is what makes control system design interesting and difficult. In broad terms, a feedback system designer will try to shape the loop gain as a function of frequency so that the low-frequency, high-gain requirements are met without infringing the high-frequency, low-gain limits imposed by plant model errors, sensor errors and actuator limits (see e.g., [GL95], [SP96], [ZDG96], [Mac89]).

4.1.1 General control configuration In this section, our aim is to analyze various performance criteria and limits using singular values. The closed-loop configuration that we will use for much of this discussion is shown in Figure 4.1.

4. ATTAINABLE PERFORMANCE AND FREQUENCY DOMAIN APPROXIMATION

58

Figure 4.1 : Control configuration with three compensators K, R, and F (3 CDOF) In Figure 4.1 K, R, and F represent the possible compensators for the general plant P consisting of the nominal transfer function matrix G and the uncertainty matrix ∆. The diagonal transfer function matrices MK and MF can be used to represent the dynamics and the failures of the measurement equipment. The dynamic part of the matrix MK is usually included in the plant dynamics. Thus, the elements on the diagonal of MK and MF have just two possible values: 1 for normal operation and 0 for failure. The diagonal transfer function matrix A is used to describe the actuator dynamics, saturations, and failures, again the dynamic part can be included in the general plant P , so that A represents saturation and failures only. Moreover, for feedback with integral action it is recommended to implement it in A, i.e., A = s-1I. The input signals d, r, and n represent the process disturbances, reference values (setpoints), and noise signals, respectively. The output signals are the controlled variables y, the feedback error r-y, and the control actions u and u'. The transfer function matrix from the inputs to the outputs signals for the compensator structure in Figure 4.1 (assuming for simplicity MK = MF = A = I and P = G, i.e., the transfer function for the nominal model G) is given by  y   S(Gd + GF )    r − y  =  − S(Gd + GF )  u   SI ( F − KGd )

SGR −T  d    I − SGR T  r  SI R − SI K   n 

where S, SI , and T were already defined in chapter 3:

(4.1)

59

4.1 PERFORMANCE REQUIREMENTS AND LIMITATIONS S = SO = ( I + GK )

−1

SI = ( I + KG)

−1

(output) sensitivity function

(4.2)

input sensitivity function

(4.3)

T = I − S = TO = GK ( I + GK ) = ( I + GK ) GK . −1

−1

(4.4)

(output) complementary sensitivity function1 The complementary sensitivity function T is often referred to as the closed-loop transfer function. Usually, the output sensitivity function S is just called sensitivity function and is more often used in the analysis than input sensitivity function SI. SI only appears in the last row of (4.1), i.e., SI is used to calculate the control action. As SIK = KS, if F = 0 and R = K we can substitute SI by S in (4.1). In many (perhaps most) applications the prefilter is R = K and no feedforward action (F = 0) is included. We then have a one degree-of-freedom compensator (1 KDOF) design problem, whereas the general problem represented by Figure 4.1 is known as the three degree-offreedom compensator (3 CDOF) problem. In the case where the F action is not applied or R is the same as K, the general 3 CDOF problem reduces to the two degree-of-freedom problem with prefilter R (2 RDOF) or with feedforward action F (2 FDOF), respectively. The standard unity feedback situation is equivalent to 1 KDOF. For many processes, this configuration is sufficient to achieve the desired performance. Of course, this is the easiest structure to design and to implement. In the next subsection we will discuss when the inclusion of an additional degree-of-freedom is necessary. This analysis can be done by the examination of the influence of the controllers K, F and R in shaping (manipulating) the effect of the input signals d, r and n in the output signals y, r-y and u. In general the sensor noise, the actuator constraints, and closed-loop stability requirements are the decision criteria to evaluate whether the 2 or 3 CDOF structures are indispensable for a good performance, since these 3 factors limit the feasible disturbance attenuation and tracking characteristics.

4.1.2 Disturbance attenuation and feedforward action The signal d represents an exogenous disturbance such as a feed stream variation in a distillation column that affects the output y of the system via a transfer function matrix Gd. The disturbance attenuation problem is to find some means to reduce or eliminate the influence of d on the output y. Before embarking on the design of a feedback controller, it is worthwhile to note that the disturbance attenuation problem may also be addressed by other means. It may be possible to modify the system in such a way that the disturbance is eliminated or reduced in magnitude; for example, the effect of the feed flow may be reduced or eliminated using a feedtank between two distillation columns. Another possibility is to use a control 1

The name complementary sensitivity comes from the algebraic relation between S and T, i.e., S + T = I.

4. ATTAINABLE PERFORMANCE AND FREQUENCY DOMAIN APPROXIMATION

60

structure that automatically compensates the disturbance. In the case of distillation columns, if a ratio control structure (e.g., L/D,V/B-structure [Luy92]) is used, the influence of the flow disturbance is almost automatically compensated. If plant modifications are not possible (or practical), one can measure the disturbance and compensate its effect via a feedforward controller F as shown in Figure 4.1. In the open-loop situation with K = 0, the transfer function matrix from d to y is Gd + G F so that the effect of the disturbance may be eliminated if G is square and has no right-half-plane (RHP) zeros or deadtime by choosing F = -G-1Gd. Complete cancellation is not possible if G has RHP-zeros or deadtime. In this case, the effect of d may be reduced by making Gd + G F small. Specifically, σ ((Gd + GF )( jω )) ≤ ξ (4.5) ensures that y ( jω )

2

≤ ξ d ( jω ) 2 .

(4.6)

Now we consider perfect feedforward control, which requires that the plant G is invertible. Solving for the control action u gives u = G-1r. However, for the actual plant P we have y = P u and the actual control error e = r-y = r-PG-1r. Then we obtain for multiplicative input and output uncertainties e=GEIG-1r and e=EOr, since P is given by G [ I+EI ] and [I+EO]G, respectively. These results show that the performance of the pure feedforward control is very dependent on model uncertainties. One of the main reasons for applying feedback control rather than pure feedforward control is to reduce the effect of uncertainty. For example, with integral action in the controller we can achieve zero steady-state control error e even with large model errors (provided det(G(0)) does not change its sign). The inclusion of the feedback control will reduce the effect of the uncertainty by a factor S' (sensitivity function of P) relative to that with pure feedforward control. This conclusion can easily be obtained by the analysis of the transfer function between d and y for feedforward and feedback control, since

σ ( S [Gd + GF ] ( jω )) ≤ ξ ⇒ y( jω ) 2 ≤ ξ d ( jω ) 2 .

(4.7)

Often, feedback control action alone is enough to achieve the desired system performance. When it is not the case, the feedforward (2 FDOF) controller structure can be a very effective means of improving the dynamic response of a control system. For feedforward control it is necessary to measure some of the major disturbances that drive the process away from the desired state. If one knows how a process responds to these disturbances, one can, in principle, generate a control signal that will compensate for the predicted response to a disturbance before it occurs, thereby holding the process at the desired state. Examining (4.7) in the frequency domain, we see that the most useful contribution of feedforward is to reduce the effect of those disturbances that cannot be effectively suppressed ωc. Control can be by feedback; that is, at frequencies around the crossover frequency _ significantly improved if the feedforward is adjusted such that σ (Gd+GF) is small in the frequency range 01 . ω c < ω < 10ω c . There is no particular advantage of feedforward control at very low frequencies where feedback is effective (i.e., where σ (S ( jω )) ni, G-1 must be replaced by a right inverse. Then we approximate the frequency response of the "ideal" controller Kid by a low order controller using the method presented in section 4.4.2. Note that this approach will only work for systems with small RPnumber (i.e., RPN 0). In this case, equation (5.20) reduces to z = ( ζ B −1 − 1) r1′− f − r3′ .

(5.21)

We can see that z is positive for small values of the selectivity ζB, since f, r1', and r3' are always positive so that (ζB-1 - 1) r1' can be bigger than f + r3' for some large value of ζB-1. For given cAin and Tin, z is only a function of ζB and T. Figure 5.2 shows this dependence for cAin = 5.1 mol/l and Tin = 115.0 °C. The plot of the concentration cB ( Figure 5.2b and Figure 5.3 ) reveals an interesting behavior of the system. The reactor exhibits a change of the sign of the static gain at the peak of the reactor yield ( i.e., where the concentration cB achieves its maximum value ), and displays nonminimum phase behavior for operation to the left of this peak and minimum-phase behavior for operating points on the right. This fact can be seen in Figure 5.2a. Another interesting fact is that the variation of z on ζB-1(or ζB) increases with the reactor temperature. Moreover, Figure 5.2c and 5.2d show that the manipulated variables exhibit large variations close to the maximal attainable selectivity value ζB = 0.5. Figure 5.3 was obtained by combining Figures 5.2b and 5.2c. The points shown in Figure 5.3 represent the selectivity ζB for different reactor temperatures T. Figure 5.3 shows more clearly the dependence between cB and f.

5. CONTROL OF A CSTR WITH THE VAN DE VUSSE REACTION SCHEME

90

300

z

1.1

200

cB 100

1 0.9 0.8

0 0.7 140

−100 10

0.6 120

140

8

0.5

130

6

120 4

−1 B

ζ

100 2

90

10

T [ ° C]

80

100

0.4

110

9

8

7

6

4

ζ

(a)

3

2

T [° C]

(b)

100

f

5 −1 B

80

4

x 10 0.5

80

Q

60

0

K 40

−0.5

20

−1

0 0.5

−1.5

130

0.4 120

0.3

ζB

−2

140

110 120

100

0.2

90 0.1

80

−2.5 10

T [ ° C]

100 9

8

7

6

ζ

(c)

5 −1 B

4

3

2

T [° C]

80

(d)

Figure 5.2: Operating point curves as a function of the reactor temperature T and selectivity ζB assuming cAin = 5.1 mol/l, Tin = 115.0 °C, and the nominal values presented in Table a5.1 and a5.2: (a) transmission zero when only cB is measured, (b) concentration cB [mol/l], (c) f = Fin /VR [h-1] ( Note that f was plotted as a function of ζB instead of ζB-1 ), and (d) QK [kJ/h] 120 °C

cB 1 105 °C 0.8 95 °C 0.6 85 °C

0.4

ζ B = 0. 5

0.2

0 0

5

10

15

20

25

30

35

f [ h−1 ]

Figure 5.3 : cB vs. f obtained by combining Figures 5.2b and 5.2c

91

5.2 ANALYSIS OF DIFFERENT OPERATING POINTS

5.2.2 Input saturation Input saturation is an important point to be considered in determining the best operating point. Here we calculate the necessary variation of the manipulated variables to compensate the influences of the inlet disturbances cAin and Tin in the controlled variables cB and T corresponding to a given steady-state operating point. It is useful to compare the required variation with the possible range. Mathematically, it means that we must analyze the following expressions:  f − fCmin   fCmax − f   f _ S = max  ,  3  f −   35 − f   

(5.22)

 Q − QK ,Cmin   QK ,Cmax − QK   QK _ S = max K  ,  Q + . 8 500    0 − QK   K 

(5.23)

where the possible ranges are (3 ≤ f [h-1] ≤ 35) and (-8500 ≤ QK [kJ/h] ≤ 0) and fCmin , fCmax, QK,Cmin, and QK,Cmax represent the minimal and maximal calculated values of f and QK. Summarizing, what we need to do is to calculate f and QK for given cB, T, cAin, and Tin and take the maximal and minimal possible values. Figure 5.4 shows the variation of f and QK versus cAin and Tin for cB = 0.8827 mol/l and T = 104.15 °C (these values correspond to the 6th operating point of Table 5.3). In this figure, two facts are important. First, the surfaces are monotonic. Second, the extreme point (Tin-5, 4.5) corresponds to fCmin and QK,Cmax and the other extreme point (Tin+5, 5.7) to fCmax and QK,Cmin, so that our analysis can be reduced to these two points.

Q

f

26

0 K −2000

24 22

−4000

20 −6000

18 16

−8000

14 110 6

105

−10000 4.5

5.5

T

100

5

in 95

4.5

c

5

c Ain

Ain

5.5 6

(a)

96

98

100

102

104

106

108

T in

(b)

Figure 5.4: Steady-state values of the manipulated variables as functions of cAin and Tin assuming cB = 0.8827 mol/l, T = 104.15 °C: (a) f = Fin /VR [h-1] and (b) QK [kJ/h]


92

5.2.3 Operating points For the nonlinear reactor model presented in section 5.1 and a given reactor volume VR, one can chose 4 variables from the set { cAin , Tin , Fin , QK , cA , cB ,T , TK } to be calculated, the other 4 variables must be given, assumed or optimized. The subset { cAin , Tin , Fin , QK } represents a good selection for the design task, since they are the physical inputs to the system. In principle, all 4 variables could be optimized for some objective function using selectivity, reactor yield, etc. Of course, cAin is usually given. Although Tin is also a disturbance, it could be changed by the inclusion of a heat exchanger in the inlet stream. Table 5.2: Performance index of the different operating points presented in Table 5.3 ( all values are calculated assuming VR = 10 l ) Reactor Controllability indices performance indices cB cB Trans. f_S* QK_S* _ _ ζB = c -c ψB = c σ Gr(0)] σ[Gh(0)] [ Zero (z) (5.23) (5.22) Ain A Ain

OP

Maximization Criterion

1

[EnKl93]

0.2329

0.1765

131.27

0.44

0.85

10.3

1.50

2

[CKA95] ψB

0.3733

0.2137

0.00

0.65

2.04

12.7

1.01

3

ψB

0.3763

0.2162

2.38

0.53

0.70

12.0

1.03

4

ζB

0.5037

0.1009

-32.10

0.94

6.76

7.5

0.34

5

ζB +ψB

0.4714

0.1813

-28.38

1.64

8.44

11.1

0.67

1 ζB +ψB - |z| 1 ζB +ψB - |z| for cAin=4.5

0.4708

0.1731

-19.71

0.34

1.11

11.3

0.64

0.4888

0.1520

-26.54

0.56

1.35

10.2

0.54

6 7 *

f_S and QK_S are the necessary changes of the manipulated variables to compensate the influences of the inlet disturbances cAin and Tin on the controlled variables cB and T.

Table 5.3: Optimization and process variables for the different OPs of Table 5.2 Optimization variables Tin Fin OP cAin f=V [mol/l] [° C] R

Process variables cC cD

QK

cA

cB

T

TK

[kJ/h]

[mol/l]

[mol/l]

[mol/l]

[mol/l]

[° C]

[° C]

1

5.1

130.00

18.83

-4496.0

1.235

0.900

2.419

0.546

134.14

128.95

2

5.1

104.38

14.19

-1113.1

2.18

1.090

1.090

0.740

113.61

112.33

*

*

2.169

1.103

1.140

0.688

125.98

117.33

*

4.079

0.514

0.074

0.433

96.23

87.59

*

3.138

0.925

0.386

0.650

111.72

103.06

§

3.225

0.883

0.332

0.660

104.15

99.71

25.50§ -2912.1§

3.515

0.775

0.219

0.591

103.51

100.15

3 4 5

5.1 5.1 5.1

123.52 30.00 99.27

*

30.00

*

111.37 30.00

6

5.1

101.69 20.00

7

5.1†

99.40

§

-7500.0 -7498.0 -7500.0 -3850.4

The applied optimization ranges for f and QK were: 10 ≤ f ≤ 30 , -7500 ≤ QK ≤ -1000, § 10 ≤ f ≤ 20 , and -4500 ≤ QK ≤ -1000. † optimization for cAin= 4.5 mol/l. *

93

5.2 ANALYSIS OF DIFFERENT OPERATING POINTS

The first and second operating points (OP) in Table 5.2 were taken from [EnKl93] and [CKA95]. The 3rd, 4th, and 5th OPs were determined by maximization of the reactor yield, reactor selectivity, and reactor yield plus selectivity using as the optimization range 10 ≤ f ≤ 30 and -7500 ≤ QK ≤ -1000 without considering the controllability of the system. These three OPs represent the maximal achievable values for the corresponding optimization criteria. The last OP represents a tradeoff between reactor performance and controllability. The reactor performance is almost optimal and the controllability is acceptable. In the next three sections, controllers for the 1st, 2nd, and 7th OP will be developed. These points correspond to points to the left of the peak, on the peak, and to the right of the peak on the concentration curves in Figure 5.3. _ _ Remark 5.1: In Table 5.2, σ[ Gr(0)] and σ[ Gh(0)] represent the maximal singular value of the transfer matrix_Gr = (A, Br, C, D) and Gh = (A, Bh, C, D) defined in Table a5.3 at ω = 0. _ σ[ Gr(0)] and σ[ Gh(0)] given an idea how sensitive an OP is to uncertainty of kinetic parameters and reaction enthalpies, respectively. Small values means smaller sensitivity and are therefore preferred. Remark 5.2: Table 5.3 shows that the increase of the selectivity in this system is achieved by the reduction of cC. The concentration of the component D is almost unchanged in all operating points. This fact is a consequence of the relative low activation energy of the parallel reaction. It means that the reduction in reactor temperature influences more the series reaction than the parallel reaction. Remark 5.3: As already mentioned, the selectivity is only the adequate performance index for the reactor when recovering of the desired product from one of the unwanted products is not possible. For the chemical system analyzed here, i.e., where cyclopentenol (B) is produced from cyclopentadiene (A) and the by-products cyclopentanediol (C) and dicyclopentadiene (D) are produced in unwanted side and consecutive reactions, it is not the case, since all reactions are in fact reversible (see, e.g., [BW91, p. 63, 117, and 399]), i.e., low T

low T

A

  → ← 

high T low T

B

  → ← 

high T

C .

(5.24)

2 A → D ←  high T Equation (5.24) shows qualitatively the influence of the temperature. Reversibility of the reactions was not analyzed here. Therefore, the reader should consider the kinetic parameters of Table a5.1 more as an example for the Van de Vusse reaction scheme than the real kinetic parameters for the cyclopentadiene reactions. For an irreversible reaction, the 7th OP is the optimal reactor condition, whereas the 1st OP would be the optimal operating point for the CSTR when, for example, the reaction B ↔ C in (5.1) is reversible. Here, in another reactor, the reaction C to B could be favored given a process with optimal yield of the component B. The 2nd OP, in principle, should be only considered in a situation where no recycle streams are possible. In section 5.4, these three OPs will be considered in detail. But first we will design a µ-optimal controller for the 1st OP.


94

5.3 µ-Optimal Controller for the First Operating Point In Chapter 4, it was shown that for a µ-optimal controller design, it is necessary to have an uncertainty representation and a performance weighting function. Many kinds of uncertainty models can be applied. The choice of the appropriate representation depends on the system characteristics. The rule is to choose the uncertainty as simple as possible. Here we show how a standard representation can give a high order controller which can efficiently be approximated by a low order controller using the frequency response approximation method discussed in chapter 4. The main part of this section was presented in [MT95] and [TM95]. In section 5.4 we will see that with much less effort the same performance can be achieved using the results of RPN analysis.

5.3.1 Uncertainty description Table 5.4 indicates that from a practical point of view, the pole uncertainty for this CSTR system seems to have very little importance if compared with the variation that occurs in the RHP-zero (see also Figure 5.2a ). This implies that describing the system uncertainty in parametric form via an affine parameter-dependent model is not worthwhile. The main uncertainty, which is related to the RHP-zero, could be captured in a more simple and efficient manner by other representations. For example, the system uncertainty could easily be factored to the output of the system. Moreover, if we take into account that the RHP zero is an almost output pinned zero  the linear model of table a5.5 has the output zero direction yz = [-0.999999, 0.000049 ]T ≈ [-1 0]T  this description could be further simplified. Table 5.4: Poles and zero for three different concentrations of cB, cAin=5.1 mol/l, Tin = 130°C, and T = 135.0°C cB RHP-zero Eigenvalues

0.7 mol/l

0.825 mol/l

0.95 mol/l

245.4

174.8

114.7

-10.66 -51.06 -80.87 -120.90

-14.55 -54.85 -85.56 -121.79

-19.87 -62.46 -90.78 -122.99

Often the uncertainty is most easily described in terms of uncertainties of the individual transfer matrix elements. This kind of uncertainty description may arise from an experimental identification of the system or from a set of linearized models. Particularly, for ill-conditioned system, this uncertainty description is not a good representation of the actual sources of uncertainty (see [MZ89, example 11.2-2] for an illustrative example). Nevertheless, we use it here, since the uncertainty of the system consists of two major parts. The first part can be modeled by a family of linearized plants corresponding to the different steady-state points. The other part is due to the uncertain chemo-physical parameters (e.g., rate coefficients, activation energies, reaction enthalpies) which are known only within bounds, with a relative uncertainty ranging from 1.3% to 56%. Due to the large number of uncertain parameters it is most convenient to model the complete uncertainty as unstructured bands for each element of the transfer matrix. Linearization of the CSTR model for many different operating points showed that most of the possible variations

95

5.3 µ-OPTIMAL CONTROLLER FOR THE FIRST OPERATING POINT

can be included in the uncertainty bands obtained by linearizing the model for the extreme values of cB. The Nyquist diagrams of the nominal plant are shown in figure 5.5 with the uncertainty plotted as circles around the nominal values. The radii of the uncertainties in each channel were approximated by 3rd order transfer functions which are summarized in table a5.4. figure 5.6 shows the block-structure of the nominal plant with uncertainties that is used for the design of the µ-optimal controller. Channel 11

Channel 12

-5

x 10

0.015 0.01 4

Cb=0.7 0.005

2 Cb=0.95

-0.01

Cb=0.95

IMAG

-0.005

-0.015 0 -0.02 -0.025 Cb=0.7 -0.03 -0.01

-0.005

0

0.005

0.01 0.015 REAL

0.02

0.025

0.03

-2 -7

0.035

-6

-5

-4

-3

-2

-1

0

1

REAL

Channel 21 1

0.1

2 -5

x 10

Channel 22

-3

0.2

x 10

0.5

0 0

IMAG

-0.1

IMAG

IMAG

0

Cb=0.95

-0.5 Cb=0.95

-0.2 -1 -0.3

-1.5

-0.4

Cb=0.7

Cb=0.7 -0.5 -0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

-2 -0.5

0

0.5

REAL

1

1.5

2

REAL

Figure 5.5: Nyquist diagram of each channel with uncertainty bounds

Figure 5.6: Nominal plant with uncertainty

2.5

3 -3

x 10


96

5.3.2 Performance weighting function The full-feedback µ-optimal controller for the general plant was computed by DK-iteration using the performance weights 10 4 (0.002954 s 2 + 0.05435 s + 1) 10 4 (0.00189 s 2 + 0.04348 s + 1)  , W p = diag  . (0.04941 s + 1)(597.8 s + 1) (0.04348 s + 1)( 434.7 s + 1)   

(5.25)

With the second order performance weights in (5.25) the resulting responses were faster than with a first order weight for the same minimal µ-value. The linearized nominal plant used by the µ-controller design is given in table a5.5. After five D-K iterations we obtained a controller with 78 states and a µ-norm of 1.23 (see figure 5.8).

5.3.3 Controller performance and order reduction The simulation results of product concentration reference steps which are shown in the sequel were calculated for the educt concentrations cAin = 4.5 mol/l, cAin = 5.1 mol/l, and cAin = 5.7 mol/l. Furthermore a deviation of +2% in the activation energy of k1 and k2 in (5.1) from its nominal value is also simulated. figure 5.7 shows the responses with the 78-state controller obtained by DK-iteration. The diffence between the step responses for the considered educt concentrations are small. The simulation with the augmented activation energy which causes 40% greater reaction rates shows that the controller is insensitive to this deviation, even though it was not included in the uncertainty model. 1

0.95

Cb [ mol/l ]

0.9

0.85

0.8

0.75 Ca0 = 4.5 mol/l Ca0 = 5.1 mol/l Ca0 = 5.7 mol/l Ea(1,2) = +2%

0.7

0.65 0

10

20

30

40

50 60 time [ min ]

70

80

Figure 5.7: Simulation of the 78 state controller

90

100

97

5.3 µ-OPTIMAL CONTROLLER FOR THE FIRST OPERATING POINT

Controller reduction by frequency response approximation To check which µ-value can be achieved with low-order controllers, first the frequency response approximation of section 4.4.2 is applied to get full-feedback controllers. Optimization of a PI-controller yields −15.5 s + 571  1  11.9 s + 926 C1 =  . s  2652 s − 213000 10900 s + 198000

(5.26)

An increase of the controller complexity to 4 states, yields no improvement. Approximation of each column with order 3 (6 states) gives  0.00069 s 3 + 0.497 s 2 + 31.8 s + 956  (0.000048 s 2 + 0.026 s + 1) s C2 =   114 . s 3 + 18.8 s 2 − 1500 s − 194000  (0.000048 s 2 + 0.026 s + 1) s 

−0.00029 s 3 + 0.37 s 2 + 31.2 s + 1110   (0.0000023 s 2 + 0.14 s + 1) s . 4.24 s 3 + 983 s 2 + 32000 s + 292000   (0.0000023 s 2 + 0.14 s + 1) s 

(5.27)

1.5

1

78 states 6 states 2 states

0.5 −4 10

−2

10

0

10 Frequency [rad/h]

2

10

4

10

Figure 5.8: µ with the full-feedback controllers

The µ-curves of the approximated full-feedback controllers are compared in figure 5.8 with the controller resulting from DK-iteration. It can be seen that even with PI-controllers the approximation is quite good and with order 6 a µ of 1.28 is achieved. The significantly smaller µ-values of the approximated controllers at low frequencies result from the integral part of the controller, which could not be enforced exactly with µ-synthesis. The simulations with the 6 state full-feedback controller are practically the same as with the original one.


98

1

0.95

Cb [ mol/l ]

0.9

0.85

0.8

0.75 Ca0 = 4.5 mol/l Ca0 = 5.1 mol/l Ca0 = 5.7 mol/l Ea(1,2) = +2%

0.7

0.65 0

10

20

30

40

50 60 time [ min ]

70

80

90

100

Figure 5.9: Simulation of the full-feedback controller C2 Application of the approximation method described in section 4.4.2 to optimize a decentralized PI-controller yields 0  1 12.5 s + 786 C3 =   0 12500 s + 142000 . s

(5.28)

The result for two 2nd order controllers is  0.23 s 2 + 11.9 s + 761  (0.0043 s + 1) s C4 =   0  

   2 42.2 s + 5700 s + 230000   . ( 0.002 s + 1) s  0

(5.29)

The achieved µ of the decentralized controllers and a one-way decoupling controller consisting of two PI-controllers in the first column and a single PID-controller in column 2 are shown in figure 5.10. The decentralized PI-controller achieves already a µ-norm of 2.3; for the decentralized controller having 4 states a µ−norm of 1.89 is obtained. table 5.5 summarizes the µ−values for all designed controllers. Table 5.5 : Comparison of the achieved µ-values full-feedback

decentralized

DK

C1

C2

C3

C4

states

78

2

6

2

4

µ

1.23

1.47

1.28

2.3

1.89

99

5.4 ANALYSIS AND CONTROLLER DESIGN USING RPN

The simulation results with the decentralized PI-controller are shown in figure 5.11. The controller is very robust. Only for cA0 = 4.5 mol/l the responses are remarkably slower than with the centralized controllers. The step responses with C4 are only slightly better and therefore not shown. 2.5

1

0.95 2

Cb [ mol/l ]

0.9

1.5

0.85

0.8

0.75 1 decentralized 2 states decentralized 4 states one way decoupling 3 states 0.5 −4 10

−2

10

0

10 Frequency [rad/h]

Ca0 = 4.5 mol/l Ca0 = 5.1 mol/l Ca0 = 5.7 mol/l Ea(1,2) = +2%

0.7

2

10

Figure 5.10 : µ with the decentralized controllers

4

10

0.65 0

10

20

30

40

50 60 time [ min ]

70

80

90

100

Figure 5.11 : Simulation of the decentralized PI-controller C3

5.4 Analysis and Controller Design using RPN Before we start with the controller design procedure, it is necessary to determine what kind of controller is necessary (e.g., linear vs. nonlinear), which uncertainty model should be applied (e.g., complex vs. real, structured vs. unstructured), which controller structure should be used (e.g., decentralized vs. centralized), and so on. In Chapter 1, the answer to these questions was called controller structure & order (CSO) selection. In the last section we showed that control of the CSTR with a satisfactory performance is possible with a simple decentralized PI-controller (cf. figure 5.11). Perhaps this PI-controller would be chosen for a practical implementation on a process control system due to its simplicity and possibility of easy online tuning (remember that the model is just a model). Here we show how we can answer the questions related to CSO selection in a simple and systematic manner via RPN and RPPN. By such an analysis, a time-consuming design of many different controllers can be avoided.

5.4.1 Analysis of the OPs 1, 2, and 7 using RPN and RPPN A polytopic model consisting of 6 linearized models determined as shown in Table 5.6 was calculated for each OP. In [CKA95] is suggested to consider the parameters sets given in Table 5.7 for the analysis of robustness against parametric uncertainty. These sets are related to two extreme cases chosen by physical considerations representing worst-case deviations from the nominal values.


100

Table 5.6 : Set of linearized models used to generate the polytopic models for each OP Model

N

Steady state point conditions N

1

{cB , T , cAin = 4.5 mol/l , TinN} and the nominal values of Table a5.1 and a5.2

2

{cBN, TN, cAin = 5.7 mol/l , TinN} and the nominal values of Table a5.1 and a5.2

3

{cBN, TN, cAin = 5.1 mol/l , TinN - 5 } and the nominal values of Table a5.1 and a5.2

4

{cBN, TN, cAin = 5.1 mol/l , TinN + 5 } and the nominal values of Table a5.1 and a5.2

5

{cBN, TN, cAin = 5.1 mol/l , TinN} and case 1 of Table 5.7 and a5.2

6

{cBN, TN, cAin = 5.1 mol/l , TinN} and case 2 of Table 5.7 and a5.2

N

means nominal value corresponding to an OP from Table 5.3

Table 5.7 : Two extreme cases for parametric uncertainty [CKA95] Parameter

Case 1

Case 2

k10 [h-1]

1.327 x 1012

1.247 x 1012

k20 [h-1]

1.327 x 1012

1.247 x 1012

k30 [l mol-1h-1]

8.773 x 109

9.313 x 109

∆HAB [kJ/mol]

6.56

1.84

∆HBC [kJ/mol]

-9.08

-12.92

∆HAD [kJ/mol]

-40.44

-43.26

To streamline our discussion, we will here use the same desired closed loop transfer function Td = diag{Td2[0.01, 0.0234], Td1[0.03, 5%]}, where Td2[0.01,0.0234]=1/(0.01s+1)/(0.0234s+1) and Td1[0.03, 5%]=1/(0.00018s2+0.0184s+1). This transfer function was selected without much analysis and is almost equivalent to the second order performance weights (5.25). Figure 5.12 shows the condition numbers for the 1st OP. The comparison of γ*(GN) and γ (GN) allows to say that the original system is quite ill-scaled. Here GN represents the nominal model corresponding to the steady-state conditions from Table 5.3. Moreover, Figure 5.12 shows that γ#(P) and γ#*(P) (i.e., γ#(P) calculated with scaling matrices L and R that minimize γ( LGNR ) at each frequency ω ) are almost the same indicating that the polytopic model for the 1st OP is almost round and centered at GN. Figure 5.12 also illustrates the condition number for the scaled system GS and the scaled polytopic model PS, i.e., γ (LSGNRS) and γ# (Ls#PRs#) respectively. Here the scaling matrices LS and RS were determined using the RPN-scaling procedure of section 3.4.2. Ls# and Rs# are scaling matrices that minimize γ# (Ls#PRs#) at the same frequency ωsup used in the RPN-scaling procedure.

101


4

10

3

10

unscaled

2

10

1

10

0

10 −1 10

0

10

1

2

10 10 Frequency [ rad/h ]

3

10

4

10

Figure 5.12 : Condition numbers for the 1st OP: γ*(GN) (solid line), γ#(P) (dashdot line), γ#*(P) (dotted line), γ (LSGNRS ) (lower dashed line), γ# (Ls#PRs#) ( upper dashed line), and γ (GN) (upper solid line) 1

10

0

10

−1

10

−2

10

−1

10

0

10

1

10 Frequency

2

10 [ rad/h ]

3

10

4

10

Figure 5.13 : RPN- and RPPN-plots for the 1st OP: Γ(GN) (solid line), Γ# (P) (dashdot line), ΓLR(GN) (dashed line), and ΓLR#(P) (dotted line)


102

Figure 5.13 shows the plot of Γ(GN), Γ#(P), ΓLR(GN), and ΓLR#(P) calculated from equations (3.75), (3.76), (3.78), and (3.79), respectively. Observe that for time-invariant parameters the system presents no problem, since Γ(GN) and Γ# (P) are not too different. The comparison of Γ(GN) and Γ#LR(P) indicates that only at very high frequencies fast time-varying parameters can present problems for this system. Moreover, the RPN-scaling procedure of section.3.4.2 will be very successful for the controller design procedure, since Γ(GN) and ΓLR(GN) are of the same magnitude indicating that the frequency dependence of L and R has not to be considered in the controller design. In conclusion, the controller design for the 1st OP will not present a big problem. 4

2

10

10

3

10

2

10

1

1

10

10

0

10

−1

10

−2

0

10 −1 10

10 0

10

1

10 Frequency

2

10 [ rad/h ]

3

10

4

Figure 5.14 : Condition numbers for the 2nd OP: γ*(GN) (solid line), γ#(P) (dashdot line), γ#*(P) (dotted line), γ (LSGNRS) (dashed line), and γ# (Ls#PRs#) ( upper dashed line)

−1

10

10

0

10

1

10 Frequency

2

10 [ rad/h ]

3

10

4

10

Figure 5.15 : RPN- and RPPN- plots for the 2nd OP: Γ(GN) (solid line), Γ# (P) ( dashdot line ), ΓLR(GN) (dashed line), ΓLR#(P) (dotted line) 1

2

10

10

0

10

1

10

−1

10

−2

0

10 −1 10

10 0

10

1

10 Frequency

2

10 [ rad/h ]

3

10

4

10

Figure 5.16 : Condition numbers for the 7th OP: γ*(GN) (solid line), γ#(P) (dashdot line), γ#*(P) (dotted line), γ (LSGNRS) (dashed line), and γ# (Ls#PRs#) ( upper dashed line)

−1

10

0

10

1

10 Frequency

2

10 [ rad/h ]

3

10

4

10

Figure 5.17 : RPN- and RPPN- plots for the 7th OP: Γ(GN) (solid line), # Γ (P) (dashdot line), ΓLR(GN) (dashed line), ΓLR#(P) (dotted line)

103


Figure 5.14 shows the condition numbers for the 2nd OP. Note that the plant behaves nicely in the crossover frequency range, i.e., γ*(GN) ≈ 2. The same cannot be said at low frequencies where the plant becomes singular so that γ*(GN(0)) = ∞. This fact is related to the fact that the transmission zero of GN is 0. The situation is even more critical for γ#(P), since the sign of the process gain for the channel 11 is not the same for all elements of the polytopic model of the 2nd OP. The analysis of Figure 5.15 indicates that the system, when only the concentration cB is measured, is very difficult to control with a fixed time-invariant linear controller. To control this system satisfactorily, it is necessary to apply two controllers with different sign in channel 11 to compensate the sign change of the plant. The problem then is to determine when the controller sign must be changed. The situation is completely different for the 7th OP, which is the easiest OP to control from the 3 OPs as shown by Figures 5.16 and 5.17. In the sequel we will design a low order controller for the 1st and 7th OPs and propose a new control structure to solve the control problem related to the 2nd OP.

5.4.2 Controller design for the 1st and 7th OPs using RPN The RPN and RPPN analysis suggests that no problems are expected in the controller design of a linear controller for the 1st and 7th OPs. Moreover, an inverse-based controller will work quite well for these OPs. To verify if these indications are true, here we design the controller for these OPs via the procedure presented in section 4.4.3 using the attainable closed-loop transfer function T. The "ideal" controller was approximated by the full-feedback PI controller 1  14.4 s + 896.2 −0.3951 s + 50.97 C5 =  . s 1870 s − 158378 2126 s + 17297 

(5.30)

1 CAin = 4.5 mol/l CAin = 5.1 mol/l CAin = 5.7 mol/l Ea(1,2) = +2%

0.95

Cb [ mol/l ]

0.9

0.85

0.8

0.75

0.7

0.65 0

10

20

30

40

50 60 Time [ min ]

70

80

90

100

Figure 5.18: Simulation of the full-feedback controller C5 for the 1st OP


104

Note that the controller C5 in contrast to C1 can be easily reduced to an one-way decoupling structure, because the element (1,2) is very small (in comparison to the other ones) and can be therefore neglected. Of course, we could also directly discard this element by the approximation procedure of section 4.4.2. Simulation results with the controller C5 are shown in Figure 5.18. For the 7th OP, the RPN and RPPN are small. We can therefore design an inverse-based controller for the system as already done for the 1st OP. Moreover, by an analysis of the inverse of the scaled plant model we can get an idea about the necessary controller structure and order. Note that this analysis only works when the inverse-based controller will produce good results, i.e., when the RPN is small. Figure 5.19 illustrates this analysis for the 7th OP. We can conclude that the element (1,2) of the controller is not necessary and the gain of channel 21 at the crossover frequency is not too large, indicating that a decentralized controller will achieve good performance. The analysis of the slope also suggests that a PI controller will be enough to produce a good approximation. Taking these facts into account, we designed the decentralized PI controller 0  1  −17.52 s − 1917 C6 =  . 0 1502 s + 19560 s

(5.31)

Figure 5.20 shows the simulation with controller C6 which confirms our suppositions. Channel 11

2

Log Magnitude

10

1

10

0

10

−1

10

−1

0

2

10

Channel 21

2

0

2

10

10 Channel 22

2

10

0

1

10

10

−2

0

10

10

−4

10

10

10

10 Log Magnitude

Channel 12

0

10

−1

0

2


10

0

2


Figure 5.19: Plots of the inverse of the scaled nominal model for the 7th OP

105


1

0.95

CAin = 4.5 mol/l CAin = 5.1 mol/l CAin = 5.7 mol/l Ea(1,2) = +2%

Cb [ mol/l ]

0.9

0.85

0.8

0.75

0.7

0.65 0

10

20

30

40

50 60 Time [ min ]

70

80

90

100

Figure 5.20: Simulation of the decentralized controller C6 for the 7th OP

5.4.3 A new control structure for the 2nd OP If the CSTR must be operated at the 2nd OP, the best practical solution is to try to measure and/or control the composition of another component (e.g., cA), since any other concentration depends monotonically on the manipulated variable f and therefore will not exhibit the nonminimum phase behavior. When the composition of the component cB is important, the control of any other component presents no problem, since a cascade control configuration can be used, where the setpoint for the internal loop is determined by the controller related to the cB composition control. 1

2

10

10

0

10

1

10

−1

10

−2

0

10 −1 10

10 0

10

1

10 Frequency

2

10 [ rad/h ]

3

10

4

10

Figure 5.21 : Condition numbers for the 2nd OP when cA is measured : γ*(GN) (solid line), γ#(P) (dashdot line), γ#*(P) (dotted line), γ (LSGNRS) (dashed line), and γ# (Ls#PRs#) (upper dashed)

−1

10

0

10

1

10 Frequency

2

10 [ rad/h ]

3

10

4

10

Figure 5.22 : RPN- and RPPN-plots for the 2nd OP when cA is measured : Γ(GN) (solid line), Γ#(P) ( dashdot line ), ΓLR(GN) (dashed line), ΓLR#(P) (dotted line)


106

Figures 5.21 and 5.22 show the condition numbers and RPN-plots when the concentration of the component A is used as the controlled variable instead of cB for the 2nd OP. Note that the original control problem becomes a very simple problem by the choice of a suitable controlled variable. Another interesting and efficient alternative is to control the transmission zero z instead of a composition. Remember that for the optimal reactor yield the transmission zero z, when only the composition cB is measured, is equal to zero. The transmission zero z can be analytically calculated from equation (5.21), which is rewritten below  c − cA z =  Ain −1 cB 

  E   E   k10 exp 1  − 2c A k 30 exp 3  − f .  T [K ]  T [K ] 

(5.21)

Observe that the transmission zero z is a function of cB, T, f, cA, and cAin. If the concentration cAin is measured, the new controlled variable can be seen as an implicit feedforward control action. Moreover, if the concentration cA is also measured, the control of z represents an efficient way to take this additional information into account. When only the variables cB, T, and f are known, we can assume the nominal values for the variables cAin and cA of Table 5.3. Doing so, (5.21) can be written as ∆ 3 −1 z A =   cB

  E   E   k10 exp 1  − 4k 30 exp 3  − f .  T [K ]  T [K ] 

(5.32)

With this new controlled variable zA, which is only a function of cB, T, and f, the control at the 2nd OP becomes an easy problem as shown in Figures 5.23 and 5.24. Another additional feature of the control of zA instead of cB is that zA is directly related to reactor yield which was the optimization criterion for the determination of the 2nd OP. 1

2

10

10

0

10

−1

1

10

10

−2

10

−3

0

10 −1 10

10 0

10

1

10 Frequency

2

10 [ rad/h ]

3

10

4

10

Figure 5.23 : Condition numbers for the 2nd OP when zA is controlled: γ*(GN) (solid line), γ#(P) (dashdot line), γ#*(P) (dotted line), γ (LSGNRS) (dashed line), and γ# (Ls#PRs#) (upper dashed line)

−1

10

0

10

1

10 Frequency

2

10 [ rad/h ]

3

10

4

10

Figure 5.24 : RPN- and RPPN- plots for the 2nd OP when zA is controlled: Γ(GN) (solid line), Γ#(P) ( dashdot line ), ΓLR(GN) (dashed line), ΓLR#(P) (dotted line)

107


Analogously, we can develop an expression based on equation (5.20) instead of (5.21). For example, for C11 = 1, C12 = 1, cAin-cA = 4, and cA = 1, we obtain the controlled variable zB E  E E  E   − 4  f + k10 exp  1  + k20 exp  2   − cB  f + k10 exp  1  + 2 k30 exp  3   ∆ T  T  T  T    zB = . 4 − cB

(5.33)

Figure 5.25 and 5.26 show the condition numbers and RPN-plots when zB is used as the controlled variable instead of cB for the 2nd OP. Considering these figures no control problem is expected. 1

2

10

10

0

10

1

10

−1

10

−2

0

10

10 −1 10

0

10

1

10 Frequency

2

10 [ rad/h ]

3

10

4

10

Figure 5.25 : Condition numbers for the 2nd OP when zB is controlled: γ*(GN) (solid line), γ#(P) (dashdot line), γ#* (P) (dotted line), γ (LSGNRS) (dashed line), and γ# (Ls#PRs#) (upper dashed line)

−1

10

0

1

10

10 Frequency

2

10 [ rad/h ]

3

10

4

10

Figure 5.26 : RPN- and RPPN-plots for the 2nd OP when zB is controlled: Γ(GN) (solid line), Γ#(P) ( dashdot line ), ΓLR(GN) (dashed line), ΓLR#(P) (dotted line)

Figure 5.27 shows the simulation of the new control structures with the controllers C7 and C8. The controller C7 was obtained by sequential design closing first the temperature channel and applying the procedure presented in section 4.4.3 to each sequential step, whereas the controller C8 was calculated by the direct application of the procedure described in section 4.4.3 to the attainable closed-loop transfer function T. The controllers C7 and C8 are given by for zA  −0.04958s 3 − 11.28s 2 − 2447s − 52630  0   2 s s + 37 . 45 s + 1057 ( )   C7 = (5.34)  1861s + 21220  0     s for zB  −0.01339s 3 − 19.56s 2 − 795.3s − 4614  s 3 + 40.43s 2 + 245.8s C8 =  3 2  −293.6 s − 34840 s − 203200 s + 229800  s 3 + 40.43s 2 + 245.8s

   5545s + 52820  . s 0

(5.35)


108

(a) zA and CB 1.2

4

1.1

zA

2

1

0 zA

CB [ mol/l ]

CB

0.9

−2 0

50

100

150

200

250

300

0.8

(b) CAin and Tin 110

CAin [ mol/l]

5.5

105 5 CAin

Tin [ C ]

Tin

100

4.5 0

50

100

150 200 Time [ min ]

250

300

(c) zB and CB −40 1.2

zB

1.1 −44

1 zB

−46 −48 0

50

100

CB [ mol/l ]

CB

−42

0.9

150

200

250

300

0.8

(d) z 10 zB controlled

z

0

−10 zA controlled −20 0

50

100

150 200 Time [ min ]

250

300

Figure 5.27 : Simulation of different control structures based on transmission zero control: (a) controlled variable zA given by (5.32) and concentration cB (dashdot line) using the controller C7 (equation (5.34)), (b) disturbances cAin and Tin, (c) controlled variable zB given by (5.33) and concentration cB using the controller C8 (equation (5.35)), (d) true transmission zero z calculated using (5.21). Note that z = 0 means that the reactor produces the maximal possible concentration cB for a given operating point defined by the actual values of T, f, cA, and cAin.

109


We concentrate our attention on Figure 5.27d, where the transmission zero z given by equation (5.21) is plotted. The analysis of this figure must be done with respect to the variation of z. The absolute value plays no role here, since it reflects the offset produced by a given choice of cAin-cA and cA to make the new controlled variables dependent on cB, T, and f only. The mean value can easily be improved by a different choice of the set point, e.g., for zA in Figure 5.27 the change of the set point to 1.5 instead of 0 will move the mean of z close to 0 (cf. Figure 5.29d). The control performance can also be improved by the choice of other values for cAin-cA and cA in the calculation of zA and zB. The problem here is that relatively exact values of the kinetic constants must be known. For the case where cA can be measured, another alternative could be to estimate (cAin-cA) by (5.9) (i.e., (cAin-cA) = (r1+r3)/f ) instead of using a constant value. However, this is not recommended when a value for cA must be assumed, since in this case (cAin-cA) varies less than cA. As in Figure 5.27d z is almost 0 under zB control until 260 min, until this time cB is also maximal for nominal reactor temperature. Thereafter, if the control objective is to achieve the maximal reactor yield with constant cB, the reactor temperature T must be modified. Remember that the nominal steady-state reactor temperature is only optimal for nominal values of cAin , Tin , and cB. Thus, a different reactor temperature must be determined, when cAin and/or Tin change. To set the new set point for the reactor temperature, we can apply a cascade controller that uses the measured composition cB to determine the set point of the reactor temperature loop. Analyzing Figure 5.27d, one could conclude that the control structure based on zA is more sensitive to variations of the feed composition cAin than the structure where zB is used. Moreover, the situation seems to be reversed for inlet temperatures below the nominal inlet temperature (i.e., for time > 260 min). Although these conclusions are correct, the dependence on the control structure is not too strong as suggested, since the performance also depends on the controller structure, tuning, and manipulating ranges. Figure 5.28 shows the control action produced by the controllers C7 and C8. Note that the channel 22 of the controller C8 is faster than C7 producing thus better performance in the temperature control (cf. Figures 5.28a and 5.28c). However, when the inlet temperature is below the nominal inlet temperature, the controller C8 when applied to control zB presents a bad performance. Figure 5.28d shows that the input saturation of QK is responsible for the inferior performance. Without input saturation the reactor temperature could be maintained and consequently, the performance problem would be solved automatically. If the inlet temperature Tin can be measured or when the input saturation of QK can be detected, a corrective action can easily be implemented and in this case the controlled variable zB should be preferred. Figure 5.29 shows that the same saturation problem also happens when the controller C8 is applied to control zA, but in this case the effect on z and cB is insignificant (i.e., z ≈ 0 and cB is practically maximal). As expected, the better temperature control improves the performance against inlet temperature disturbance, but the sensitive to variations of cAin is similar to the sensitive produced by the controller C7 (cf. Figure 5.29d). Concluding, the structure zB has very good performance when the temperature control is also good. On the other hand, the performance of the structure zA is practically independent on temperature control. Thus, when


110

less precise information about the system is available, the controlled variable zA and a decentralized controller should be preferred. (a) Controlled variables ( zA and T ) 120 4

T 115

T [C]

zA

2 110 0 zA

105

−2 0

50

100

150

200

250

300

100

(b) Manipulated variables ( f and Qk ) 25

−1

f [1/h]

Qk [ MJ/h ]

−1.1

Qk 20

−1.2 f

−1.3

15

−1.4 10 0

50

100

150 200 Time [ min ]

250

300

−1.5

(c) Controlled variables ( zB and T ) −40

116 115 T

114

−44

113 112

zB

−46

T [C]

zB

−42

111 −48 0

50

100

150

200

250

300

110

(d) Manipulated variables ( f and Qk ) 22

0 Qk

−1

18

−2

16

−3

f

14 12

Qk [ MJ/h ]

f [1/h]

20

−4 50

100

150 200 Time [ min ]

250

300

−5

Figure 5.28 : Control action corresponding to the simulations of Figure 5.27 (a) controlled variables zA (solid line) and T (dashdot line) with set points, (b) manipulated variables f (solid line) and QK (dashdot line) for (a) (c) controlled variables zB (solid line) and T (dashdot line) with set points, and (d) manipulated variables f (solid line) and QK (dashdot line) for (c). The ranges for the manipulated variables are 3 ≤ f [h-1] ≤ 35 and -8.5 ≤ QK [MJ/h] ≤ 0.

111


(a) Controlled variables ( zA and T ) 120 6 T

zA

2

T [C]

115

4

110

0

105

zA

−2 0

50

100

150

200

250

300

100

(b) Manipulated variables ( f and Qk ) 25

0 Qk

20

f [1/h]

−2 −3

15

−4

f 10

Qk [ MJ/h ]

−1

50

100

150 200 Time [ min ]

250

300

−5

(c) CB and CAin 5.6 5.4

1.1

5.2 CB

1

5

CAin

4.8

0.9

CAin [ mol/l ]

CB [ mol/l ]

1.2

4.6 0.8 0

50

100

150

200

250

300

4.4

(d) z 10 zA and C8

z

0

−10

−20 0

zA and C7

50

100

150 200 Time [ min ]

250

300

Figure 5.29 : Simulation of the controlled variable zA with the controller C8 for the disturbances of Figure 5.27b. The set point for zA. is 1.5. (a) controlled variables zA (solid line) and T (dashdot line) with set points, (b) manipulated variables f (solid line) and Q (dashdot line) (c) concentration concentration cB and disturbance cAin (d) comparison of the true transmission zero z calculated using (5.21) for zA with the controller C8 and for zA with the controller C7 ( from Figure 5.27d). Note that the change of the set point of zA to 1.5 instead of 0 moves the mean of z close to 0.


112

Appendix A5

Table A5.1: Chemical kinetic parameters for the Arrehnius equation [EnKl93] Reaction

Unit Activation Reaction Enthalpy of ki0 Energie Ei [K] ∆Hi [kJ/mol]

Collision factor ki0

k1 A  → B ∴ r1= k1(T ) cA

(1.287±0.04)×1012

h-1

-9758.3

4.2 ± 2.36

k2 B  → C ∴ r2 = k2 (T ) cB

(1.287±0.04)×1012

h-1

-9758.3

-(11.0 ± 1.92)

(9.043±0.27)×109

liter mol h

-8560

-(41.85 ± 1.41)

k3 2 A  → D ∴ r3= k3 (T ) c A

2

For the 3 PDOF problem, where the reactor volume can be manipulated, we need the dependence of the coolant surface AR on the reactor volume VR . For a cylindrical reactor, where only the base and the side area are cooled, we can write the relation between AR and VR as AR = (πDR2/4) + (4 VR/DR ) .

Table A5.2: Physico-chemical parameters and reactor dimensions [EnKl93] Parameter Name

Symbol

Value

Unit

density of mixture

ρ

0.9342 ± 0.0004

heat capacity of mixture

Cp

3.01 ± 0.04

heat transfer coefficient for cooling jacket

kW

4032 ± 120

kg l    kJ  kg K    kJ  m2 h K  

surface of cooling jacket

AR

0.215

[m2]

nominal reator volume

VR

10

[l]

reactor diameter

DR

0.2312 or 0.3678§

[m]

coolant mass

mK

5.0

[kg]

heat capacity of coolant

CpK

2.0 ± 0.05

 kJ  kg K  

§

The reactor diameter was not given in [EnKl93]. The values given here are the positive solutions of the equation AR = (πDR2)/4 + (4 VR )/DR for the nominal values AR and VR given in this table.

113

APPENDIX A5

Table A5.3: Linearized model of the CSTR with the Van de Vusse reaction scheme  x! = A x + Bu u + Bd d + Br r + Bh h   y = C x + Du where y T = ∆ c B ,∆ T ,∆ VR

[

[ = [ ∆F

Definition of the auxiliary variable f and constants :

]

x T = ∆c A ,∆ c B ,∆T ,∆TK ,∆ VR uT

in

,∆ QK ,∆Fout

[

d = ∆c Ain ,∆Tin T

   T r =   

[

]

]

]

Differentials of the reaction rates:

 ∆k10 ∆k20 ∆k30  , ,   or k20 k30   k10  ∆E1 ∆E2 ∆E3  , ,    T [K] T [K] T [K] 

h T = ∆H1 ,∆H 2 ,∆H3 ,∆kW

π DR Fin 1 1 4 , C1 = , C2 = ,C3 = ,C4 = ρ Cp VR m K C pK DR 4 2

f =

]

r1′=

 E  dr1 = k10 exp 1  , dc A  T [K] 

r1′′= −

r2′ =

 E  dr2 = k20 exp 2  , dc B  T [K] 

r2′′= −

r3′=

 E  E3 r3 dr3 dr = 2 c A k30 exp 3  , r3′′= − 3 = dc A dT (T [K]) 2  T [K] 

E1 r1 dr1 = dT ( T [K]) 2 E2 r2 dr2 = dT ( T [K]) 2

A=  − f − r1′− r3′ 0   − f − r2′ r1′    − C ( r ′ ∆ H + r ′ ∆H ) − C r ′ ∆H 1 3 3 1 2 2  1 1   0 0   0 0  ( c Ain − c A ) VR   −c B VR Bu =  ( Tin − T ) VR  0   1  f 0  Bd =  0  0  0

0 0  f,  0 0 

0 0 0 C2 0

0  0 0,  0 −1

r1′′+ r3′′

0

− r1′′+ r2′′

0

(

− f ( c Ain − c A ) VR

 C1 ( r1′′∆H1 + r2′′∆H2 + r3′′∆H3 ) C3 + C4 VR   C k 1 W − f − C k C + C V V  VR 1 W 3 4 R R  

(

)

)

C2 kW ( C3 + C4 VR )

− C2 kW ( C3 + C4 VR )

0

0

− r3 0  − r1    − r2 0 r1   Br = −C1 r1 ∆H1 − C1 r2 ∆H2 −C1 r3 ∆H3  ,   0 0 0     0 0 0

   f c B VR     − + F T T ( ) −1  in in  VR 2  C1 C3 kW ( TK − T )   C2 kW C4 ( T − TK )    0 

C11  C= 0  0

0 0 0  0   0  0 0 0   + C C V 4 R) −C r −C r − C r C ( 3  − T T ( ) Bh =  1 1 1 2 1 3 1 K , VR   C2 ( C3 + C4 VR )( T − TK )  0 0  0  0  0 0 0

C12 0 0

0 0 0  1 0 0 , 0 0 1 

0 0 0    D = 0 0 0  0 0 0 

Remarks: 1. All variables in the matrices A, Bu, Bd , Br , Bh, and C correspond to a steady-state solution of the nonlinear model for the physico-chemical parameters of Tables a5.1and a5.2. 2. The upper left corner in all matrices represents the linearized model corresponding to the 2 PDOF problem. In this case, the system reduces to four equations, i.e., ∆VR = 0. 3. The elements C11 and C12 of the matrix C depend on the controlled variable. For example, if only cB is measured and controlled, we will have C11 = 0 and C12 = 1; if cA is the output, then C11 = 1 and C12 = 0; for the case where cA/cB is measured, C11 and C12 are given by 1/cB and -cA/cB2, respectively; and so on. ∆ki0 ∆Ei 4. The set of variables   ki0  and  T  has the same matrix Br, so they can be analyzed together. 5. Gr = (A, Br, C, D) and Gh = (A, Bh, C, D) are used in Table 5.2.


Table A5.4: Uncertainty transfer functions used in the µ-controller synthesis l (s )= g11

0.004805 (0.001001s + 1)(0.009398 s + 1)(0.3586 s + 1) (0.001001s2 + 0.03855s +1)(0.158s +1)

l (s )= g12

0.00001877 (0.0003883s +1)(0.00361s +1)(0.04372s +1) (0.004281s2 + 0.0547s +1)(0.1484s +1)

l (s )= g21 l (s )= g22

0.1887 (1.06e - 006s2 + 0.001471s +1)(0.8382s +1) (0.01169s +1)(0.1176s +1)(0.8222s +1) 0.0007439 (1.888e - 006s2 + 0.001023s +1)(0.2156s +1) (0.008591s2 + 0.1264s +1)(0.2895s +1)

Table A5.5: Nominal linearized model used in the µ-controller synthesis  x! = A x + Bu  y = C x + Du , where  kJ u T0 = [ f Q! ] = [ 16.2943 h −1 - 3555 ], K h Tin = 130 ° C and c Ain = 51 .

mol l

mol mol T ] = [1.078 0.825 135.0° C 130.9° C ] K l l 0 - 3.7806 0 -84.7158  53.2354 - 69.5297  0.7875 0 , A= 146.5003 208.2508 - 35.8206 30.8285    0 86.6880 - 86.6880   0  4.0225 0  0 1 0 0  -0.8250 0  C =   0 0 1 0  , B= -5.0000 0    D=0 0 0.1  

x T0 = [ c

A

c

B

T

114

Chapter 6 Control of a Pilot Plant Distillation Column In this chapter, the idea of hierarchical control is applied to a multipurpose packed distillation column. Special attention is given to the heating system, which is an experimental plant for a process with recycle streams. Another point also analyzed in this chapter is the additional dynamic characteristics presented by packed distillation columns that are not found in the corresponding tray columns. Here, experimental results of the pilot plant are interpreted by a simplified dynamic model and special attention is given to the properties important for feedback control. This chapter also shows how the process dynamics can be improved by applying a feedback controller. A new controlled variable is proposed to take the nonequilibrium temperature dynamics into account.

6.1 Process Description 6.1.1 Purpose of the experimental pilot plant distillation column Trays and packings are the most common internals used in distillation columns. Recently, packed distillation columns are increasingly used in the industry, therefore a better understanding of the dynamic and steady-state behavior and reliable control of packed columns is important. In packed columns, due to their smaller holdup, the interaction in the column in the middle and high frequency range is stronger than in columns with trays where the tray holdups cause decoupling of the fast transients. At the University of Dortmund, a Multi-Purpose Packed Distillation Column (MPPDC) was built in cooperation by the Thermal Separation Processes Group (TV) and the Process Control Group (AST) to analyze the static and dynamic behavior and the control of packed distillation columns.

6.1.2 Description of the pilot plant The pilot plant (see Figure 6.1) has a diameter of 100 mm. Different packings can be used. The column is divided into three sections of 1 m active height each. It is made of glass and has an electrical heating jacket to reduce the heat losses. The preheated feed can be introduced into the column at two points between the active sections depending on the setting of the

6. CONTROL OF A PILOT PLANT DISTILLATION COLUMN

116

manual valves V2 and V4. At the same positions, a vapor side stream can be withdrawn via the manual valves V3 or V5. The vertical thermosiphon reboiler is heated by warm water under a pressure of up to 10 bar. The water temperature and the volumetric flow rate are manipulated by adjustable electrical heating elements and a control valve (RV3), respectively. The column is equipped with an oil-cooled total condenser. The cooling oil temperature and the volumetric flow rate can be adjusted with a thermostat and a control valve (RV2). The reflux ratio is manipulated by a reflux divider, which is operated electromagnetically. For vacuum-operation the column can be evacuated by means of a rotary vane pump which is controlled by a bypass control valve (RV4). The temperature is measured along the column by means of eleven resistance thermometers Pt 100. Each packing section has three thermometers placed equidistantly. In addition, probes can be taken at the same heights, which can be analyzed off-line with a gas chromatograph. Pressure transducers provide the measurement of absolute pressure and pressure drop in the column. The distillate and side stream flow rates are measured by a coriolis-mass-flow-meter and a differential pressure flow-meter (V-Konus), respectively. The feed and bottom stream flows are determined by the frequency setting of the corresponding diaphragm pump. The unit is equipped with a process control system from HiTec Zang, Herzogenrath, Germany. The column can be operated under a wide range of conditions including vacuum conditions and with different mixtures of very different boiling temperatures. All instruments are certified for operation in explosion areas (EExi). Table 6.1 summarizes all instruments of the MPPDC that are connected to the process control system. Table 6.1: List of input and output signals of MPPDC ! 19 14 1 1 1 1 1 1 1 1

Analogue Inputs Pt 100 (EExi) Pt 100 coriolis-mass-flowmeter density (coriolis) PDI PI PDI (V-Konus) PI (V-Konus) electromagnetic flowmeter float chamber to measure liquid level 1 wheels flowmeter 42

! 4 1 1 2 8

Analogue Outputs control valves (RV) offset for RV3 electrical heating (el.3) diaphragm pumps

! Digital Outputs 5 electrical heating jackets 1 reflux divider (RD) 6

! EExi- barriers 12 zenerbarriers ( ⇔ 20 Pt 100 EExi) 5 isolating amplifiers 17 ! Additional 1 three-point controller (for RV3) 3 flowmeters (FI 171, FI 172, and FI 307) 2 PI (PI 303 and 400) 2 TI (TI 306 and 132)

Figures 6.1, 6.2, 6.3, and 6.4 show the P&I diagram of the experimental packed distillation column (MPPDC); a photo of the condenser and main column part; a photo of the lower part of the column with the vertical thermosiphon reboiler; and the heating system and process control system, respectively.

117

6.1 PROCESS DESCRIPTION

Figure 6.1: P&I diagram of the experimental packed distillation column (MPPDC)


Figure 6.2: Condenser and main column part

118

119

6.1 PROCESS DESCRIPTION

Figure 6.3: Lower part of the column with the vertical thermosiphon reboiler


Figure 6.4: Heating system and process control system

120

121

6.2 HEATING SYSTEM

6.2 Heating System The heat duty Q is an important manipulated variable for composition control. In our pilot plant, Q belongs to the lower layer of the control hierarchy. The setpoints for Q are given by the composition control positioned in the second layer of the control hierarchy or by the operator in the manual mode. Therefore, the performance of the heat duty control is crucial for the functionality of the column. The control goals to be achieved are fast closed loop response to setpoint change and keeping the heat duty to the column as constant as possible. The first goal make decoupling between the composition and the heat duty control possible. The second objective reduces the temperature variations in the column produced by the nonequilibrium temperature dynamic as we will see later.

Figure 6.5 : Heating system of the MPPDC The heating system (HS) is per se interesting, since it can be seen as an experimental plant for a process with recycle streams. Figure 6.5 shows the heating system in detail. The centrifugal pump P3 causes warm water to circulate in closed loop. To avoid that the water evaporates the pressure in the heating system can be adjusted manually in the range of 1-10 bar, being 5 bar the usual pressure. The tank in Figure 6.5 has double functionality. It prevents that the pump P3 runs dry and it increases the energy holdup in the HS what reduces the coupling between the heat inflow and outflow to the system. The energy is introduced into the HS through adjustable electrical heating elements (EL3) in the electrical heating tank (EHT). The water


122

leaves the EHT following two parallel paths: one is via the tube side of the vertical thermosiphon reboiler giving part of its heat to the column and the other one goes via the manual bypass valve V6. Both streams come back to the tank through the plate heat exchanger which is not utilized in normal operation1. The equipment dimension and properties are summarized in Table 6.2. Table 6.2: Equipment dimensions and characteristics Tank Volume

170

l

Maximal capacity Total Head ∆pmax

7 54 5.25

m3/h m bar

60 78

kW l

Centrifugal pump (P3)

Electrical heating tank (EHT) Maximal Power Tank Volume Vertical thermosiphon reboiler (Reb.) 1.4 Total tube surface Total volume of warm water in the 9 vertical thermosiphon reboiler Reynolds number in the tube ≈ 9000

m2 l

Based on the measured temperatures TI_301, TI_302, TI_114, and TI_162, and the volumetric flow FI_305 and using the manipulated variables RV3 (the position of the control valve RV3) and EL3 (the adjustable electrical heating elements), we will propose a control strategy that achieves the mentioned control goals for all possible distilling mixtures. Moreover, we show how the startup of the vertical thermosiphon reboiler can be automated for all possible mixtures. In the next subsection we show how a quite simple model can be used to understand the control problem.

6.2.1 Nonlinear grey-box model Although the model of each equipment item is simple and has no special dynamic characteristics, the total system exhibits complex dynamics. ODE-System based on energy balances As the system is in closed-loop, i.e., there is no change of mass in the HS, the system dynamics are described by the energy balances in each part of the system. • The energy balance in the Tank: Assuming that the tank is adiabatic and has constant volume VT and homogeneous temperature TT, the energy balance is given by: dTT dT F dT = Fρ w cw (TTin − TT ) ⇔ T = (TTin − TT ) ⇔ T = K1 F(TTin − TT ) (6.1a) dt dt VT dt where ρw and cw are the water density and heat capacity, respectively.

ρ w cw VT

1

The plate heat exchanger is only used to speed up the shutdown of the pilot plant.

123

6.2 HEATING SYSTEM

• The energy balance in the electrical heating tank (EHT): This equipment is modeled by two differential equations, one related to the heating tank and the other to the heating elements. The assumptions of adiabatic tank, constant tank volume VH, homogeneous tank temperature TH , and uniform electrical element temperature TE yields: dTH F U A dT = (TT − TH ) + H H (TE − TH ) ⇔ H = K2 F(TT − TH ) + K3 (TE − TH ) ρ w cw VH dt VH dt

(6.1b)

dTE U A EL3 dT = − H H (TE − TH ) + ⇔ E = − K 4 (TE − TH ) + K5 EL3 dt M E cE M E cE dt

(6.1c)

where ME and cE are the mass and the heat capacity of the heating elements, respectively. • The energy balance in the vertical thermosiphon reboiler (Reb.): For the thermosiphon reboiler we have dTR F2 U A dT = (TH − TR ) − R R (TR − TB ) ⇔ R = K6 F2 (TH − TR ) − K 7 (TR − TB ) ρ w cw VR dt VR dt

(6.1d)

where TB, TR, and VR are the boiling temperature of the distilling mixture, the homogeneous water reboiler temperature, and the volume of the reboiler, respectively. Due to the small volume of the vertical thermosiphon reboiler in comparision to the other equipment items (see Table 6.2), it is not necessary to model the thermosiphon reboiler as a distributed system (i.e., with partial differential equations). To complete the model the inlet tank temperature TTin in (6.1a) is given, assuming ρw and cw as constant, by F T + F2 TR TTin = 1 H , F = F1 + F2 . (6.1e) F For the system of equations (6.1a-e), the constants K1, K2, and K6 are directly determined by the inverse of the corresponding volumes in Table 6.2. For the other constants some additional considerations must be made. Heat transfer in the vertical thermosiphon -reboiler The modeling of thermosyphon reboilers is complicated by the fact that, unlike a forced convection reboiler, the fluid circulation rate cannot be determined explicitly. The circulation rate, heat-transfer rate, and pressure drop are all interrelated, and iterative calculation procedures must be used to determine the heat flux. The typical procedures found in the literature for the calculation of heat flux need usually many physical and thermodynamical properties of the mixture (e.g., vaporization entalphy, viscosity, density, etc.) (see, e.g., [Sin93, page 665] or [Wal88, Table 8.10]). Moreover, all correlations presented in the literature are appropriate for the case where the boiling mixture is in the tube side of the thermosyphon reboiler. Our column presents the reverse situation, since the column is made of glass which could not support the 5 bar pressure used in the HS. Figure 6.6, taken from [Sin93, page 667], shows that the iterative solution and the high number of physical properties can be quite well represented as a function of the reduced temperature.


124

Thus, based on Figure 6.6 an usuful expression for K7 will have the following form: K7 =

K07 , a = 4.3 ,b = 3.6 ( a − b ⋅ Tr )

(6.2)

where Tr is the reduced temperature of the organic mixtures or equal to 1 for aqueous solutions. For the determination of the parameters a and b, we have used two different distilling mixtures: the aqueous mixture Methanol/Acetonitril/Water (MAW) and the organic mixture Chlorobenzene/Ethylbenzene (CE) (Tr = 0.64). The representation of the heat flux in this simple way is the basis of the control strategy that will be presented later. Figure 6.6 : An example of heat flux as a Note that the reduced temperature Tr takes the function of the temperature difference for a effect of changing mixtures and/or operating conditions (e.g., vaccum conditions) into vertical thermosyphon reboiler account automatically. (from [Sin93, page 667] ) Total flow rate and the influence of bypass In the system of equations (6.1a-e) the main system nonlinearity comes from the volumetric flows F, F1, and F2 of Figure 6.5. Only F2 of the three flows is measured by FI_305. The other two can be estimated using the pump characteristic curve. For the pump P3 only two parameters are known: the maximal volumetric flow (Fmax = 7 m3/h = 1.944 l/s) and the maximal pressure difference (∆pmax = 5.25 bar). With these known values we have assumed the following pump characteristic curve:  F  ∆p  = 1−  ∆pmax  Fmax 

  ∆p  2    ⇔ F = Fmax 1 −    ∆pmax    

(6.3)

As the valves V6 and RV3 are in parallel, ∆p produced by both valves is the same (see Figure 6.7). Therefore, if we are able to determine ∆p we can also calculate the flow F by (6.3). Now, if we consider that ∆p through the control valve RV3 can be written as (see, e.g., [Wal88]) Figure 6.7 : Schematic ∆p drop in the HS

∆p = ( K a + Kb F2 ) F2

(6.4)

where the constants Ka and Kb are a function of the valve position RV3 and the medium properties (such as viscosity and density). Note that the medium properties are a function of

125

6.2 HEATING SYSTEM

the temperature which is of course not constant in the HS. Nevertheless, its effect can be neglected when compared with the effect of the valve position RV3, which can be represented in the following form: K a = K a1 +

Ka2 K + a3 2 RV3 RV3

, K b = K b1 +

Kb2 K + b3 2 . RV3 RV3

(6.5)

Note that when the bypass valve V6 is closed, F = F2 and ∆p calculated by (6.3) and (6.4) must be the same. This situation is shown in Figure 6.8 where the points 'a' and 'b' illustrate the intersection of both equations for different valve positions RV3. This idea can be applied to determine the parameters Kai and Kbi in (6.5). For it, we need only to make an experiment with closed bypass valve V6. This experiment gives several points like 'a' and 'b' of Figure 6.8. Through a linear regression the following values for Kai and Kbi were obtained: K a1 = −0.60121 , K a 2 = 26.244 , K a3 = 609.17 , Kb1 = 0 , K b 2 = 0.17025 ,and Kb 3 = 0.

(6.6)

The above parameters give an akzeptable approximation for values of F2 > 0.6 l/s only.

Figure 6.8 : The pump characteristic curve given by (6.3) (solid line) and two pressure drop curves (dashed lines) calculated by (6.4) for two different valve positions ( for the curve corresponding to the point 'a', RV3 is more open than for the point 'b') When the bypass valve V6 is open, we need only to determine ∆p by (6.4) which is a function of the valve position RV3 ( equation (6.5) ) and the measured volumetric flow F2. The total volumetric flow F is calculated by the pumpe characteristic curve, i.e., by (6.3). The dynamics of the control valve RV3 As one of the manipulated variables of the HS is the valve position RV3, it is important to know its dynamic characteristics and its relation to the volumetric flow F2 (which is the corresponding physical variable that appears in our equation system (6.1a-e) ). Due to the


126

bijective relation between RV3 and F2, for each valve position RV3 there is only one F2. Moreover, the original quadratic characteristic curve of the valve RV3 was linearized so that at a steady-state point ∆F2/∆RV3 is almost constant for all ∆RV3 variations. The valve RV3 is a motor valve with constant rotation speed. The constant rotation speed is responsible for the special dynamic characteristic of RV3, since the necessary time to make a variation on the volumetric flow depends on the total variation size. This special dynamic is captured by the following irrational transfer function: ∆F2 (s ) 1 − e −τs  ∆F2  60 s = , τ= ⋅ ∆RV3    ∆RV3(s)  τs  ∆RV3  steady − state 100%

(6.7)

where τ is the time necessary to achieve the end position for a given ∆RV3. Note that the valve RV3 needs 60s to go from total closed to total open position (100%). It means that for the choice of the operations conditions this fact must be taken into account, since here small variations of RV3 are favorable for the controller performance. Later, we discuss this point in detail. Final model structure The final model equation is given by (6.8a-d) dTT F F   K = K1 ⋅ F ⋅ 1 − 2  TH + 2 TR − TT  − 08.5 (TT − T∞ )   dt F F   F

(6.8a)

K19 K16 K dTH K14  K15 0 .67   F  = K2 ⋅ F(TT − TH ) + K3 (TE − TH )  2 + 3 + K20 F    − 09.5 (TH − T∞ ) (6.8b)     1.9 dt F F F

K K dTE F K19 K = − K 4 (TE − TH ) 14  152 + 163 + K20 ⋅ F 0 .67    + K5 ⋅ EL3 F   1.9  dt F

(6.8c)

K /(4.3 − K17 ⋅ Tr ) ⋅(TR − TB ) dTR = K6 F2 (TH − TR ) − 07 . 1 1 dt + K11 K12 ⋅ F20.8 −100 K13

(6.8d)

([

] )

This model is based on (6.1a-e) with some additional terms which are necessary to increase the accuracy of the model. For example, the experimental data showed that the adiabatic tank and heating tank assumptions were wrong. Moreover, the pipe line also contributes to the heat loss2. The last term of the equations (6.8a) and (6.8b) takes the heat loss in the HS into account. An expression for UH in (6.1b) is given in the final model by the second term of the equation (6.8b). Finally, the second term of (6.8d) includes the effect of the variation in the volumetric flow in the heat flux based on standard expressions for determination of the heat 2

Almost 40% of the inserted heat in the heating elements are lost to the environment.

127

6.2 HEATING SYSTEM

transfer resistance found in the literature [Wal88]. A detailed derivation of this expression is presented in [Sc95]. Two sets of data will be used to validate the quality of the model. The first corresponds to open loop experiments for the distilling mixture CE, the second set was obtained in closed loop for the distilling mixture MAW. Note that the following experimental data was obtained for the case where the bypass valve V6 set such that the maximal F2 flow was 1.3 l/s. In comparison, for the case where V6 is closed, the maximal achievable flow F2 is 1.88 l/s (of course, this value is temperature dependent, but a maximal variation of 0.02 l/s was observed for 100°C temperature variation). Parameter optimization with open-loop data Here the results obtained for input variations of RV3 are shown only. In [Ma95] several additional experiments are presented indicating that the nonlinear model (6.8a-d) represents variations of EL3 very well. Experimental results show that reproducing the experimental data obtained by variations on RV3 is a lot more difficult. The optimized model parameters are summarized in Table 6.3. The parameters K3 , K4 , K15 , and K16 in (6.8a-d) were optimized. The other parameters were determined from first principles and standard correlations (see [Sc95] for details). The result of simulation with these parameters is presented in Figure 6.9. Observe that the simulation and experimental results agree very well.

(a) TI_301 and TI_302 142

T [°C]

141.5 141

TI_301

140.5 TI_302

140 139.5 0

500

1000

1500

2000

2500

3000

3500

4000

4500

(b) FI_305 and EL3 1.4

13.5 FI_305 13

EL3

1

EL3 [ kW ]

FI_305 [ l/s ]

1.2

12.5

0.8

12

0.6

11.5

0.4 0

500

1000

1500

2000 2500 Time [ s ]

3000

3500

4000

4500

11

Figure 6.9 : Comparison of the experimental data for CE mixture in open loop (dotted line) and simulation with nonlinear model (solid line): (a) temperature measurements (b) manipulated variables


128

Figure 6.10a shows the simulation for the same experimental data as in Figure 6.9 considering only the vertical thermosyphon reboiler. The experimental results are exceptionally well reproduced indicating that improvements of the model can only be achieved with a better model of the other parts of the system, what could only be done with additional measurements, such as a flowmeter for F or F1 and a sensor for the temperature TTin. Note that our goals are related to the control of the heat duty meaning that the model quality is good enough for our control purposes. Of course, if we want to develop a correlation for the heat flux in the heating tank, model improvements would be necessary. With the model obtained here for CE, we designed a controller which was applied to control the MAW distilling mixture. The corresponding experimental results are presented in the next point. (a) CE mixture

(b) MAW mixture

140.6

90

140.5

89

140.4 140.3

TI 302 [ ° C ]

TI 302 [ ° C ]

88 140.2 140.1 140

87

86

139.9 139.8

85 139.7 139.6 0

500

1000

1500

2000 2500 Time [ s ]

3000

3500

4000

4500

84 0

1000

2000

3000

4000

5000 6000 Time [ s ]

7000

8000

9000

10000

Figure 6.10 : Simulation of the vertical thermosyphon reboiler (equation 6.8d) without the other components (resp. equations): (a) CE mixture (experimental data of Figure 6.9) and (b) MAW mixture ( experimental data of Figure 6.11) Table 6.3 : Parameters for the set of equations (6.8a-d) CE

MAW

CE

MAW

K1

1.0684e-2

1.0684e-2

K12

856.07

856.07

K2

2.4853e-2

2.4853e-2

K13

2.0667

2.0667

K3

4.4689e-2

3.7684e-2

K14

1

0.2400

K4

4.3938e-2

4.3315e-2

K15

0.1070

0

K5

3.1514e-3

3.1514e-3

K16

1.0180

0

K6

1.1542e-1

1.1542e-1

K17

3.6127

3.6127

K07

4.7277e-4

4.7277e-4

K18

0.7500

0.7500

K8

1.0646e-4

1.0646e-4

K19

0.24038

1.1006

K9

8.1797e-5

7.4193e-5

K20

0

0.8700

K11

1.3727e+3

1.3727e+3

129

6.2 HEATING SYSTEM

Parameter optimization with closed-loop data The optimized parameters for the MAW mixture are represented in Table 6.3. Observe that with the MAW mixture, the HS works at a lower temperature than for the CE mixture. This fact explains the different values for the parameters related to the overall heat transfer coefficients for the heating elements, i.e., the second term of equation (6.8b) (or the first term of equation (6.8c)). Parameter optimization for MAW mixture was performed using closed loop experimental data in the interval 0-3500 s. The rest of the experimental data of Figure 6.11 were used to validate the model. Here, the parameters K3 , K4 , K14 , and K20 in (6.8b) were optimized. The other parameters were determined from first principles and standard correlations. (a) TI_301 and TI_302

T [°C]

92

TI_301

90 88 86 TI_302

84 82 0

1000

2000

3000

4000

5000

6000

7000

8000

9000

10000

(b) FI_305 and EL3 50 FI_305

1.2

40 30

1 20

EL3 0.8 0.6 0

EL3 [ kW ]

FI_305 [ l/s ]

1.4

10 1000

2000

3000

4000

5000 6000 Time [ s ]

7000

8000

9000

0 10000

Figure 6.11 : Comparision of the experimental data for the MAW mixture in closed loop (dotted line) and simulation with nonlinear model (solid line): (a) temperature measurements (b) manipulated variables. Only the experimental data until 3500 s was used for the parameter estimation. Figure 6.10b shows that the model of the vertical thermosyphon reboiler is accurate for the MAW mixture but the quality of the overall model is not as good, in particular, the model predicts fast reactions of the system while the real system shows very smoth responses.

6.2.2 Control structure design With the model presented in the last section, we can design the control structure for the HS. We start this section with the presentation of two possible control structures. After that, a polytopic model for each control structure is plotted in the frequency domain. The analysis of these results led to the conclusion that the system can be unstable due to the change of the sign


130

of the determinant at low frequencies. Finally, three possible solutions for this problem are presented.

Possible control structures To control the heat duty, we can use the temperatures TI_301 and TI_302 and the volumetric flow FI_305 as measured variables, and RV3 and EL3 as manipulated variables. One obvious candidate structure is the direct control of the desired variable, i.e., the control of the heat duty Q, which can be calculated as Q = ρcp FI _305(TI_301- TI_302) .

(6.9)

As the variable Q is formed by the combination of three measured variables (i.e., FI_305, TI_301, and TI_302), the sensor noise can be amplified considerably, especially if FI_305 and (TI_301-TI_302) have the same relative sensor noise magnitude. In this case, if we choose ∆Τ ( = TI_301-TI_302 ) as the controlled variable instead of Q, we can significantly reduce the measurement noise, since in this case the noise contribution of FI_305 and ∆T enter in different points of the control loop (see Figure 6.13). Another advantage of the ∆T structure is the controller order necessary to achieve a given performance, which is smaller than for the case where Q is controlled. As the HS has two process degrees of freedom (PDOF), we can choose one other variable to be controlled. Here, the natural chose is the control of TI_301, since this variable and the maximal volumetric flow give the maximal possible heat duty into the column. Moreover, the temperature difference between TI_301 and the column bottom temperature is very important for the good operation of the vertical thermosyphon reboiler. Thus, considering the last discussion, the following two alternative structures are investigated:

Structure name:

Controlled variable by RV3

Controlled variable by EL3

Q-structure

Q = ρcp FI _305(TI_301- TI_302)

TI_301

∆T- structure

∆Τ€=€TI_301-TI_302

TI_301

Understanding the ∆T - structure By the analysis of Figure 6.12, we can understand how the ∆T - structure can be used to control Q. In this figure, G represents a plant for the ∆T-structure, K some stabilizing controller, and u the manipulated flow rate FI_305. It can be shown that all stabilizing controllers for G in standard feedback loop will also stabilize the feedback loop of Figure 6.12. To simplify our discussion and representation, we consider that Q = u ∆Τ, i.e., ρ cp=1 or alternatively u = ρ cp FI_305.

131

6.2 HEATING SYSTEM

Figure 6.12 : Feedback loop for Q with the plant G in the ∆T-structure form The feedback structure in Figure 6.12 can be approximated by the simplified feedback structure represented in Figure 6.13. Note that this structure can be more easily implemented in the process control system. The main difference in the closed-loop performance is at high frequencies. In the low frequency range the performances of both structures are comparable.

Figure 6.13 : Simplified feedback loop to control the heat duty Q using the ∆T-structure Note that, in Figure 6.13, G represents the plant dynamics (in the ∆T-structure) and K is a stabilizing controller for G. It can be proved that the sign reversal is necessary (i.e., - K instead of K in Figure 6.13). Its origin is a direct consequence of the coupling from the actuated variable u to the reference signal ∆Tref . Linearized Model To analyze the system, we use a set of linearized models. As the main nonlinearity of the system (6.8a-d) is the volumetric flow F2 , it is logical to linearize the system for different values of F2. We chose 7 linearized models corresponding to the following values of F2 : G1=1.2, G2=1.1, G3 = 1.0, G4 = 0.9, G5 = 0.8, G6 = 0.7, and G7 = 0.6 l/s. The bypass valve V6 for these linearized models is set such that when the valve RV3 is completely open, the maximal F2 flow is 1.3 l/s. For the following discussion, Q1-Q7 and ∆T1-∆T7 represent the corresponding linearized models obtained from the 7 different flows for the structures Q and ∆T, respectively. For all transfer functions, channel 11 represents the transfer function from RV3 to Q or ∆T, channel 21 from RV3 to TI_301, channel 12 from EL3 to Q or ∆T, and channel 22 from EL3 to TI_301. In other words, the input vector is given by the vector [RV3, EL3]T and the output vector by [Q, TI_301]T or [∆T, TI_301]T.


132

Controllability analysis (RPN- and RPPN-plots) Figure 6.14 shows the RPN- and RPPN-plots for the Q- and the ∆T-structures. Based on the noise level and input constraints, we have chosen the following closed-loop transfer function:

{

}

1 1 . T = diag Td 1 [20 ,5% ],Td1[200 ,5% ] = diag  ,  2 78 . 82 s + 12 . 25 s + 1 7882 + 122 . 5 s + 1  

(6.10)

The RPN is calculated using the nominal model G4 and the RPPN using the polytopic model P = {G1,...,G7}. (b) ∆T-structure

(a) Q-structure 1

1

10

10

0

0

10

10

−1

−1

10

10

−2

−2

10

10

−3

−3

10

10

−4

10

−4

−6

10

−4

10

−2

10 Frequency

0

10 [ rad/s ]

2

10

4

10

10

−6

10

−4

10

−2

10 Frequency

0

10 [ rad/s ]

2

10

4

10

Figure 6.14: RPN-plot (solid line) and RPPN-plot (dashdot line) for Q- and ∆T-structures The RPN-plot for the Q- and the ∆Τ-structures (cf. Figure 6.14) show that the system can be easily controlled for the nominal model. Based on the RPPN-plot, we can say that the ∆Tstructure is more nonlinear than Q-structure indicating that a nonlinear controller for the ∆Tstructure could be of advantage for the control of ∆T, but when the ∆T-structure is used to control of Q, as in Figure 6.12, the final controller can be interpreted as a gain-scheduling controller. This fact makes the nonlinear degree of both structures equivalent.

System dynamics Figure 6.15 clearly shows that there is a large difference between the time response related to the manipulation of RV3 (or F2) and of EL3. This difference is not only a consequence of the relative small dimensions of the vertical thermosyphon reboiler in comparison to the other parts of the HS (see Table 6.2). Its origin is the positive feedback structure produced by the recycle of energy. Note that, if the HS would not have the recycle stream, the expected response in the system would be almost as fast as the channel 21 in Figure 6.15. To understand this very important effect, Figure 6.16 illustrates the positive feedback mechanism that governs the response to EL3.

133

6.2 HEATING SYSTEM (b) ∆T4

(a) Q4 5

10

5

Log Magnitude

Log Magnitude

10

0

10

−5

10

0

10

−5

10

−10

10

−10

10

−4

10

−3

10

−2

10

−1

0

10

10

1

10

Phase [degrees]

Phase [degrees]

−3

10

−2

10

−1

10

0

10

1

2

10

10

200

200

0

−200 −4 10

−4

10

2

10

−3

10

−2

10

−1

10 Frequency [ rad/s ]

0

10

1

10

2

10

0

−200 −4 10

−3

10

−2

10

−1

10 Frequency [ rad/s ]

0

10

1

2

10

10

Figure 6.15 : Bode-plot of Q4 (a) and ∆T4 (b) for the CE mixture: channel 11 (solid line), channel 21 (dashdot line), channel 12 (dotted line) and channel 22( dashed line)

Figure 6.16 : Schematic representation of the positive feedback between EL3 and TI_301 Each block in Figure 6.16 can be represented by a low order transfer function. To keep the discussion as simple and illustrative as possible, we assume that the effect of the bypass, the vertical thermosyphon reboiler, and the heat losses can be captured by a factor k, i.e., the transfer function G5 in Figure 6.16 is given by G5 (s) = k =

EL3 − Q − Qheat losses EL3

, 0 ≤ k ≤1.

(6.11)


134

In words, equation (6.11) says that the tank inlet temperature TTin is smaller than TI_301 by a factor k, i.e., TTin= k TI_301. The factor k depends on how much energy remains in the system (EL3-Q-Qheat losses) in comparison to the energy introduced(EL3). Now, based on theoretical model given by equations (6.1a-b), we can write G1and G2 as G1 (s) =

1 1 ,G2 (s) = . τ 1s + 1 τ 2s + 1

(6.12)

With these transfer functions the 'closed-loop' transfer function from EL3 to TI_301 is given by τ 1τ 2 s2 + (τ 1 + τ 2 )s + 1 TI_301(s) = G (s) . (6.13) τ 1τ 2 s2 + (τ 1 + τ 2 )s + 1 − k 3 EL3(s) For k close to 1, equation (6.13) represents a system with slow response and large steady-state gain. For k=1, the system has a pole at the origin. The high frequency response of (6.13) is independent on k indicating that a fast closed loop control will reduce the variations of the dynamics. This is confirmed by the frequency responses of channel 22 in Figures 6.17 and 6.18a. The curves in Figures 6.17 and 6.18 represent the linearized model for different volumetric flows or, equivalently, different k values. Channel 12 0

1.5

−0.1

1

−0.2

IMAG

IMAG

Channel 11 2

0.5 0 −0.5 0

−0.3 −0.4

2

4

−0.5 −0.5

6

0

REAL

0.5

1

REAL

Channel 21

Channel 22

1

0

−0.5

0

IMAG

IMAG

0.5

−0.5

−1

−1 −1.5 −2

−1

0 REAL

1

−1.5 −1

0

1 REAL

2

3

Figure 6.17 : Nyquist-plot of the Q-structure for the CE mixture: Q1 (points), Q2 (solid line), Q4 (dashdot line), Q6 (dashed line), and Q7 (dotted line). All curves rotate clockwise with increasing frequency.

135

6.2 HEATING SYSTEM

(a) CE mixture Channel 11

Channel 12

0.8

0

0.6

−0.05 IMAG

IMAG

0.4 0.2

−0.1 −0.15

0 −0.2 −1.5

−1

−0.5 REAL

−0.2 −0.2

0

0

0.2

0.4

REAL

Channel 21

Channel 22

1

0

−0.5

0

IMAG

IMAG

0.5

−0.5

−1

−1 −1.5 −2

−1

0

−1.5 −1

1

0

1 REAL

REAL

2

3

(b) MAW mixture Channel 11

Channel 12

1.5

0 −0.05 IMAG

IMAG

1 −0.1

0.5 −0.15 0 −4

−3

−2 REAL

−1

−0.2 −0.2

0

Channel 21

0.4

Channel 22 0

3

−0.1

2

−0.2

IMAG

IMAG

0.2 REAL

4

1 0 −1 −10

0

−0.3 −0.4

−5

0 REAL

5

−0.5 −0.5

0

0.5

1

REAL

Figure 6.18 : Nyquist-plot of the ∆T-structure for (a) CE and (b) MAW mixtures: ∆T1 (points), ∆T2 (solid line), ∆T4 (dashdot line), ∆T6 (dashed line), and ∆T7 (dotted line). All curves rotate clockwise with increasing frequency.


136

In Figures 6.17 and 6.18a, the transfer function from RV3 to TI_301 (channel 21) at low frequencies changes its sign twice when the flow increases from 0.6 l/s to 1.2 l/s. The steadystate gain is positive for G7( 0.6 l/s ); almost zero for G6 ( 0.7 l/s); negative for G2, G3, G4, and G5; and again positive for G1 (1.2 l/s). To understand this effect, we need to consider two competing effects that act in opposite directions. The first effect is related to the tank inlet temperature TTin. The second is related to the effect of the factor k in (6.13). The influence of the manipulation of RV3 can schematically be interpreted as follows: ↓ ∆T ,↑ TI_302 ⇒ ↑ TTin ⇒ ∆TI_301 > 0 (effect A ) ∆ RV3 > 0 ⇒ ↑ F2 ,↓ F1 ,↑ F ⇒  (6.14) ↑Q ,↓ k ⇒ ∆TI_301< 0 (effect B) .  The effect A has a positive gain and the effect B has a negative gain. Considering this fact, we can say that for G7 effect A > effect B; for G6 A ≈ B; for G2, G3, G4, and G5 A < B; and G1 A > B. Now, we turn to the analysis of the two parts of Figure 6.18. In part 'a', the corresponding grey-box nonlinear model was obtained by open loop identification with input signals concentrated at middle and low frequencies. The part 'b', on the other hand, was obtained by closed-loop identification, i.e., with the input signals concentrated at high and middle frequencies. The comparison of parts 'a' and 'b' shows that at high and middle frequencies both models agree to some extent, but at low frequencies there is a large difference, specially for the channel 21. The disagreement at low frequencies between the two models in Figure 6.18 can have two interpretations: (i) the MAW mixture modifies the response of the thermosyphon reboiler so that the effect A in (6.14) is always smaller than the effect B or (ii) the order of the mathematical model is too low to permit the correct representation of the system dynamics at high and at low frequencies. In the next subsection, based on experimental data, we show that the second possibility is the correct interpretation. Instability with integral control action A stable nominal n × n system GN subject to real, stable, independent, channel perturbations δi will be robustly stable under integral control if and only if the condition µ ∆ ,real (W1 GN−1W2 ) < 1

for ω = 0 ,∆ real = diag{ δ 1 ,! ,δ n × n } , and

 g11x − g11N    " 0   W2 = ( In × n )1 ,# ,( In × n )n  x  N gn1 − gn1 , W1 =  $ x N  ( In × n is the identity matrix g1n − g1n   of dimension n × n)   0 "   x N gnn − gnn  

[

is satisfied.

]

(6.15)

137

6.2 HEATING SYSTEM

This result can easily be obtained using the results of Chapter 3 considering an additive uncertainty at each channel and the fact that the complementary sensitivity function T(0) is equal to the identity matrix for controllers with integral action ( i.e., T(0) = I ). In (6.15) gijN and gijx are the elements ij of GN and of Gx ∈ P={G1,...,G7}, respectively. Table 6.4 presents the evaluation of equation (6.15) for Q models obtained for the CE mixture. Based on these results, we can conclude that if we use an inverse-based controller that uses G1, G2, G3, and G4 as the nominal model the closed loop system will not be stable for all flows F2 (or models Gx). Tables 6.4 also shows that Q5 produces the largest stability margin. Moreover, the closed loop system stays stable at low frequencies for an inverse-based controller developed with the models G5, G6, and G7. Observe that these results are strongly related to the change of the sign at low frequencies that occurs in channel 21 (cf. Figures 6.17 and 6.18a), but this sign change is not enough to explain the low frequency instability, since Q1 has a good stability margin for the nominal model Q4, whereas Q7 has not, and for both the channel 21 has another sign than Q4 at low frequencies. Therefore, for the stability analysis is necessary to consider the effect in all channels as done here. Nevertheless the sign change in channel 21 is a good indicator of the instability and gives us insights about its physical origin, which is related to the bypass flow F1. Similar conclusions hold for the ∆T-structure (not shown here, see [Sc95, section 4.2]). Table 6.4 : Steady-state RS analysis using equation (6.15) for the Q-structure Model

Nominal Model

" Gx

Q1

Q2

Q3

Q4

Q5

Q6

Q7

Q1

0

0.6040

0.6698

0.4558

0.4120

0.4312

0.6288

Q2

0.6898

0

0.1459

0.2249

0.4152

0.7525

0.9359

Q3

0.8856

0.1685

0

0.1887

0.5074

0.8219

0.9885

Q4

0.7089

0.2811

0.2127

0

0.3398

0.6697

0.8541

Q5

0.6944

0.5850

0.6451

0.3839

0

0.3685

0.5930

Q6

0.7582

1.1778

1.1740

0.8531

0.4157

0

0.2730

Q7

1.2104

1.6608

1.6080

1.2393

0.7601

0.3102

0

MAX

1.2104

1.6608

1.6080

1.2393

0.7601

0.8219

0.9885

An equivalent analysis was done for the MAW model (not shown here) and it showed that the Q and ∆T models obtained for the MAW mixture do not predict the unstable behavior. This fact allows us to conclude that if in the real plant shows instability also for the MAW mixture, the model for MAW is not correct at low frequencies. This was done and experimental results confirmed our supposition that the MAW model identified here is not correct at low frequencies. Experimental verification of the instability Simulations with the nonlinear model suggest that the unstable behavior will indeed occur. In [Sc95, Figure 4.9] a simulation using a controller based on the inverse of Q4 for the CE


138

mixture is shown. The simulation becomes unstable, when the flow F2 reaches the value 0.3 l/s (or lower). Of course, it is not surprising, since the 'family' of linear models was obtained from the nonlinear model. Thus, this question can only be answered with experiments. Figures 6.19 and 6.20 present two experimental results obtained with the following decentralized controllers:  − 0.1219 5.216 s 2 + 0.254s + 0 .0001071  K ∆T 4 = diag ,  s s 2 + 0 .01775 s   K ∆T 6

(6.16a)

 − 0.09617 5.652 s 2 + 0.2071s + 0.00005526  = diag  , . s s 2 + 0.01353 s  

(6.16b)

The controllers K∆T4 and K∆T6 were calculated based on the nominal models ∆T4 and ∆T6 using the sequential design procedure presented in section 4.4.4. The desired closed-loop function used for both controllers is Td = diag{ Td1[20,5%], Td1[200.5%] }. The implementation of these controllers was in the ∆T configuration as illustrated in Figure 6.13.

(a) Q [ kW ] (solid line) | Qref. (dotted line) 20

Q [ kW ]

15 10 5 0 0

100

200

300

400

500

600

700

800

900

(b) FI_305 [ l/s ] (solid line) | TI_301−TI_302 (dashdot line) 8 7

0.6

6 0.4 5 0.2 0 0

4 100

200

300

400

500 Time [ s ]

600

700

800

900

TI_301−TI_302

FI_305 [ l/s ]

0.8

3

Figure 6.19 : Experimental verification of the instability for CE mixture using a controller based on the inverse of ∆T4.

139

6.2 HEATING SYSTEM

(a) Q [ kW ] (solid line) | Qref. (dotted line)

Q [ kW ]

15

10

5

0 0

500

1000

1500

2000

2500

(b) FI_305 [ l/s ] (solid line) | TI_301−TI_302 (dashdot line) 8

FI_305 [ l/s ]

0.8

6

0.6 4 0.4 2

0.2 0 0

500

1000

1500 Time [ s ]

2000

2500

TI_301−TI_302

1

0

Figure 6.20 : Experimental verification of the stability for CE mixture using a controller based on the inverse of ∆T6. Figures 6.19 and 6.20 show that the simple model predicts the system instability quite well. Observe that only for the controller based on ∆T4, the HS becomes unstable for low flows (i.e., smaller than 0.2 l/s). Moreover, these results confirm our supposition that the instability for this system is related to the low frequency characteristics of the HS, since in Figure 6.19 the system needs almost 2 minutes before it becomes unstable. Figure 6.20 confirms the prediction that the controller design based on the inverse of G6 (or G5) will not produce instability in the system. Observe that in this figure although the volumetric flow achieves a very low value (0.1 l/s), the HS does not become unstable. Figure 6.21 shows that the instability of the HS is not a particularity of the ∆T-structure and the CE mixture, but it is caused by the bypass flow F1. Note that also for the Q-structure the heat duty Q achieves first its setpoint and stays there for almost 2 minutes before becoming unstable. Figure 6.21c shows that the instability is not directly caused by the control of the temperature TI_301, i.e., the instability is due to the control of Q as we can see by the comparison of the controllers Κ∆Τ4 and Κ∆Τ6 in (6.16). Observe that the channel 11 of the controller Κ∆Τ6 is slower than Κ∆Τ4 guaranteeing the stability of the HS for low volumetric flows. Theoretically, Q can be increased by increasing F2 and/or ∆T (see (6.9)). Due to the


140

recycle stream, increasing F2 will always decrease ∆T (cf. Figures 6.19 and 6.20). The instability occurs when the variations in F2 are fast in comparison to the ∆T-variations. Experimental results indicate that the maximal rate of variation of F2 to avoid the instability is a function of the bypass setting. (a) Q [ kW ] (solid line) | Qref. (dotted line)

(c) TI_301 (solid line) | TI_301ref. (dotted line) 99

TI 301 [ ° C ]

Q [ kW ]

15

10

5

0 0

500

1000

1500

98.5 98 97.5 97 0

2000

500

(b) FI_305 [ l/s ] (solid line), TI_301−TI_302 (dashdot line)

3 0.5 2

1500

2000

1

EL3 [ kW ]

4 1

1000 Time [ s ]

2000

1500

2000

30 TI_301−TI_302

FI_305 [ l/s ]

5

500

1500

(d) EL3 [ kW ]

1.5

0 0

1000

20

10

0 0

500

1000 Time [ s ]

Figure 6.21 : Experimental verification of the instability for MAW mixture using a controller based on the inverse of Q4. Solution of the instability problem For the correct operation of the HS the instability must be avoided. One solution to the instability problem could be to apply a controller like K∆Τ6, but this controller has a bad performance (cf. Figure 6.20). Here we present three solutions for this problem that preserve the system performance and solve the instability problem. 1. Increasing the controller order. Increasing the controller order is equivalent to use a PID controller instead of the PI controller. Control action due to the derivative mode occurs only when the error is changing. The presence of the derivative mode contributes an additional output, KD d(y-yref)/dt, to the final control element when there is any change in error. When the error ceases to change, derivative action no longer occurs. The effect of this is similar to having a proportional controller with a high gain when the measured variable is changing rapidly, and a low gain when it is varying slowly. In other words, a controller with a derivative part will have a performance equivalent to K∆T4 in the middle frequency range and at low frequencies similar to K∆T6. The only problem with this solution can be the noise level, since the derivative action will exacerbate the situation by magnifying small rapid changes in the noise into large and unnecessary controller output signals. Additionally increasing in the controller order can be recommended to solve this problem avoiding the high frequency disadvantage caused by a pure derivative action. The difficulty by the additional increase in the controller order is that we could not use any more the standard PID controller implemented in our process control system.

141

6.2 HEATING SYSTEM

2. Addition of a degree of freedom in the control configuration. As discussed in Chapter 4, the 2 RDOF control configuration could solve the problem with the noise level while preserving the performance to the setpoint changes without instability. Loosely speaking, the controller in this case will have a performance equivalent to K∆T4 for setpoint changes and to K∆T6 for disturbance compensation. 3. Reduction of the manipulating range of RV3. The system instability can only occur when the flow through the bypass valve V6 is larger than through RV3. More specifically speaking, for the experimental results analyzed here, only when the valve RV3 is less than 30% open. It means that if we reduce the operating range of the valve RV3 from 0-100% to 30-100%, the instability problem will be automatically solved. As we will see in the next section, this reduction of the operation range causes no reduction of the controllability.

The third solution was the easiest alternative to be implemented in the process control system and, therefore, was our choice. The experimental results are shown in the next section, where we present the final control configuration for the HS. But first, it is interesting to analyze the choice of the operating conditions of the HS. Here the bypass valve V6 and the temperature difference (TI_301-TI_114) play an important role. • The bypass valve V6 has a particular influence on the system dynamic of the HS. Experimental and simulation results show that if the bypass valve V6 is closed, the unstable behavior produced by an inverse-based controller (as discussed above and which is strongly related to the bypass stream) will not occur. At this point one could ask about the functionality of the bypass stream in the HS. The main role of the bypass is to guarantee a minimal flow through the electrical heating tank (EHT). Of course, this minimal flow could be guaranteed if we limit the operation range of the valve RV3 (e.g., from 0-100% to 20-100%). Note that this 'possible' solution is a bad solution, since the main nonlinearity in the EHT is caused by the flow F=F1+F2 (see Figure 6.5). It means that the variation in the flow F increases with the reduction of the flow F1 , i.e., by closing of the bypass valve V6. Therefore, for the EHT, opening of valve V6 is favorable. The opposite situation is found in the thermosyphon reboiler, since here the range of the flow F2 is reduced by opening V6. This causes a reduction in the range in which we can change the heat duty Q by manipulation of RV3. • The temperature difference (TI_301-TI_114) determines the gain from RV3 to Q and is fundamental for the correct operation of the thermosyphon reboiler. Due to the special dynamics of the valve RV3 which is dependent on ∆RV3 ( see equation (6.7) ), small ∆RV3 variations are preferred implicating that only a reduced variation range should be used at normal operation. Therefore, the gain, i.e., the temperature difference, must be chosen so that the usually necessary variations of the valve RV3 are not too large.

Now, we have all necessary information to propose a control configuration for the HS.


142

6.2.3 The final control concept and the reboiler startup procedure In this subsection, we present the final practical implementation of the control strategy used to control the HS for all possible mixtures and operating conditions, such as operations under vacuum or at low pressures. Another very important point is the startup procedure, especially for pilot plants, where the startup and shutdown usually represent a considerable time of the total operation time. Here we also present a totally automated startup procedure for the vertical thermosyphon reboiler and later we complement this procedure including a startup procedure for the composition control. Final control concept The main goal of HS is to control the heat duty Q to the column. For this, we have two manipulated variables RV3 and EL3. Until now, we have seen that RV3 has a very fast effect on Q, whereas EL3 gives a very slow response. Loosely speaking, the HS can be seen as 'large tank' where the energy holdup depends on the recycle flow. In this large tank picture the energy transfer to the reboiler comes from EL3, i.e., EL3 is the inlet stream to the 'tank'. Remember that for short periods somewhat more energy can be transfered to the reboiler ( i.e., Q > EL3 ) using part of the internal energy in the HS. It means that the internal energy in the HS is the 'tank' holdup. Finally, the valve RV3 regulates the outlet stream. Therefore, it is also interesting to use EL3 as a manipulated variable in the final concept. Here we can use a type of feedforward control. So if more Q is required, EL3 is automatically increased avoiding that the 'tank' becomes 'empty'. Another point to be considered is that the temperature difference between the HS and the bottom of the column must be adjusted to maintain effective operation of the vertical thermosyphon reboiler. Here we present a control concept that satisfied all points discussed in a simple and efficient fashion. Remember that we can write the heat duty as Q = U R ⋅ AR ⋅ ∆Tlog. mean temperature ≈ U R ⋅ AR ⋅ (TI _301 − TI_162).

(6.17)

For a given Qref , we can express T_301ref as T_301ref =

Qref U R ⋅ AR

+ TI_162 .

(6.18)

Now, if we consider the expression for UR AR given by equation (6.2), the last equation can be written as follow: T_301ref =

(4.3 − 3.6 ⋅ Tr ) Q K07

ref

+ TI_162 .

(6.19)

This equation can be implemented in the control system to give the setpoint for the control loop EL3 # TI_301. To take the reboiler incrementing temperature (i.e., TI_114 ≈ TI_162+1) into account, we include an additional parameter mB, so that the final formula implemented in the process control system is T_301ref = m A ⋅ Qref + TI_162 + mB .

(6.20)

143

6.2 HEATING SYSTEM

The parameter mA is given by the nonlinear model as mA = 0.7 (4.3-3.6Tr). Here Tr takes the mixture and the operating conditions into account automatically. The 'theoretical' value for mA was slightly modified in the practical implementation of this strategy to allow using the same value of mB for all mixtures and to change the operating position of RV3 to its optimal value (i.e., 60% open). The practical and theoretical values are listed in Table 6.5 for MAW and CE mixtures. Table 6.5: Parameters used in (6.20) for the determination of the setpoint of the control loop EL3 # TI_301 mA (theorectical)

mA (practical)

mB

MAW

0.63

0.8

1

CE

1.34

1.25

1

Figure 6.22 shows the final control configuration for the HS using the ∆T-structure, but it could also be implemented using the Q-structure. Note that only one setpoint must be given (i.e., Qref). The setpoint for the temperature TI_301 is automatically calculated by (6.20). Figures 6.23 and 6.24 illustrate how this control configuration works in the practice. During the experiment shown in Figure 6.23, the heavy boiling point component (i.e., water) was fed into the column, what increases the bottom boiling temperature. Note that the RV3 is almost always in the optimal position of 60% making fast variations in Q of ± 2 kW possible. To avoid the possibility of instability we used a lower bound of 30% for RV3. The reduction of the maximal possible range for RV3 makes no practical difference. In Figure 6.24, the experimental results are presented for the case where the feed composition was held constant, but due to the internal composition variations the temperature suffered some variation before achieving the steady-state profile. Observe again that the RV3 valve is always at the optimal position.

Figure 6.22 : Final control strategy for the HS using the ∆T-structure


144

(a) Q [ kW ] (solid line) | Qref (dotted line) 10

Q [ kW ]

9 8 7 6 0

500

1000

1500

2000

2500

3000

3500

4000

(b) Tempeatures ( left side ) | RV3 ( right side ) 100 TI_301

90

RV3 85

60

RV3 [ % ]

T [°C]

80

TI_162 40

80 0

500

1000

1500

2000 2500 Time [ s ]

3000

3500

4000

Figure 6.23: Performance of the final control configuration for the MAW mixture and the ∆T-structure (a) Q [ kW ] (solid line) | Qref (dotted line)

Q [ kW ]

9.5

9

8.5

8 0

1

2

3

4

5

6

7

8

9

(b) TI_301 (solid line), TI_301ref (dotted line) | RV3 (dashed line) 150

90

TI_301

80

149

70

RV3

148.5

60 148 0

1

2

3

4

5 Time [ h ]

6

7

8

9

Figure 6.24: Performance of the final control configuration for the CE mixture and the Q-structure

50

RV3 [ % ]

TI 301 [ ° C ]

100 149.5

145

6.2 HEATING SYSTEM

Other experiments with the column at low pressure also confirmed the sucess of the proposed control configuration. Startup procedure for the Heating System During the startup phase we want to spend as little time as possible. So, the maximal power in the HS should be used as long as possible. Here the determination of the point when we should reduce the maximal power to the normal operating conditions is critical. Too late power reduction causes the flooding in the column and bad operation of the vertical thermosyphon reboiler. When the maximal power in HS is used at the startup phase, the resistance to the heat duty to the column is concentrated in the thermosyphon reboiler, since at this phase the natural convection and the phase change, which are responsible for the high heat transfer rate in the thermosyphon reboiler, are reduced. Moreover, both factors happen simultaneously, i.e., when the phase change happens, the density in the reboiler is reduced putting the fluid in reboiler into movement by a natural convection mechanism. (a) Temperatures 200

T [°C]

150 100

TI_114 TI_162 TI_103 TI_301

50 0 0

10

20

30

40

50

60

(b) Q ( solid line ), Qref ( dotted line ) | EL3 ( dashed line ) 15

60

10

5

Q

40

20

maximal power

EL3

controllers off 0 0

EL3 [ kW ]

Q [ kW ]

controllers on

0 10

20

30 Time [ min ]

40

50

60

Figure 6.25 : Experimental verification of the startup procedure for CE mixture. On the left side of the vertical dotted line (until 32 min) the valve RV3 was 100 % open and EL3 set to the maximal power (60 kW). On the right side, the controllers for Q and TI_301 became operating. Thus, the startup problem is solved, if we can determine when the changeover point occurs, i.e., the point where natural convection is sufficiently high to permit the correct working of the


146

thermosyphon reboiler. How can the changeover point be determined? The answer to this question is very simple. For it, we use two temperature measurements, one at the input and the other at output of the thermosyphon reboiler, as TI_162 and TI_114 in Figure 6.22. In normal operation, TI_114 is slightly higher ( ≈ 1°C) than TI_162, but at the beginning, when the natural convection is too low, the opposite situation is encountered, i.e., TI_162 is higher than TI_114 (cf. Figure 6.25). When the natural convection increases, the difference between TI_114 and TI_162 decreases. Figure 6.25 shows that in the final phase ( ≈ 30 min) the temperature TI_114 has an inflection. This point represents the ideal time to put the controllers into operation. Although the determination of the inflection point for TI_114 is a very simple task, since we only have to analyze the variation rate of TI_114, we chose to implement a simpler criterion based on the temperature difference between TI_162 and TI_114. When TI_162-TI_114 < 4.5°C the controllers are switched on. This startup procedure works quite well for the MAW mixture also. Only for the case when the mixture consists of components with quite different boiling points, the temperature difference of 4.5°C must be somewhat increased, since in this case after the inflection point both temperatures will get close to each other more rapidly. Note that the startup procedure presented here works for all mixtures. Moreover, this procedure is especially successful and important when the column works under low pressure conditions. For this operating condition, the operators can hardly determine the correct time to put the controllers into operation.

147

6.3 COMPOSITION CONTROL

6.3 Composition Control In this section, we will first present the mathematical model based upon the HETP (height equivalent to theoretical plate) concept. In the second subsection we compare the results of dynamic experiments at the pilot plant with the model. We discuss the limitations of the model with respect to controller design. In the third subsection, the selection and the design of the composition control loops are discussed and experimental results for closed-loop operation are shown. Finally, we show how the time and energy consuming startup of the column can be improved using closed-loop composition control. The material of this section was initially presented in [TRE96].

6.3.1 Process model How good must the model be? What kind of information must be taken into account? Questions like these often arise when we start modeling of a system. One can try to model all possible effects, considering radial and axial distribution, statistical maldistribution of the liquid flow, heat losses, etc. Of course, for process design sometimes such details can be very important and often empirical formulae are used to get around some modeling problem. It is more useful to reformulate the question how exact the model must be by what kind of problem one wants to solve using the mathematical model. For example, if one wants to apply the model in pure feedforward control, the model needs to consider heat losses for the correct description of low frequency characteristics of the plant. For feedback control, this kind of information is often unnecessary, since the integral part of the controller handles these low frequency uncertainties quite well. Another important point concerns the high frequency dynamics which are always present in a detailed model but which in a practical sense cannot be distinguished (or identified) from measurement noise. In conclusion, for feedback control the model must be exact in the middle (crossover) frequency range. Particularly, for packed columns it is interesting to polarize the discussion about the model in equilibrium vs. nonequilibrium and discrete vs. continuous models. Equilibrium vs. nonequilibrium stage models The standard model used in separation process simulation is the equilibrium stage model. The model is very simple conceptionally and has been used to simulate and design most real columns. Nevertheless, the trays of an actual column are not equilibrium stages. The usual way of getting around this problem is to use an efficiency factor of some kind (e.g., Murphree efficiency). It is common practice to assume that the efficiencies are the same for all components on any given tray even though there is abundant experimental and theoretical evidence to show that the efficiencies for different components are not the same (see e.g., [Gór95], [TK93]). The equilibrium stage model is often also used to simulate packed columns. The efficiency is replaced by the Height Equivalent to a Theoretical Plate (HETP). Of course, the HETP for multicomponent systems suffers from the same problems as tray efficiencies. A nonequilibrium stage may represent either a single tray or a section of the packing in a packed column. The same basic equations can be used to model both types of equipment and


148

the only significant difference between these two models is that alternative expressions must be used to estimate the mass transfer coefficients and hydrodynamic parameters (see e.g., [ChemSep]). Instead of HETP (or efficiencies) values, as for the equilibrium model, the nonequilibrium model needs transfer coefficients, which can be estimated from theoretical models, but are usually determined by experiments. In general, for two component mixtures the HETP and the nonequilibrium approach give equivalent results. The advantage of the nonequilibrium model is that it allows an extrapolation from experimental results obtained for binary systems to multicomponent mixtures. Discrete vs. continuous models Packed columns are, of course, continuous contact devices and must in principle be modeled by partial differential equations, usually with the time variable t and the axial distribution variable z. The radial dispersion term can often be neglected, since the radial profiles are generally nearly flat. For the solution of this system of partial differential equations the variable z is usually discretized by, e.g., finite element and finite difference methods. Doing so, the original system is converted into a number of nonequilibrium stages. When the HETP concept is used we bypass the discretization part and the nonequilibrium effect. Of course, the simplicity of the HETP model is paid for by the limited extrapolation range and by the correctness of results. Nevertheless, the HETP model may be useful for our goal, i.e. for feedback control. In the next subsection, we present an extension of the standard model used in the simulation of tray columns and in section 6.3.2 we verify that this model satisfies the requirements of feedback control. Model equations The equilibrium stage model is based on material and energy balances and equilibrium relationships. The following assumptions are made: the liquid phase is ideally mixed and the holdup in the vapor phase is negligible with regard to the liquid phase holdup. The equations are formulated for each stage and for the reboiler and condenser system. For a stage j and component i the following equations are obtained:

Vj yj, i

Mj

stage j M,j xj, i, yj, i Vj+1 yj+1, i

0 = Vj +1 + L j −1 − L j − Vj

Lj-1 xj-1, i

Mj

dx j ,i dt

dhL ,j dt

= Vj +1 y j +1 ,i + L j −1 x j −1 ,i − L j x j ,i − Vj y j ,i

(6.21b)

= Vj +1 hV ,j + i + L j −1 hL ,j −1 − L j hL ,j − Vj hV ,j

(6.21c)

N

Lj xj, i

(6.21a)

∑ y j ,i = 1 , i =1

y j ,i = K VL j ,i ⋅ x j ,i

(6.21d)

Figure 6.26: A theoretical stage Packing hydraulics (from Stichlmair et al. [SBF89]): H L = 0 ,555 ⋅ FrL

13

2   ∆pirr   ⋅ 1 + 20 ⋅     Zρ L g   

(6.21e)

149


∆pirr  1 − ε ⋅ (1 − H L ε )  =  ∆pd  1− ε 

2+c 3

⋅ (1 − H L ε )

−4 ,65

.

(6.21f)

Here ∆pd is the pressure drop of the dry packing: ∆pd 3 1 − ε u2 ⋅ ρ = ⋅ f ⋅ 4 ,65 ⋅ V V . ε Z 4 dP

(6.21g)

The system of equations (6.21a-g) was implemented in [SpeedUp] with the physical properties calculated by PROPERTIES PLUS [Aspen+]. For more details the reader is referred to [Roß95]. Determination of HETP using steady-state experimental results The dependence of the column performance on the packing load (represented by the F-factor = uv ρv0.5) is not described by the model. It assumes a constant number of equilibrium stages. The adequate number of stages can be determined by a comparison of simulation and experimental results for column operation at total reflux.

Figure 6.27: Concentration profiles: Comparison of experimental results and simulations with different values for NTSM (Number of Theoretical Stages per Meter, NTSM = HETP-1) for F-factor = 0.64 Pa0.5 For our experiments, a narrow boiling ideal mixture of chlorobenzene and ethylbenzene is used. The normal boiling points are 131.7°C and 136.2°C respectively. The vapor-liquid-


150

equilibrium was modeled by the Peng-Robinson equation of state. Due to the relatively low number of stages, for this mixture the attainable purities are also low, so that no strong nonlinearities are expected. Four experiments were carried out using the MontzPak A3-500 [Montz]. The results of the simulation with 7 theoretical stages per meter packing (TSM) show the best agreement with the experimental values (see Figure 6.27). This result fits the 6.5 TSM given by [Montz] for the same F-factors of our experiments. All our simulations and results are generated with 7 TSM giving a total of 21 theoretical stages.

6.3.2 Open-loop verification of the mathematical model The direct comparison of experimental data and the nonlinear model is not an easy task. For this comparison, consistent experimental data is necessary. Moreover, many additional factors (such as heat losses, dynamics of measurement devices and actuators, etc.) are present in the real system. Building a model that takes all these additional factors into account usually requires a very large effort that is often not worthwhile. In this section we examine if the relatively simple model presented in the last section is good enough for feedback control. Subsequently our analysis will concentrate on two issues: the dynamics of each channel in isolation, and the interaction (relative dynamic) characteristics. Nonlinear simulation (comparison in the time domain) The chosen manipulated variables are the setting of the reflux-divider RD = L/(L+D) and the reboiler heat duty Q. To study the dynamic characteristics, we have chosen input signals with different properties (see Figure 6.28). (a) RD = L/(L+D) [−] 1

RD [−]

0.8 0.6 0.4 0.2 0 0

5

10

15

20

25

30

35

30

35

(b) Reboiler Heat Duty Q [ kW ] 9

Q [ kW ]

8

measurement approximation

7

6 0

5

10

15 20 Time [ min ]

25

Figure 6.28: The manipulated inputs used for the validation experiment: (a) reflux divider RD=L/(L+D); (b) heat duty Q [kW] (dashdot line), approximation for the nonlinear simulations (solid line)

151


First, Q was chosen as a ramp signal followed by a step change. Until minute 21 only variations of Q were performed. Between minutes 21 and 27, input variations were applied simultaneously. The remaining interval contains only manipulations of RD. Other experiments were used to identify a low order linear black-box output error (OE) model [IdTB92]. The input signals were PRBS for Q and RD, so that the experimental results presented here are also used to verify the black-box model. For the simulation, the input Q was approximated by piecewise linear functions as shown in Figure 6.28. This approximation is only used in the nonlinear simulations. All other simulations were done with the original signal. The column was operated without side draws and with an almost constant feed composition of 0.64 mol of Chlorobenzene per mol of mixture. The feed was inserted into the column through the manual valve V4 and was subcooled by 16.4 °C. In the simulations, this subcooling was not considered. Its practical effect is equivalent to an additional vapor condensation at the feed height. Moreover, in our simulations 1 bar top pressure was assumed. Increasing this value would make the curves in Figure 6.29 closer to each other. A very important aspect in model verification is to ask which property we want to verify. For feedback control, the dynamic part of the model is more important and can be analyzed by considering the deviations from the steady-state values. This is done in Figures 6.30 and 6.31 for the same temperatures as in Figure 6.29.

135.4

135.2

135

T [°C]

134.8

134.6

134.4

TI_103 TI_107 TI_111

134.2

134

133.8 0

5

10

15 20 Time [ min ]

25

30

35

Figure 6.29: Comparison of the experimental results (upper set), and the simulation results with the nonlinear model (lower set) for three temperatures of Figure 6.1


152

0.1

∆T [°C]

0.05

0

−0.05

−0.1

0

measurement nonlinear model linearized model

5

10

15 20 Time [ min ]

25

30

35

Figure 6.30 : Comparison of deviations of temperature TI_107 obtained from experimental data (dotted line), nonlinear simulation (solid line), and the linearized model (dashdot line) In Figure 6.30, a comparison of the nonlinear model, the linearized model, and experimental data is shown. From this result we can infer, as we were already expecting, that the system is almost linear for this control structure and this mixture. The same plots for the other temperatures (not shown here) give similar conclusions. Since the temperatures TI_103 and TI_111 will be used for feedback control later, it is interesting to compare the nonlinear model with the identified black-box model. Figure 6.31 shows the results of this comparison. Note that the experimental data in this figure has not been used in the identification procedure. This allows a fair comparison of the two models. The analysis of Figures 6.30 and 6.31 allows us to conclude that: • The nonlinear and the identified model are able to predict the deviation of the temperature TI_103 caused by manipulations of RD (see Figure 6.31 (a) after 20 min). On the other hand, the nonlinear model yields poor predictions of the effect of variations in Q. Here, the identified black-box model seems to represent this effect better. • For the temperature TI_111, as illustrated in Figure 6.31 (b), the nonlinear and the identified model give acceptable results for both inputs. Note that the initial response to variations in Q is reproduced by both models quite well. The smaller gain observed in the experimental measurements can be explained by the subcooling of the feed stream that was not taken into account in our simulations.

153


( a ) TI_103

0.1

∆T [°C]

0.05

0

−0.05

−0.1

0

measurement nonlinear model ident. model

5

10

15 20 Time [ min ]

25

30

35

25

30

35

( b ) TI_111

0.1

∆T [°C]

0.05

0

−0.05

−0.1

0

measurement nonlinear model ident. model

5

10

15 20 Time [ min ]

Figure 6.31: Comparison of deviations of temperatures TI_103 (a) and TI_111 (b) obtained from experimental data (dotted line), nonlinear simulation (solid line), and the identified black-box model (dashdot line)


154

• The analysis of the model quality for the channels Q-TI_103 and RD-TI_111 is more difficult than for the channels Q-TI_111 and RD-TI_103. This is due to the relatively small gains of these channels as a consequence of the (RD,Q)-control-structure and the narrow boiling mixture. In addition, the number of possible disturbances increases with the distance between the manipulated and the controlled variables. Linearized model vs. black-box model (comparison in the frequency domain) In the last subsection, the black-box model and the nonlinear model were compared in the time domain. Here we extend this comparison to the frequency domain in order to analyze the structural properties of the model. Since the process, for this mixture, purity and control structure, is almost linear (cf. Figure 6.30), we can concentrate our attention on the structural properties of the linearized models. As was discussed in chapter 2, the RGA is a useful tool to study the relative dynamic properties of the different channels. (a)

RGA

1.5

1

0.5

0 −4 10

−3

10

−2

10

−1

10

0

10

1

10

(b) Minimized Condition Number 4

3

2

1 −4 10

−3

10

−2

−1

10 10 Frequency [ rad/min ]

0

10

1

10

Figure 6.32: Comparison between the linearized (solid line) and the identified (dashdot line) model : (a) RGA (Relative Gain Array) and (b) minimized condition number Figure 6.32 shows that the black-box model exhibits more interaction in the range 0.2 - 2 rad/min than the linearized model. The faster initial response of TI_103 to heat duty variations in Figure 6.31a predicted by the black-box model explains this fact. Note that in the low frequency range, the linearized model shows more interaction than the identified blackbox model. Of course, the black-box model cannot give good results in this frequency range, since all input perturbations used by the identification procedure were concentrated in the frequency range 0.1 - 2 rad/min corresponding to signals with variation periods between 3 min and 63 min. To improve the identification at low frequencies, it would be necessary to apply a

155


periodic signal with a period of 50 hours. It is helpful that for feedback control we do not need this information. This would only be necessary if one wants to make the closed loop response slower than the process! Since the nonlinear model is based on an equilibrium stage model, the temperature can only change if the equilibrium is changed. This means that the temperature calculated by this model develops in a cascade fashion, i.e. first the equilibrium point of the first stage is changed, then the next and so on. The experimental data indicates that this mechanism is not responsible for the temperature deviations caused by input variations of Q. In fact, it seems that the time lag for the temperature TI_103 is infinitesimal. This temperature variation can thus only be captured by a nonequilibrium model. At this point the question arises: Is the HETP equilibrium model still useful for our goal? To answer this question, remember that our primary aim is composition control. The temperatures are only secondary variables that allow us to determine the composition via phase equilibrium relations. Thus, we should put the question as: Is the composition dynamics similar to the experimental temperature dynamics? If the composition dynamics is different from the experimental temperature dynamics, we can say that knowing this dynamics exactly will not give an advantage for process control purposes. To verify this, we formulate the hypothesis that the temperature deviations caused by input variations in Q cause a direct and equal initial response for all temperatures, that is Tj = Teq,j + ∆, where Teq,j is the corresponding equilibrium temperature for the stage j and ∆ is equal for all stages. This means that if the difference between two measured temperatures like TI_107 - TI_103 is equal to the difference of the corresponding equilibrium temperature, i.e. Teq,2 - Teq,1, our hypothesis is true. This comparison is done in Figure 6.33 for TI_103 and TI_107, where the equilibrium temperatures are given by the nonlinear model. Figure 6.33 indicates that our hypothesis is true.

0.25

0.2

0.15

0.1

0.05

0 0

5

10

15 20 Time [ min ]

25

30

35

Figure 6.33: Temperature difference TI_107 - TI_103 calculated from experimental data (dotted line) and predicted by the nonlinear model (solid line)


156

6.3.3 Closed-loop validation of the mathematical model The minimized condition number in Figure 6.32(b) and the fact that the system is almost linear indicates that the distillation system analyzed here will present no special problems for control. Moreover, the interactions are not strong, so that we chose to implement a decentralized PID controller. The corresponding controller was designed via sequential design presented in section 4.4.4.

Controller Performance The controller design was based on the identified model and was directly implemented in the process control system. The controller performance is illustrated in Figure 6.34. At 60 min, the channel TI_111-Q of the controller was detuned to get around the oscillatory response in the temperatures. The cause of this oscillation can be attributed to a synergetic effect between sensor noise and the nonequilibrium temperature dynamic effect discussed in the last subsection. The use of a temperature difference, e.g., TI_107-TI_103, as the controlled variable instead of TI_103 could solve this problem without controller detuning.

(a) OUTPUTS 136

T[°C]

135.5 TI 111 135

TI 107 TI 103

134.5 0

50

100

150

200

20

1

15

0.95 Feed Flow [l/h] Q [kW]

10

0.9

5

RD [−]

F [l/h] − Q [kW]

(b) INPUTS

0.85 RD [−]

0 0

50

100 150 Time [ min ]

200

0.8

Figure 6.34: Closed-loop performance of a decentralized controller applied during the startup phase.

157


The discussion of Figure 6.34 can be divided into: • The startup phase, from 0 until 100 min, • The regulatory part (disturbance compensation), from 100 to 170 min. Observe that the controller can compensate a feed flow variation of 100%, • The servo part (setpoint changing), from 170 min to the end. The permissible setpoint variation for this narrow boiling point mixture is relatively small. Here a setpoint variation of 0.25 °C in the temperature TI_111 is shown. Similar results were obtained for TI_103 (not shown here).

(a) OUTPUTS

T[°C]

135.5

TI 111

135

TI 107

134.5

TI 103

0

1

2

3

4

5

6

7

8

9

(b) INPUTS 1 Feed Flow [l/h]

16

0.95 14

RD [−] 0.9

12 Q [kW]

10

RD [−]

F [l/h] − Q [kW]

18

0.85

8 0

1

2

3

4

5 Time [ h ]

6

7

8

9

0.8

Figure 6.35: Column startup with constant inputs

Improving the startup with a controller The first part of Figure 6.34 illustrates that the column can achieve the steady-state within 1 hour. The same column without feedback control needs approximately 9 hours to achieve the same steady-state point when Q and RD are held constant (see Figure 6.35). This significant difference illustrates how even a simple control scheme can improve the process dynamics. It shows that process control and dynamics play a very important role even if our


158

goal is just steady-state studies. Almost the same reduction could be achieved by one-point composition control (for example, a controller between TI_103 and RD). Observe that a onepoint composition controller can be tuned on-line quite easily. So with little work one can save much time (7 hours or more) and energy. Moreover, if we use the closed-loop system, we can analyze the column under different conditions on the same day. Thus, it is possible to do the experimental work of a week (without feedback control) in just one day. Many people in the process industry and at the universities seem to be unaware of this important fact.

Chapter 7 Conclusions and Directions for Future Works Most books and papers in the control literature have only considered the controller design step of the control design task. In this thesis, we explored the full dimension of the control design concept, i.e., control structure design (CSD) + controller design. Although our emphasis is in CSD, contributions were also made to controller design step. Chapters 2, 3, and 4 of this thesis addressed the development of a systematic approach to CSD, whereas Chapters 5 and 6 exemplified the application of the CSD procedure. Moreover, Chapters 5 and 6 show different aspects of operability and control problems that are not fully captured by the current control theory.

7.1 Conclusions The main contributions of this thesis can be summarized as follows: Chapter 2 presented a seven-step procedure that permits to analyze all different aspects that must be considered in CSD, such as model uncertainties, nonlinearities of the process, input saturations, interactions between the control loops and process units, failure sensitivity, sensor noise, etc. The first step is the determination of the number of process degrees of freedom and the understanding of the main process dynamics. The second step uses some heuristics or "feelings" to make a preselection. The third, fourth and fifth steps are the determination of the IO-scaling, the nominal performance, and nonlinearities robustness issues, respectively, and can successfully be analyzed by the RPN and RPPN. In steps 1 to 5, the number of possible control structures is reduced to a few possible structures that are then analyzed for failure sensitivity and controller structure and order in step 6. After performing these six steps, often, we obtain more than one good solution, so that additional criteria must be considered. These additional criteria are more or less subjective and strongly dependent on the particular problem at hand. This is in the seventh step of our procedure and was also discussed. In Chapter 3, the robust performance number (RPN) and the robust performance number for a plant set (RPPN) were defined. With these new indices, the controllability of the system can be analyzed systematically. Therefore, RPN and RPPN are the basis of a systematic

7. CONCLUSIONS AND DIRECTIONS FOR FUTURE WORKS

160

approach to control structure design presented in this work. Another important contribution in Chapter 3 is the interpretation of the role of the minimized condition number and of the RGA in the determination of controllability. It was shown that the conjectured upper bound for γ*(G) is wrong in general. We also presented a strategy for system scaling based on RPN that permits a correct analysis of the feedback system properties. The crossover frequency range (or equivalently the desired performance) plays a very important role for the determination of the optimal scaling. Moreover, different types of uncertainty description are also discussed. A criterion for the determination when a block diagonal input uncertainty can be substituted by a full block representation without increasing the conservativeness in controller design is also presented. Chapter 4 shows how the attainable performance T, for a given nominal model G, can be calculated based on the performance limitations produced by the closed loop stability requirements, sensor noise, and actuator constraints. Note that the attainable performance T is an important part of the definition of RPN and RPPN. A general 3 CDOF control configuration was analyzed and criterions for the choice of the appropriated control configuration were presented. At the end of Chapter 4, it is shown how different approaches to performance weight selection can be applied for H∞-controller design methods. As the H∞ and µ synthesis usually give high order controllers, the efficient closedloop model reduction method proposed by Engell and Müller [EnMü93] is discussed, which can also be used for controller design. Here our main contribution is to show that the application of this procedure to scaled systems using the scaling procedure based on RPN gives very good results. The fact that we have spend more time to simulate a given controller than to design it gives an idea about the efficiency of the method. Of course, our method will not work always. A criterion based on RPN is stated to identify when the procedure will produce good results. For the cases, where the procedure is not applicable, more time consuming procedures (e.g., DK-iteration, H∞/H2-synthesis, etc.) must be used. In this case the final controller can be reduced using the proposed order reduction method. In Chapter 5, three operating points (OPs) of a CSTR reactor for a Van de Vusse reaction kinetic scheme were analyzed. First, it was shown how the operation condition can change the reactor performance and controllability considerably. Considering the case where only the composition of the desired product (B) is controlled, the OPs correspond to a nonminimum phase ( zero > 0, 1st OP ), a steady state singular system ( zero = 0, 2nd OP ), and a minimum phase ( zero < 0, 7th OP ) OP. The choice of the OP depends on the possibility of obtaining the desired product from the unwanteded product (C). If this is possible, the nonminimum phase OP should be preferred. In the other case, the minimum phase OP should be our choice. For the situation, where the reactor works stand alone, which is an unrealistic condition, the 2nd OP represents the alternative with maximal product yield and should therefore be chosen. An analytical expression for the transmission zero was derived and used as basis for a new control structure for the control of the reactor at the 2nd OP. Moreover, Chapter 5 showed the susccessful application of RPN and RPPN for the controllability analysis and controller design procedure. The performance of RPN controller is comparable to a µ optimal controller. It illustrates that much work can be saved, if we first analyze the system properties using RPN, before applying controller synthesis methods.

161

7.2 DIRECTIONS FOR FUTURE WORKS

Chapter 6 shows the application of the hierarchical control design to a multi-purpose packed distillation column (MPPDC). This experimental plant could be divided into two experimental systems: the heating system (HS) and the packed column unit. Particularly, the HS exhibits very special system dynamics due to the energy recycle stream. We show how a simple mathematical model can be used to predict and understand the process dynamics. The process understanding produced by our simple mathematical model allowed us to formulate a good control strategy. The final control configuration works for a width range of mixtures and shows excellent performance. Moreover, we have shown how the startup procedure of the HS can be fully automated for all mixtures and column pressures. In Section 6.3, experimental data of the pilot plant obtained by experiments for the low purity separation of an ideal binary mixture is used to show that the HETP-model can be applied for the prediction of the composition dynamics. The experimental results indicate that the initial response to the heat duty is almost instantaneous and equal for all temperatures. However, this nonequilibrium temperature dynamics is not important for composition control, since it corresponds mainly to nonequilibrium effects on the temperatures. We have shown that the composition dynamics follows a cascade mechanism which is well described by a simplified HETP dynamic model. Note that this situation can be quite different for a strongly nonideal mixture where the interaction between the energy and composition balances can play an important role. Remember that for the ideal mixture presented in Chapter 6, it would suffice to consider only the material balance with constant relative volatility for satisfactory modeling of the system's dynamics. To compensate the effect of the nonequilibrium temperature dynamics we proposed to use a temperature difference as the main controlled variable instead of one temperature alone. The advantage of this new controlled variable is that the composition dynamics are represented better. In addition, this transformation reduces the interaction in the crossover frequency range. Section 6.3 also illustrates that much time in the startup phase can be saved by applying a feedback controller. It seems that in the industry and university communities, there exists a misunderstanding about steady-state experiments: for a steady state experiment all inputs (e.g., manipulated variables and external disturbances) should be as constant as possible. Nevertheless, this is only correct if the system is already at the steady-state point. Of course, best way to reach this point is not to apply a constant input from the beginning of the experiment, as it is often done. The input trajectory can be determined by quite complex and numerically intensive simulations or, in a more simple, elegant, and efficient way, by means of a simple feedback controller as it was discussed in Chapter 6.

7.2 Directions for Future Works Studies of plant wide control In this thesis, we have proposed a CSD methodology that can be applied to any process. Here we have only analyzed systems of low dimensions. A natural extension of the applications presented here is the analysis of a plant wide control problem. Many points related to the plant wide control problem were already discussed in Chapters 1, 2, 3, and 4. By the inclusion of the separation trend and the possibility of reversible reactions, the CSTR example discussed in Chapter 5 could be the basis of a benchmark problem for plant wide control and optimization

7. CONCLUSIONS AND DIRECTIONS FOR FUTURE WORKS

162

studies. Another possibility is the analysis of the plant wide control of a process with three distillation columns and two recycle streams proposed by Luyben and Luyben [LL95]. Integrated process and control design The conventional approach to synthesizing of the control system is to analyze the process after it was completely designed. The process design step assumes steady-state conditions, even the control system must cope with the dynamic situation. This may lead to a poor match between the process characteristics and the control system characteristics. In order to design a better controlled plant, the process design and the process control design should be considered simultaneously or sequential-interactively (i.e., process design, control structure analysis, process redesign, control structure analysis, etc.). Here the concept of RPN and RPPN could be inserted in the process design task aiming to make the process and the control design simultaneously. Future theoretical analysis of RPN and RPPN The RPN and RPPN permit a determination of how easily a system can be controlled. In this work only semi-quantitative values are used to interpret these indices. With a more quantitative criterion based on RPN and RPPN, it may be possible to draw more precise conclusions, such as how much performance can be obtained by the application of a nonlinear controller instead of a linear controller. Controller design for the 3 CDOF control configuration In Chapter 4, we discussed when a 3 CDOF control configuration and its variations, i.e., 2 FDOF and 2 RDOF, will increase the system performance. To design the 3 CDOF controller, developing an efficient low order controller design algorithm is still necessary. Here special attention must be given to system with large RPN values, since for these systems an inverse-based controller will not produce good results. Experimental verification of the "multi-effect batch distillation system" (MEBAD) Although batch distillation generally is less energy efficient than continuous distillation, it has received increased attention in the last few years. More recently, in [HKH95] a "Multi-Effect Batch Distillation" systems based on total reflux operation was proposed. There are at least two advantages of this multivessel column compared to conventional batch distillation where the products are taken over the top, one at time. First, the operation is simpler since no product change-overs are required during operation. Second, the energy requirements may be much lower due to the multi-effect nature of the operation, where the heat required for the separation is supplied only to the reboiler and cooling is done only at the top. Hasebe et al. [HKH95] show that for some separations with Nc components the energy requirement may be similar to that for continuous distillation using Nc-1 columns. The control strategy for this system is crucial for the good and automated function of this system. In recent work, Skogestad et al. [SWSL96] have proposed a new control configuration based on the control of Nc-1 temperatures via the manipulation of Nc-1 holdups of a Nc MEBAD. The simulations presented in [SWSL96] indicate the feasibility of this simple and efficient control strategy. The experimental verification of this control strategy could be done in our MPPDC described in Chapter 6. Note that MPPDC can work with all kind of mixtures, since all equipments are certified for explosion areas.

Appendix A Preliminaries This appendix summarizes and generalizes some fundamental material on linear systems theory. Most of the material can be found in the literature (e.g., [SP96], [ZDG96], [GL95], [BoBa91], [Mac89], [MZ89], among others). The main contribution is the presentation of these results in a differential algebraic equations (DAE) setting. This form presents advantages over the ordinary differential equations (ODE) form. For many systems, the DAEform is a natural and direct representation. Especially for large and stiff models the DAE representation will preserve the block-diagonal structure of the system thus improving the numerical precision and efficiency.

A.1 SVD (Singular Value Decomposition) A convenient way of representing a matrix that exposes its internal structure is known as the Singular Value Decomposition (SVD). For a no x ni complex matrix M, the SVD of M is given by n

M = U Σ V H = ∑ σ i ( M )ui viH

(A.1)

i =1

where U and V are unitary matrices with column vectors denoted by U = [ u = u1 ,u2 ,...,uno = u ] and V = [ v = v1 ,v2 ,...,vni = v ]

(A.2)

and Σ contains a diagonal nonnegative definite matrix Σn of singular values arranged in descending order:  Σn Σ =   if  0

no ≥ ni

or Σ = ( Σ n

0) if no < ni

(A.3)

APPENDIX A: PRELIMINARIES

and

164

Σ n = diag (σ 1 ,σ 2 ,...,σ n ); n = min{no, ni}

(A.4)

σ = σ 1 ≥ σ 2 ≥...≥ σ n = σ .

with

(A.5)

_ The maximum (σ) and minimum σ singular values can alternatively be defined by

σ (M ) = max u ≠0

Mu u

2

and σ (M ) = min u ≠0

2

Mu u

2

.

(A.6)

2

_ σ and σ can be interpreted geometrically as the least upper bound and the greatest lower bound on the magnification of a vector by the matrix operator M. Definition A.1.1 ( Euclidean ) Condition Number (γ). γ). The condition number of a matrix is defined as the ratio ∆

γ ( M )=

σ(M) . σ(M)

(A.7)

! In numerical analysis, the condition number measures the difficulty of inverting a matrix (e.g., [Li93]). It has been argued that it has a control-theoretic significance, in that it measures the inherent difficulty of controlling a given plant. The direct examination of γ alone however gives no conclusive information about the system's controllability, since all systems can be scaled to get a very large (infinite) condition number. On the other hand, the minimal attainable condition number is a finite value that depends on system characteristics only. Thus, for control purposes the minimized condition number over all scaling matrices is more useful to analyze the inherent controllability of the system. Definition A.1.2 Minimized ( Euclidean ) Condition Number (γγ∗). The minimized condition number is obtained by minimizing the condition number over all possible scalings and is defined by ∆

γ * (M )= min γ (L M R ) , L ,R

(A.8)

where L and R are real, diagonal, and nonsingular scaling matrices. If only one side is scaled then we get the input and output minimized condition numbers: ∆

∆

* γ* I (M )=minγ (M R ) and γ O (M )= min γ ( L M ) . R

L

(A.9)

A.2 Function Spaces for Systems and Signals As control objectives are usually stated in the form of desired properties of systems and signals, it is natural to introduce a mathematical classification of both systems and signals. It is useful to have numbers which indicate the size of a vector or a matrix, of a signal or a

165

A.2 FUNCTION SPACES FOR SYSTEMS AND SIGNALS

system. We here introduce some frequently used complex matrix function spaces. For more details the reader is referred to [ZDG96]. Definition A.2.1 R denotes the space of all real-rational transfer function matrices. Definition A.2.2 The L2/H2 norm of G is given by ∆

1 G 2= 2π

∞

1 ∫ Trace(G ( jω ) G( jω )) dω = 2π −∞ *

∞

n

2 ∫ ∑ σ i (G( jω )) dω

(A.10)

−∞ i =1

(a) RL2 denotes the space of all real-rational transfer function matrices with no poles on the imaginary axis with finite L2 norm (i.e., G is strictly proper). (b) RH2 denotes all transfer function matrices in RL2 which have no poles in right-half-plane (RHP, i.e., Re(s)>0). (c) H2 is the set of transfer function matrices which are analytic in Re(s)>0 with finite L2norm. The H2 space is the generalization of RH2 to irrational causal transfer functions ! Definition A.2.3 The L∞/ H∞ norm of G is given by ∆

G

=supσ [G ( jω )] . ∞ _

(A.11)

ω∈ℜ

(a) RL∞ denotes the space of all real-rational transfer function matrices which have no poles on the imaginary axis with finite L∞-norm ( i.e., G is proper). (b) RH∞ denotes all transfer function matrices in RL∞ which have no poles in Re(s)>0. RH∞ is the space of all stable, real-rational transfer function matrices. (c) H∞ is the set of transfer function matrices which are analytic in Re(s)>0 with finite L∞norm. The H∞ space is the generalization of RH∞ to irrational causal transfer functions. ! An important property of the H∞-norm is that it is submultiplicative: (A.12) GH∞≤ G∞ H∞ (The other Hp-norms do not have this property.) The spaces L2 and H2 are often specialized to the case of vector-value of transfer functions, and for the case of standard control system signals, H2 can be shown (e.g., [Fr87], Theorem 1) to be the set of Laplace transforms of signals with bounded energy for t ≥ 0. The next theorem states an important result for H∞-optimization theory, relating the H∞-norm of a transfer function matrix to the H2-norms of its associated input and output transfer function vectors. Theorem A.2.4 If G ∈ H∞ , then GH2 ⊂ H2 and G

∞

{

}

= sup Gu 2 : u∈ H 2 , u 2 = 1 .

(A.13)

! If y = G u, then this theorem states that for any input u, of unit energy, the energy in y for t ≥ 0 is bounded by ||G ||∞ , the H∞ -norm of G.


166

A.3 Linearized dynamic model representation The linearized differential algebraic equations (DAE) of index 1 (or less) given by (A,B,C,D,E)1 can be directly used in the system analysis. Note that in principle we can transform the (A,B,C,D,E)1 to the (Ar ,Br ,Cr ,Dr) form. To illustrate this transformation we write the (A,B,C,D,E) index 1 DAE-system in the factored form  E11  0 

E12  d  x1   A11 = 0  dt  x2   A21 y = [C1

A12   x1   B1  u + A22   x2   B2 

(A.14)

 x1  C2 ]  + Du  x2 

where x1 is the vector of differential variables (those for which we have explicit differential equations), x2 is the vector of algebraic variables, and the submatrices Aij, Bi, Cj, and Eij are adapted to the dimensions of x1 and x2. From (.14) we find, if A22 is invertible, that x2 = − A22−1 ( A21 x1 + B2 u) so that (.14) can be reduced to

(E

11

−1 A21 − E12 A22

) dxdt = ( A − A A A )x + (B − A y = (C − C A A )x + Du . 1

11 1

−1 12 22 21

−1 2 22 21

1

1

−1 12 A22 B2

)u

(A.15)

1

Finally, the equivalent system (Ar ,Br ,Cr ,Dr) is given by A B !######"r ######$ !######"r ######$ −1 −1 dx1 −1 −1 −1 −1 A21 A11 − A12 A22 A21 x1 + E11 − E12 A22 A21 B1 − A12 A22 B2 u = E11 − E12 A22 dt −1 y = C1 − C2 A22 A21 x1 + Du . %##&##' Cr

( (

) (

)

)

(

) (

)

(A.16)

In the development of a mathematical model, one should take care with the index of the DAE model. The index is equal to the number of times we must differentiate the set of algebraic equations to obtain continuous differential equations for all unknown variables (see, e.g., [Sch95, chapter 2]). An index of zero denotes a set of differential equations without any algebraic equations. Higher values of the index result when the algebraic and differential equations are solved simultaneously. Index 0 problems are much easier to solve numerically than those with index 1. Indices greater than 1 are more difficult to solve. Therefore, if we simulate a differential algebraic model, we need to be careful to pose the model so that the index does not exceed 1. Usually, when one specifies the values of the manipulated variables ( e.g., heat duty, reflux) the corresponding DAE will be of index 1, whereas for the case where the controlled variables (e.g., temperatures) are specified the DAE index is often 2 or more. Observe that this problem can easily be solved through the transformation of an index 2 problem into an equivalent index 1 problem. For it, we need only to insert, for example, a PI controller between the controlled and manipulated variables. Doing so, we specify the controlled variable indirectly via the setpoint specification.

167

A.4 ZEROS AND POLES OF MULTIVARIABLE SYSTEMS

Nevertheless, there exist systems where this easy procedure is not sufficient to reduce the index of the problem. A broad class of chemical processes modeled by high index DAE systems (i.e., index > 1) consists of multiphase system where the individual phases are in thermodynamic equilibrium. Typical examples of such systems are vapor-liquid systems (e.g., distillation columns, multiphase reactors, etc.) when the vapor holdup is considered in the mathematical model. Of course, a common approach to avoid modeling this process by a high-index DAE system is to assume a negligible vapor holdup compared to the liquid holdup, thus eliminating the need to model the vapor dynamics. Clearly, the accuracy of the model depends on the validity of the assumption that the vapor holdup is negligible. These assumptions will not hold at high pressures when the vapor holdup becomes comparable to the liquid holdup. Distillation columns usually work with low and moderate pressures (e.g., 8 bar are typical for double effect distillation or heat integrated columns [Luy92, chapter 24]) making this modeling simplification uncritical. However, for two phase reactors (e.g., a vapor-liquid reactor under high pressure or liquid-liquid reactors) the hold up in both phases must be considered producing typically an index 2 problem (see [KD95] for an example). The high-index problem for a linearized DAE-system is equivalent to a rank loss of the matrix A22 in (.14). In this case, since A22 is a singular matrix, (.16) cannot be applied to obtain the standard (A,B,C,D) system representation. An algorithm to solve this problem was presented by Kumar and Daoutidis [KD95,KD96].

A.4 Zeros and Poles of Multivariable Systems Zeros of a system may arise when competing internal effects of the system are such that the output is zero even when the inputs (and the states) are not themselves identically zero. The notion of zeros of a multivariable system adopted here is that of the transmission zeros, which are often defined via the Smith-McMillan form ( see e.g., [Mac89], [ZDG96]). Definition A.4.1 Transmission zero ( z ). Let G(s) be a transfer function matrix, z is transmission zero or simply zero of G(s) if the rank of G(z) is less than the normal rank of G(s). ! Definition A.4.2 Input ( uz ) and Output ( yz ) zero directions. Let z be a zero of G(s), then there exist an input vector direction uz and an output vector direction yz, such that uzH uz=1 and yzH yz=1; and G(z )uz = 0 and yzH G(z ) = 0 . ! We now consider the computation of transmission zeros of a rational transfer function matrix ∆  A − sE B G(s) with a minimal realization (A,B,C,D,E)1. The matrix R(s ) =  is known as D  C Rosenbrock's system matrix [Mac89]. If (A,B,C,D,E)1 is a minimal realization, then the system matrix R(s) loses rank only at the transmission zeros. A numerically stable algorithm for finding transmission zeros of a rational transfer function matrix consists of solving generalized eigenvalue problem given by  A − sE  C 

B   x z ,I  0  . = D  uz  0 

(A.17)


168

where xz,I and uz are the zero input state directions and the input zero directions, with uz satisfying the condition uzH uz=1. The vector xz,I depends on the system realization and it is therefore not unique. Similarly one can compute the zeros z, the ouput zero directions yz, and the output zero state directions xz,O by solving the generalized eigenvalue problem

[x

H z ,O

 A − sE yzH   C

]

B = [0 0] . D

(A.18)

Another possibility for calculating the zero directions is via the SVD of G(z). In this case the zero directions uz and yz are given in the columns of V and U corresponding to the singular value which becomes zero at s = z. The SVD approach can be applied without modifications for irrational transfer function matrices. Remarks about zeros: 1. The presence of zeros implies the blocking of certain input signals ( [SP96], [MZ89] ): If z is a zero of G(s), then there exist an input signal of the form uz ezt 1+(t) where uz is the input vector direction and 1+(t) is a unit step, and a set of initial conditions xz such that y(t) = 0 for t > 0. 2. Pinned zeros. A zero is pinned to a subset of the outputs if yz has one or more elements equal to zero, which occurs whenever a zero z is associated with a particular output. In Chapter 5 we present an example where the zero is pinned to one of the outputs. Similarly, a zero can be pinned to a subset of the inputs if uz has one or more elements equal to zero. Definition A.4.3 Poles. The poles pi of a system (A,B,C,D,E)1 are the generalized eigenvalues of the pencil (A,E), i.e., the nontrivial solutions of Ax = piEx. If E is nonsingular, the poles pi can be calculated by the eigenvalues of the matrix ( E-1A ). ! By using the Smith-McMillan form, it can be shown that p is a pole of G(s) if and only if it is a zero of G-1(s) [Mac89]. Therefore, we will call the input and output zero directions of G -1(s) the output and input pole directions of G(s), i.e., G -1(p) yp = 0 with ypH yp=1 and upH G -1(p) with upH up=1, respectively. If G(s) is a proper rational transfer function matrix with a statespace realization (A,B,C,D,E)1 with a nonsingular matrix D, then, the poles of G(s) and the corresponding output/input directions may be computed as for the case of zeros, i.e., by using a minimal realization of G -1(s) and solving the generalized eigenvalue problems (.17) and (.18). For most systems, the D is singular (normally D = 0), then to apply the generalized eigenvalue for the zero problem to calculate the input and output directions we need to substitute D by D=εI, where ε is a small number that makes small changes in the system dynamics ( usually ε < 10−4 ). A more direct and better approach consists in determining the pole directions from the eigenvectors of the following generalized eigenvalue problems [SP96] At p = p E t p and AT q p = p E T q p (A.19) where tp and qp are the corresponding eigenvectors of the pole p. Thus, the pole directions are given by y p = C t p and u p = BT qP . (A.20)

169

A.5 INPUT AND OUTPUT BLASCHKE FACTORIZATION

Another alternative again is the SVD approach. If G( p) is nonsingular, then it follows from SVD of G-1( p) that the pole directions yp and up are given in the columns of V and U corresponding to the singular value which becomes zero at s = p.

A.5 Input and output Blaschke factorization Here we present the factorization of rational transfer functions in inner and outer transfer matrices, i.e., the factorization of the transfer matrix in the product of an allpass (nonminimum phase ) and a minimum phase factor. It is well-known that a nonminimum phase transfer function admits such an input and output factorization. Here we present a procedure to determine this factorization for RHP-zeros, being also applicable to RHP-poles as will be shown. Input (Blaschke) factorization Let a system G(s) with nz RHP-zeros denoted by zi, then the following procedure can be applied to factorize G(s) as G(s)BI,z(s) : 1. Let (A,B,C,D) be a minimal realization of G(s) and B(0) = B. 2. Repeat for i = 1 to nz  A − zi I B( i −1)   x z ,i  0     = D  uz ,i  0   C B (i ) = B (i−1) − 2 Re(zi )x z ,i u zH,i .

(A.21) (A.22)

Then, the input (Blaschke) factorization of G(s) is given by ) ( 1) G(s) = Gm (s )BI( nz z (s )( BI ,z (s ) %,## &##' BI ,z

(A.23)

where Gm(s) denotes the minimum phase factor of G(s), and BI,z(i) corresponds to the all-pass factor associated with zi calculated as BI(i, )z = I −

2 Re(z i ) u z ,i u zH,i s + zi

(A.24)

and (A, B(nz),C,D) is the state-space realization of Gm(s). ! Note that uz,i does not have to coincide with the input zero directions of the original system G(s), however, it can be readily recognized that it is a linear combination of the zero directions of G(s). The uz,i must be calculated a new in each step, since in each step we have a new B(i) and therefore a new system and new directions. Of course, the first input direction is the same. A very important property of the inner-outer factorizations as the input factorization presented here is BI,z(zi)uz,i = 0, since by definition G(zi)uz,i = 0 ⇔ Gm(zi)BI,z(zi)uz,i = 0 ⇒ BI,z(zi)uz,i = 0. Output (Blaschke) factorization In a similar way we present the formulas for output factorization. Let a system G(s) with nz RHP-zeros denoted by zi then the following procedure can be applied to factorize G(s) as BO,z(s)G(s) :


170

1. Let (A,B,C,D) be a minimal realization of G(s) and C(0) = C. 2. Repeat for i = 1 to nz

[x

H z ,i

 A − zi I yzH,i  ( i −1) C

]

B = [0 0] D

C (i ) =C (i−1) − 2 Re(zi )y z ,i x zH,i .

(A.25) (A.26)

Then, the output (Blaschke) factorization of G(s) is given by G(s) = BO( 1,)z (s)( BO( nz,z ) (s) Gm (s) %##&##' BO ,z

(A.27)

where Gm(s) denotes the minimum phase factor of G(s), and BO,z(i) corresponds to the allpass factor associated with zi calculated as BO(i,)z = I −

2 Re(z i ) y z ,i y zH,i s + zi

(A.28)

and (A, B, C(nz), D) is the state-space realization of Gm(s). ! Note that the output factorization can also be calculated by transposition of the input factorization of the transposed system GmT(s). The inverse of the allpass input and output transfer matrix BI,z(s) and BO,z(s) are easy to calculate by the following equations:  BO−1,z = BO−1,z(1) (s )( BO−1,z(nz ) (s )  BI−,1z = BI−,1z(nz ) (s )( BI−,1z(1) (s )   and  −1(i ) 2 Re(z i ) 2 Re(zi )  −1(i ) H H  BO , z (s ) = I + s − z y z ,i y z ,i  BI , z (s ) = I + s − z u z ,i u z ,i i i  

(A.29)

or alternatively, in a more useful and direct form, since the inverse form is used to factor RHP-poles,  BI , p = BI(,1p) (s )( BI(,npp ) (s )  BO , p = BO(np, p ) (s )( BO(1,)p (s )   and  ( j ) . 2 Re( p j ) 2 Re( p j )  ( j) H H  BI , p (s ) = I + s − p y p , j y p , j  BO , p (s ) = I + s − p u p , j u p , j j j  

(A.30)

The state-space realizations and the procedures for input and output (Blaschke) factorization presented here are based on [ZDG96]. Proofs can be found in [HS96]. An alternative to factorize the system is by the solution of a standard optimal LQ control problem. This procedure is implemented in [RbTB92] (functions iofr and iofc). This inner-outer factorization requires that the system G(s) is stable and to have no jω-axis or infinite poles or transmission zeros. In particular D must have full rank. It means that for stable strictly proper systems changing the matrix D by Dε=εI are necessary if we want apply this factorization. Therefore, we do not prefer to use this method and consequently it is not presented here. The interested reader will found further discussion and references about this procedure in [RbTB92].

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Process

Equipment

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Selection

and

Design",

179

Curriculum Vitae Personal Information Name:

Jorge Otávio Trierweiler

Date of birth:

September 23, 1966

Place of birth:

Porto Alegre - RS / Brazil

Nationality:

Brazilian

Marrital Status:

single

Education Secondary: (1981-1983)

Nossa Senhora das Dores School - Porto Alegre

Higher: (1984-1988):

Chemical Engineering at UFRGS (Federal University of Rio Grande do Sul)

Master of Science: (1989-1990)

Chemical Process Modeling and Control at COPPE/UFRJ (Engineering Post-Graduate Programs Coordenation at Federal University of Rio de Janeiro)

Goethe Institut: (3/1991-9/1991)

German Language Course at Goethe Institut in Göttingen /Germany

Study recognition: (10/1991-10/1992)

At University of Dortmund

Ph.D.work: (since 11/1992)

At University of Dortmund with DAAD scholarship

Dortmund, July 1996