LPV-Based Gain Scheduling - CiteSeerX

Thesis for the Degree of Licentiate of Engineering

LPV-Based Gain Scheduling An H∞-LMI approach

FREDRIK BRUZELIUS

Control and Automation Laboratory Department of Signals and Systems chalmers university of technology G¨oteborg, Sweden, 2002

LPV-Baesd Gain Scheduling An H∞ -LMI approach FREDRIK BRUZELIUS Technical Report No 430L

Control and Automation Laboratory Department of Signals and Systems Chalmers University of Technology SE-412 96 G¨oteborg, Sweden Telephone +46 (0) 31 772 10 00 c 2002 Fredrik Bruzelius  Printed by Chalmers Reproservice CHALMERS UNIVERSITY OF TECHNOLOGY G¨oteborg, Sweden 2002

To my mother Kerstin

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Abstract This thesis considers a robust gain scheduling controller deign method. The gain scheduling technique is one of the most popular controller design methods in practice. However, there is a lack of theoretical results relating stability and performance to the closed loop. Here, so called parameter varying systems (LPV) are investigated and in particular their use in gain scheduling controller design is studied. Instead of the commonly used technique of mapping linear controllers into a scheduled controller, this synthesis method directly results in a parameter dependent controller. An extension of linear H∞ control methods into a parameterized H∞ method has been studied. In this framework, a constrained optimization problem with parameterized linear matrix inequalities is solved. The obtained controller has guaranteed stability and performance properties for the closed loop system. The main contributions in this thesis are extending existing results on; the characterization of the optimization problem in terms of parameterized linear matrix inequalities; practically valid controller realizations in terms of parameter dependence; an algorithm to obtain a parameter dependent state feedback controller. The motivation of this work has been driven by the problem of control design for turbo fan jet engines. The jet engine is nonlinear in its nature, and operates over a wide range in altitude and temperatures etc. Keywords: Gain scheduling, LPV systems, H∞ control, Parameterized LMIs, Jet engine control

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Acknowledgement I wish to thank a finite set, D, of people for the help during the work of this thesis where, supervisors ∈ {Claes Breitholtz, Stefan Petterson} obidicote ∈ {Bo Egardt, Claes Breitholtz, Marcus Rubenson, Stefan Petterson, Cla¨es Lindeborg} phdstuds ∈ {Knut ˚ Akesson, Jonas Fredriksson, Charlie Fransson, Anders Hellgren, Torbj¨orn Liljenvall, Marcus Rubensson, Mattias Henriksson, Petter Falkman, Anna-Karin Christiansson, Fredrik Rosenqvist, Arash Vahidi, Stefan Larsson, Peter Templin, Veronica Olesen, Adam Lagerberg, Karin Eriksson, Hugo Flordal, Birgitta Kristiansson, Peter Nordstrand, Olof Lindg¨arde, Melker H¨arefors, Dan Ring} sysadms ∈ {Lars Jansson, Berndt Andersson} faculty ∈ {Claes Breitholtz, Stefan Petterson, Torsten Wik, Bo Egardt, Martin Fabian, Bengt Lennartson} secretaries ∈ {Madeleine Persson, Catarina Forssen, Ulla-Britt Nilsson} and D = supervisors ∪ obidicote ∪ phdstuds ∪ sysadms ∪ faculty ∪ secretaries

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List of publications The presented work is based on the following publications included in the thesis: I. F. Bruzelius. Gain Scheduling using Linear Parameter-Varying Systems and H∞ Synthesis. Technical Report 003/2001, Department of Signals and Systems, Chalmers University of Technology, G¨ oteborg, Sweden, April 2001. II. F. Bruzelius. On the Computation of LPV Controllers. Technical Report R002/2002, Department of Signals and Systems, Chalmers University of Technology, G¨oteborg, Sweden, January 2002. III. F. Bruzelius. LPV-based Gain Scheduling Technique applied to a Turbo Fan Engine Model. Technical Report R003/2002, Department of Signals and Systems, Chalmers University of Technology, G¨ oteborg, Sweden, January 2002. Parts of the first report were presented at the following conferences: I. F. Bruzelius. Gain Scheduling using Linear Parameter-Varying Systems and H∞ Synthesis. 4th Swedish-Russian Control Conference, Moscow, May 2001. II. F. Bruzelius, C. Breitholtz. Gain Scheduling via Affine Linear ParameterVarying Systems and H∞ Synthesis. 40th Conference on Decision and Control, Orlando Florida USA, Dec. 2001 Parts of the last report have been submitted to the IEEE Control Systems Society Conference on Control Applications 2002.

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Contents Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v Acknowledgement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii Publications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix 1 Introduction Motivation . . . . . . . . . . . . . . Main contributions . . . . . . . . . Outline of the thesis . . . . . . . . 1.1 Robust control . . . . . . . . 1.1.1 H∞ Control . . . . . . 1.1.2 Other approaches . . . 1.2 Gain Scheduling . . . . . . . . 1.2.1 Switching methods . . 1.2.2 Interpolating methods 1.2.3 LPV methods . . . . . 1.3 Optimization . . . . . . . . . 1.3.1 LMIs . . . . . . . . . . 1.3.2 Parameterized LMIs . 1.3.3 Solving LMIs . . . . .

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2 Summary of the papers 19 2.1 Paper I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 2.2 Paper II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 2.3 Paper III . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 3 Concluding remarks 23 3.1 Future work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 Bibliography

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Paper I Gain Scheduling using Linear Parameter-Varying Systems and H∞ Synthesis 29 Paper II On the Computation of LPV Controllers

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Paper III LPV-based Gain Scheduling Technique Applied to a Turbo Fan Engine Model 77

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Chapter 1

Introduction Linear control design techniques are well developed, and the understanding of linear system theory is well established. However, in some cases, linear controllers do not meet the design specifications. Typical cases are when an object to be controlled is nonlinear, and changes its dynamical behavior (from a linear system theory point of view) during operation and the operating conditions change rapidly in the timeframe of the process itself. Linear controller design using standard techniques such as PID, may lead to acceptable performance for some conditions and unacceptably for some. Nonlinear control design techniques are in general hard to carry out, and the interpretation of design parameters is most often non-intuitive. Gain Scheduling is a method compromising between the intuitive linear system theory and the nonintuitive nonlinear theory. From another point of view, Gain Scheduling is an extension of linear control design techniques. There are two branches of the modern linear controller design paradigm, 1. Robust Control 2. Adaptive Control Under the robust control design paradigm, the idea is that the process can be described as a linear system, with additional unmodeled dynamics such as nonlinearities, immeasurable noise influences etc. A linear controller is then sought such that the designers’ specifications are met for all possible uncertainties. The adaptive control paradigm basically states that the process to be controlled may be linear, but changes in a time frame substantially slower than the dynamics of the closed loop system. The controller task is to adapt its parameters to the current linear system. A definition of adaptive control is, a linear (or possible nonlinear) controller that has a mechanism that updates the controller parameters to the changing process dynamics. The gain scheduling method may be included in this paradigm, with the distinction from traditional adaptive controllers that the updating mechanism is computed off-line rather than on-line. 1

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Chapter 1 Introduction

In this work, a design method that combines the adapting mechanism of a gain scheduling controller with the ideas of robust control, or as it will be presented here, a robust controller that with additional information about the process achieves better performance. Through out this work, the motive of practical use of the theoretical results in terms of numerical solvability is strongly enforced. This is why only affine parameter dependent systems are considered.

Motivation The main motivation of the work has been the lack of systematic controller design tools for gain scheduling. The turbo fan jet engine, which is nonlinear, and operates over a wide range of conditions such as altitude, temperature and intake velocity, has been the driving application. Gain scheduling has been an accepted technique to control jet engines for quite a while. However, the used techniques have been ad hoc and extensive simulations the way to ensure that performance specifications are met. A framework that is based on theoretical results has the potential to reduce the simulation step in the development of new engines and achieve better performance. An essential part of this work has been financed by the EURAM project OBIDICOTE, [1]. The aim of this project was to investigated possible gains of using an on-board model of the jet engine to achieve better control and diagnostics.

Main contributions The main contributions of this thesis are the following: • A novel characterization of so called parameterized linear matrix inequalities in the form of linear matrix inequalities. This result enables a better understanding of parameterized linear matrix inequalities and the possibility to solve such problems, without brute force gridding. • A less conservative scheme is deduced to achieve a practically valid controller realization in terms of operating condition dependence. • A reduction of the number of linear matrix inequalities that gives a gain scheduling controller is obtained. • A non-dynamic robust gain scheduled H∞ output feedback control design technique is presented.

1.1 Robust control

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Outline of the thesis The thesis starts with an introduction to robust control, especially H∞ control design, but other similar robust control approaches are also briefly discussed. The gain scheduling technique is discussed in section 1.2, including some different approaches to this problem, such as switching strategies, interpolating methods and linear parameter varying methods. The main tool of finding a robust gain scheduling controller in this work is based on so called linear matrix inequalities (LMIs), which are briefly reviewed in section 1.3. A special extension to LMIs, frequently encountered in this thesis, namely parameterized LMIs is also reviewed. Finally, numerical methods on how to obtain a solution to an LMI is given. The intention of the first chapter is to put this work into context and shortly describe relevant material on which the appended papers rely on. Chapter 2 summarizes the results of the three reports, which are included in the thesis. The first deals with the parameterization of H∞ control design problems in general in terms of parameterized LMIs. Also treated are some basic properties of linear parameter varying (LPV) systems, such as stability and gain. For this to be used in nonlinear systems, a transformation or approximation into an LPV system needs to be done. Issues associated with these types of questions are discussed and illustrated by example of a 3-state compressor model. The second report, considers two specific problems associated with H∞ LPV control design, namely a possible way of turning parameterized LMIs into a sufficient finite set of LMIs and how to obtain a practically valid controller. In the last report, a design example is given to a turbo fan engine model, using a special linearization technique, and, in some sense, a “static” parameter dependent state feedback controller is designed. Concluding remarks and possible directions for future work is given in Chapter 3.

1.1

Robust control

In this section, the basic ideas of robust control are reviewed. Special attention is given to the H∞ control design technique. In many applications, classical control design techniques such as PID tuning result in sufficient performance. The use of more complex techniques is then hard to motivate. However, in a situation when the process is poorly modeled, has a complex nature, or the performance specifications are particularly emphasized, robust control deign techniques can be a possible way. Here, the view of robust control in a perspective of uncertainty attenuation is given. The term uncertainty, includes here both disturbances and unmodeled dynamics. Given a nominal process description, G, and a description of the uncertainties, the basic idea behind linear robust control synthesis is to find a controller that is stable and meets some performance specification, for all possible plants and disturbances

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Chapter 1 Introduction

within the uncertainty description. Most often the uncertainties are described in the form of a feedback connection, see figure 1, with a norm bound specification on the block ∆ or as a parametric uncertainty of the coefficients of the transfer function.

- ∆

z

w G

 

y

u - K

Figure 1: A common linear robust control setup, in the so-called linear fraction form.

From this, an optimal control problem can be formulated. The ideas of norm bounds were introduced by Zames in the pioneer work [36], where the problem of H∞ control is cast as an optimization problem, subject to constraints, all in frequency domain. However, the ideas of robustness in controller design is much older. For example in the 1950’s Horowitz developed the quantitative feedback theory [17] The rest of this section is devoted to the H∞ control design approach for linear time invariant systems in the linear matrix inequality framework. The main contributions of this thesis are different extensions to parameter-varying systems in the same framework, and are presented in the included papers. Two more approaches of robust control are also very briefly presented here: quantitative feedback theory, for a historical insight and µ-synthesis, which is a more resent synthesis method.

1.1.1

H∞ Control

The foundation of H∞ control design and analysis framework is the bounded real lemma and the small gain theorem. The small gain theorem, see e.g. [21], is a general result that relates stability and feedback without making assumptions on the involved systems in terms of nonlinearities etc.

Theorem 1.1.1 Consider the connected system,

1.1 Robust control

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u1

-

6

y2 

- y1

S1

S2

 ? 

u2

assume that S1 : L2 → L2 and S2 : L2 → L2 , are individually stable, and that S1 L2 S2 L2 < γ ≤ 1 then the connected system is stable. The generality of the small gain theorem is to the cost of conservatism. For example, if S1 and S2 are single input single output stable linear time invariant systems, the small gain theory states that the corresponding Nyquist curves must be contained inside the unit circle in the complex plane. Let for example S1 represent the nominal process G with the feedback K in figure 1 and S2 the uncertainty block ∆. The idea of H∞ control design is to find a controller K by minimizing the H∞ norm of system S1 , so that as large uncertainty S2 as possible can be handled by the controller. That is, S1 L2 = Gc (s)H∞ = sup Gc (iω) ω

where Gc (s) denotes the stable transfer function matrix of the closed loop and the norms are induced norms. This implies, using the small gain theorem, that the largest stable uncertainty that can be handled by the control system is, ∆L2 ≤

1 , γ

if Gc (s)H∞ < γ. The H∞ problem is then an optimization problem, min Gc (s)H∞ K

Gc (s) stable In a state space representation, the Bounded Real Lemma represents one formulation of this optimization problem, see i.e. [37]. Theorem 1.1.2 The stable linear time-invariant system, x(t) ˙ = Ax(t) + Bu(t) y(t) = Cx(t) + Du(t) has an H∞ norm less then γ, if there exists a matrix P = P T > 0 such that AT P + AP + (B T P + γ −1 DT C)T (γI − γ −1 DT D)−1 (B T P + γ −1 DT C) + γ −1 C T C < 0

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Chapter 1 Introduction

The Bounded Real Lemma can be used on the closed loop system Gc (s), to find a controller that meets the γ specification. The uncertainty can be described by an augmentation of the nominal process. In short, this is done by picking (possibly frequency weighted) inputs w and outputs z in figure 1. A possible augmentation of the nominal plant is given in figure 2, where z1

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W1 r

w1

w2

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W2

W3

6 - W6

6

- K

u - ?- G 0

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?y

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z2

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Figure 2: Structure of an augmented plant with weighted auxiliary inputs wi and outputs zi . Wi are LTI filters. A method of choosing those filters is to think of them as inverses of upper constraints on the corresponding transfer function, see e.g. [37] or any other standard text in the field for details. These weighting filters are often chosen as design parameters without any obvious physical interpretation, and therefore is this process a nontrivial procedure. The optimization problem, expressed as the Bounded Real Lemma for the closed loop system can be reformulated as an LMI optimization problem and an LMI feasibility problem, both convex, see e.g. [13], [12],[19] for details.

1.1.2

Other approaches

Other approaches to the robust control design problem have been presented in the literature. An early method is the Quantitative Feedback Theory (QFT). Horowitz introduced this method in the 1950’s, see e.g. [18] and references therein. The main idea is to achieve specified performance despite specified range of plant uncertainty, and doing so with minimum cost of feedback. This is done by expressing the uncertainty as a parametric uncertainty in the transfer function, to get a set of possible transfer functions G ∈ {Gi }ni=1 . Consider a specification on, for example the, sensitivity function Si (jω) ,



1

≤κ, max
i 1 + K(jω)Gi (jω)


∀ω ∈ R+ ,

1.2 Gain Scheduling

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The union of the domains for which the specification is not fulfilled is called Horowitz bounds. In for example a Nichols chart, the Horowitz bounds can be plotted, together with the nominal open loop. The objective is then to shape the controller such that all Horowitz bounds are not enclosed by the open loop system. To be able to compute this, a discretization of the frequency is performed, ωk ∈ Ω. For a comprehensive and more in depth description of the QFT design method see e.g. [18]. Yet another approach to robust control design is the more recent Structured Singular Value, or µ synthesis method. These ideas were developed by Doyle and Safonov independent of each other, see [26] and [11]. The basic idea here is, instead of using the bounded real lemma and the small gain theorem to determine a controller, to use an extension of the Nyquist theorem to multi-input multi-output systems. This corresponds to finding the smallest uncertainty ∆ such that, det(I − ∆G0 (jω)) = 0 with the same notation as in figure 1. The so called structured singular value or µ of a matrix M is defined as  −1 σ (∆) : det(I − ∆M ) = 0} , µ∆ (M ) = min {¯ ∆∈∆

where ∆ is an assumed block diagonal uncertainty structure. The µ value is in general hard to compute and a scaled version of the small gain theorem is used in analysis and synthesis for an estimate of an upper bound on µ. By letting the controller having access to parts of the uncertainty, a robust gain scheduled controller can be obtained, see e.g. [16] for a more complete description of µ synthesis and its use in robust gain scheduling.

1.2

Gain Scheduling

An adaptive control system is a system in feedback that updates its feedback parameters, by use of some mechanism, to reflect the changes of the process in control. See figure 3. The main idea behind adaptive control, is to treat the process as a linear system in the time frame of the control loop, and design a linear controller in the parameters of this system. The process parameters are assumed to vary in a much slower time frame, and change the linear dynamical behavior of the system. An additional loop is applied to the feedback system, which updates the parameters of the feedback law, according to the changes in the process. This separation in time of the two loops, enables a simpler type of synthesis and analysis compared to general nonlinear control systems. The mechanism that updates the control law parameter can be designed in numerous ways, all relying on the assumption that the rates of change of the process parameters can be neglected when considering the main control loop. There are two main approaches to design the updating mechanism: model reference, and self-tuning. Model reference adaptive systems use a

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Chapter 1 Introduction

-

Setpoint

Parameter adjustment

 

Controller parameters

-

?

Controller

-

Control signal

Plant

Output -

Figure 3: Block diagram of a adaptive control system.

reference model with a desired behavior and minimize the difference between the process output and the the reference model output. Self-tuning adaptive systems, use on-line system identification to obtain the process parameters. The parameters of the controller are parameterized in the process parameters in some way and updated by the system identification. Gain scheduling controllers update its feedback parameters via an off-line schedule. The schedule is parameterized in the operating condition, which is assumed to be measured on-line. Since the updating mechanism in gain scheduling is computed off-line, analysis of such systems are much easier than for example an adaptive controller using the model reference concept. For a comprehensive text on the topic of adaptive control in general, see e.g.[2]. The idea of adaptive control began in the 1950’s in the application of autopilots for aircraft’s. The need for a controller with good performance for different altitudes led to a gain scheduling type of controller. However, the theoretical understanding of nonlinear systems, system identification and other related topics impeded the development of the adaptive control techniques. It was not until in the 1980’s commercial adaptive controllers appeared. More recently, effort has been put into merging the ideas of robust control with adaptive control, including this thesis. The development of the gain scheduling technique has, since the 1950’s, been application driven. The lack of theoretical results, such as stability analysis, explains why the method has been less accepted in the control academia, [25]. Recently, gain scheduling has drawn attention and theoretical results connecting stability to “common sense” have been made, see e.g. [27], [28]. The rest of this section is devoted to describe how gain scheduling schemes can be parameterized and how it affects properties of the controlled system. Also, a type of methods is given, which solves a parameterized problem and results in a scheduled controller.

1.2 Gain Scheduling

1.2.1

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Switching methods

The method of gain scheduling is considered to be a quasi-stationary design method. Quasi in the sense that the process is assumed to behave locally as a linear system and that the changes of the operating condition are slow. This enables a local design of linear controllers. The nonlinear process, x(t) ˙ = f (x(t), u(t)) y(t) = h(x(t), u(t)) can be linearized in stationarity (x0 , u0 ), ( f (x0 , u0 ) = 0 ), δ x(t) ˙ =

∂f (x0 , u0 )δx(t) ∂x

+

∂f (x0 , u0 )δu(t) ∂u

δy(t) =

∂h (x0 , u0 )δx(t) ∂x

+

∂h (x0 , u0 )δu(t) ∂u

where δx = x − x0 etc., and the resulting linear system can be used in the control design of a linear local controller. This is most often performed in specific points of the operating range, such that a set of linear systems is obtained. The resulting local controllers can be used in different parts of the operating range, depending on the current operating condition in a switching strategy, see figure 4. - K -Op. cond. 1 6

... ... .

?

-

G

-

- Kn -

Figure 4: Block diagram of a gain scheduling controller using a switching strategy. Despite the fact that stabilization of linearizations implies local stability of the nonlinear feedback system, there are no guarantees that stability is preserved when the operating conditions changes. The closed loop system must be verified in some post-analysis, such as simulations. Also, when implementing such a control system, one must be careful so that inactive controllers do not drift away. However, this type of methods are frequently used in practical applications, see for example [15].

1.2.2

Interpolating methods

An alternative to switching strategies to connect the linear controllers into a gain scheduled controller is instead to interpolate between the linear controllers. This

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Chapter 1 Introduction

can of course be done in numerous ways. For example, interpolation of poles, zeros and gains, is treated in [23]. Another example is to use all linear controllers and blend the output as in [20]. These ad hoc mappings suffer from the fact that there is no guarantee of stability in an operating point between the design points. Stability preserving interpolation schemes have been presented in the literature, [30], where the locally designed controllers are interpolated in such a way that the resulting scheduled controller stabilizes the process for all admissible operating conditions. With a slow varying operation argument, stability of changing operating conditions can be established.

1.2.3

LPV methods

A recent method of gain scheduling uses a process description of the so called linear parameter varying (LPV) systems,

x(t) ˙ = A(ρ(t))x(t) + B(ρ(t))u(t) y(t) = C(ρ(t))x(t) + D(ρ(t))u(t)

where ρ is the variable describing the operating conditions. The operating conditions can be described by both endogenous and exogenous signals, for example the power extraction and the intake Mach number for a Jet engine. The parameter vector ρ is assumed to be a measurable quantity, with an a priori knowledge of the bounds in the form of a connected compact set ρ(t) ∈ Γ. If the parameter of the system contains endogenous signals such as the output, the term linear in LPV becomes inadequate. To point out that this is a nonlinear system on the form of an LPV system, the term quasi-LPV is used. However, in the design process these systems are not treated differently than LPV systems. Therefore LPV systems as well as quasi-LPV systems are most often referred to as LPV systems. A parameter dependent controller is designed based on this process description. Therefore no mapping of linear controllers is necessary. However, the transformation/approximation of a nonlinear process model to an LPV system is a non-trivial task, see e.g. [22]. Most LPV methods use modern design techniques, such as H2 or H∞ control, in a parameter dependent framework, see e.g. [7],[34],[10]. The set of admissible parameter values can be treated in a direct manner. In addition bounds on the rates of change of the parameters can be incorporated to obtain a less conservative controller. The resulting controller has a stability and performance guarantee in the pre-defined operating region. However, a potential problem with these types of methods is the lack of a controller satisfying the performance specifications.

1.3 Optimization

1.3

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Optimization

In this section, a survey is given of so called Linear Matrix Inequalities (LMI) and how they can be solved. As this is an essential part of how a robust gain scheduled controller can be obtained numerically, LMIs deserve some attention. Linear matrix inequalities, were first used in the work of Lyapunov in the context of stability analysis for dynamical systems in the 1900’s, but it was not until the 1940’s Lur’e, Postnikov and others used inequalities to solve the problem by hand in control engineering problems. In the 1960’s graphical methods were developed to solve specific LMI problems in particular by Yakubovich, see e.g. [35], and in the 1970’s algebraic solutions to a related matrix equality, the Riccati equation, were found. The evolution of computation power, led to the development of algorithms to solve LMI problems using convex optimization. Recently, in the 1980’s, modern efficient algorithms, like interior-point methods, were applied to LMI problems. The efficiency of the algorithms enabled practical use of LMIs to many different fields in control theory such as robust control, optimal control, interpolation problems etc., see e.g. [24] and [8]. The following subsection is devoted to some basic properties of LMIs and to some standard results frequently used in this thesis. It is followed by a subsection on an important extension of LMIs, namely parameter dependent LMIs. For a complete picture of the use of LMIs, two numerical algorithms to solve LMI problems with a linear objective function are reviewed.

1.3.1

LMIs

A linear matrix inequality is an affine inequality with matrix coefficients, that is, F (x) = F0 + F1 x1 + F2 x2 + . . . Fn xn < 0

(1)

where Fi ∈ Rm×m , x ∈ Rn and xi denotes the scalar components of the vector x. Therefore, a more natural name for LMIs would be Affine Matrix Inequalities. However, the analogy with linear scalar inequalities, ax > b, gives a consistency. The interpretation of the inequality sign in an LMI is negative definiteness of the left hand side matrix. In other words, for all non-zero vectors, y T F (x)y < 0,

∀y ∈ Rm

Observe that it makes no sense to formulate a non-symmetric LMI, since the corresponding symmetric LMI 0.5(F T (x) + F (x)) is equivalent. The set of feasible solutions to (1), i.e. the set of x that makes the left hand side of (1) negative definite, is convex. This is due to the affinity of (1). If x and x˜ are two

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Chapter 1 Introduction

arbitrary solutions to (1) then, F (αx + (1 − α)˜ x) = αF (x) + (1 − α)F (˜ x) < 0, for 0 ≤ α ≤ 1, which implies that αx + (1 − α˜ x) also is a solution. Convexity is an important property when a solution is sought numerically. Most frequently, LMIs are represented by a matrix inequality with matrix variables. As an example the well-known Lyapunov inequality could be mentioned, AT P + P A < 0, P T = P > 0. These types of LMIs can easily be transformed into the form of (1), using a basis for the P matrix. The most obvious basis is,   x 1 x2 x4 . . .  x2 x3 x5 . . .    P =  x x x ...  4 5 6   .. .. .. . . . . . . which includes that P is symmetric, that reduces the number of decision variables from n · n to n · (n + 1)/2. However, the corresponding Fi ’s in (1) may not be independent and additional variables may be reduced depending on A, to achieve a better numerical condition for the problem. Semi-definite Fi ’s in (1) also may cause numerical problems. However, these can quite easily be reduced using orthogonal complement to LR factorization of those particular Fi , see i.e. [8] for details. Linear or affine matrix inequalities with matrix variables are called LMIs without giving the basis for the variable that turns it into the form of (1). A useful property of LMIs is that some matrix inequalities that are both quadratic in the matrix variables and convex, can be turned into an equivalent linear (affine) inequality, see e.g. [8]. This is done by use of the so-called Schur complement. The matrix inequality, R > 0, S(z) + G(z)T R−1 G(z) < 0 where R = RT , S = S T and z are the variables, has the same set of solutions as,

 S G(z) < 0. GT (z) −R The Schur complement is a result of the fact that an LMI is invariant to congruence transformations, P > 0 ⇔ TTPT > 0 for any non-singular T . It turns out that even matrix variables can be eliminated using congruence transformations. The following result is frequently encountered in for example H∞ synthesis. The LMI, G + U XV T + V X T U T < 0 where the matrix variable X, is feasible, i.e. there exist a solution, if and only if, U⊥T GU⊥ < 0, V⊥T GV⊥ < 0 where U⊥ and V⊥ are the orthogonal complements of U and V respectively, that is U⊥T U = 0 and V⊥T V = 0.

1.3 Optimization

1.3.2

13

Parameterized LMIs

The synthesis problems of robust gain scheduling and robust controllers with parametric uncertainties can be casted as a problem with parameterized linear matrix inequalities. A parameterized LMI is an extension of an ordinary linear matrix inequality with the difference that it is dependent of a parameter vector. That is, the parameterized LMI should hold for a set of parameters, F0 (θ) + F1 (θ)x1 + . . . + Fn (θ)xn < 0 ∀θ ∈ Θ.

(2)

Observe that the parameter θ is not a decision variable. For every value of the parameter θ ∈ Θ (2) represents an LMI, i.e. (2) can be represented by an infinite set of LMIs. This fact makes parameterized LMIs hard to solve numerically. In fact, it can be shown that they cannot be solved using polynomial time algorithms. In a polynomial algorithm, the growth of space (storage) and time (number of computations) follows a polynomial function in the problem size. Polynomial algorithms are considered to be efficient in contrast to non-polynomial algorithms. For the application of this work, the set Θ is always a connected compact set, a hyper-cube, Θ = {θ ∈ Rp : θi ≤ θi ≤ θ¯i ,

i = 1, . . . , p}.

(3)

For some parameter dependence in (2), a finite set of sufficient (and in some cases also necessary) LMIs can be derived , for example by evaluating the parameterized LMI for different values of θ in the set Θ. This implies that polynomial time algorithms can be used to solve such parameterized LMIs. An example of when a necessary and sufficient finite set of LMIs can be obtained is when the Fi (θ)’s in (2) are linear (or affine) in θ and the set Θ is a hyper-cube, like (3). The vertices of Θ play an important role in finding this set of LMIs. These can be described by the set of 2p elements, vertΘ = {θ|θi = θi or θi = θ¯i } With an affine parameter dependence in Fi (θ), Fi (θ) = Fi0 + Fi1 θ1 + . . . + Fip θp ,

i = 0, . . . , n

a transformation, by evaluating the Fi ’s in the vertices of Θ, Fi (θ) =

p k=1

αk Fi (vertΘk ),



αk = 1, αk ≥ 0

k

results in an LMI of the form, p

2

αk (F˜1k x1 + . . . + F˜nk xn ) < 0

k=1

where F˜ik are Fi (θ) evaluated in vertex k. Now, the parameterized LMI is feasible whenever, F˜1k x1 + . . . + F˜nk xn < 0, k = 1, . . . , 2p

14

Chapter 1 Introduction

are feasible. Linearly dependent parameterized LMIs arise for example in a stability test of a parameter uncertainty system. The autonomous system x˙ = A(θ)x, where θ is a time varying uncertainty with bounds like (3) and A(θ) = A0 + A1 θ1 + . . . + Ap θp , is stable if (4) AT (θ)P + P A(θ) < 0, P = P T > 0, ∀θ ∈ Θ. Using the same discussion as above, the parameterized LMI (4) is feasible if and only if the corresponding LMIs that are obtained by evaluating (4) in the vertices of Θ is feasible. However, if the time varying uncertainty θ has known bounds of its ˜ then (4) is conservative. That rate of change, for example θ˙ is in a hyper-cube Θ, is, the system may be stable when (4) is strictly infeasible. A less conservative test is then proposed by letting the Lyapunov matrix P depend on the uncertainty, AT (θ)P (θ) + P (θ)A(θ) +

∂P (θ) k

∂θk

˜ θ˙k < 0, P (θ) = P T (θ) > 0, ∀θ ∈ Θ, θ˙ ∈ Θ.

This is an example of a partial differential inequality that can be solved as a finite set of LMIs, by putting some restrictions on the solution. First, assuming a structure of the θ dependence of P (θ), for example by letting P (θ) have an affine θ dependence. Now, the partial differential inequality is reduced to a parameterized LMI with a quadratic parameter dependence. In fact, parameterized LMIs with quadratic dependence are important special cases of parameterized LMIs. The vertex reasoning does not apply for quadratic dependence. Additional constraints must be added to ensure negativeness of the parameterized LMI for all admissible values of θ. The problem boils down to characterize when a quadratic function is negative on a hyper-cube, via f (θ) = uT F (x, θ)u , ∀θ ∈ Θ for an arbitrary non-zero vector u and a fixed x. Keeping in mind that searching for a maximum is pointless, since this maximum is too computationally demanding to obtain. The so-called multi-convexity property of a function, which is a weaker form of convexity, guarantees that the maximum of the function is obtained in some vertex. A quadratic function is multi-convex if the second derivatives with respect to each variable are non-negative. This means that the parameterized LMI can be ˜ and the second derivative reduced to a set of LMIs formed by the vertices of Θ and Θ of the parameterized LMI with respect to each parameter vector component, see [4], in our example, AT (θ)P (θ) + P (θ)A(θ) +

p

˙ ∈ vert(Θ, Θ) ˜ θ˙k Pk < 0 , ∀(θ, θ)

k=1

ATi Pi + Pi Ai ≥ 0, i = 1, . . . , p Another approach is to make the quadratic function monotone in each axial direction of the parameter space. Using a monotone characterization, only one side of the interval of the hyper-cube and the derivative for each axial direction needs to be evaluated and represented as an LMI, see e.g. [31] and [32]. The necessary set of

1.3 Optimization

15

LMIs becomes smaller than in the case of multi-convexity. However, there are many possibilities of characterization, negative and positive monotonicity in each axial direction. For example, p ¯ T P (θ) ¯ + P (θ)A( ¯ θ) ¯ + θ˙k Pk < 0 , ∀θ˙ ∈ vertΘ ˜ A(θ) k=1

ATi P (θ) + AT (θ)Pi + Pi A(θ) + P (θ)Ai ≥ 0 , ∀θ ∈ vertΘ where θ¯ denotes the upper vertex of Θ. Both multi-convexity and monotonic characterization introduce conservatism. That is, the feasibility set is decreased in comparison to the original parameterized LMI problem. The two methods introduce different conservatism, and what may be viewed as a mixture of the two is presented in [9]. A scalar multiplier is introduced that indicates if the direction is monotone or multi-convex, that is, AT (θ)P (θ) + P (θ)A(θ) +

p

˙ ∈ vert(Θ, Θ) ˜ θ˙k Pk < 0 , ∀(θ, θ)

k=1

  αi ATi P (θ) + AT (θ)Pi + Pi A(θ) + P (θ)Ai ≥ ATi Pi + Pi Ai i = 1, . . . , p , ∀θ ∈ vertΘ where αi are scalar constants. Observe that a constant P is a special case of the three characterizations, multi-convexity, monotonicity and the mixture. This implies that the use of a constant P matrix is a conservative special case, even when the parameterized LMI is reduced to a set of LMIs. However, all three relaxation methods are in practice to conservative to use directly to the parameterized LMI. To reduce the conservatism of the relaxation methods, the methods are used on an upper bound parameterized LMI containing the original parameterized LMI plus a simple guaranteed positive quadratic matrix function, see [4],[32],[9] for details. A brute force strategy to grid the parameter space in a lattice, and evaluate the parameterized LMI on this to obtain a finite set of LMIs is possible only if the dimension of Θ is low. As the dimension grows in size, the number of LMIs explodes in quantity. Other approaches use polyhedrons in the parameter space to find necessary LMI conditions such as in [5] and [6].

1.3.3

Solving LMIs

As mentioned above, the true strength of using LMIs is that there exist efficient optimization algorithms that solve LMI problems in polynomial time (in computation). In fact, the numerical algorithms are even more efficient than most matrix manipulations such as matrix inversion and solving Riccati equations. There are a couple of software packages to solve LMI problems, for example the LMI Control Toolbox for Matlab [14], and LMITOOL which is a front end package to be used with the optimization code SP, [33], also under Matlab. All computation of LMI problems in this thesis was done using the LMI Control Toolbox. Here, two standard numerical methods are reviewed.

16

Chapter 1 Introduction

The optimization problem encountered in this thesis has the form, min cT x x

(5)

B(x) > 0

where B(x) is a linear matrix inequality of the type (1). These types of problems are in the literature often called eigenvalue problems, due to the fact that it can be recasted to an equivalent problem of minimizing an eigenvalue, see [8] for details. One of the simplest algorithms to solve problems like (5) is called the Ellipsoid algorithm. It was, developed in the 1970’s by Shor, Nemirovskii and Yudin, see e.g. [29]. The basic idea behind this algorithm is to, given a starting ellipsoid that contains the optimal solution, cut the ellipsoid in half. Then make a minimal volume ellipsoid enclosing that half that contains the optimal solution, and iterate until the optimum is reached. In more detail, this can be done as follows. An ellipsoid can be described by, E = {z | (z − a)T A−1 (z − a) ≤ 1} The algorithm is divided into a feasibility step in each minimization step. If a starting point x is infeasible, that is B(x) ≯ 0, find a vector u such that,   m xi Bi u ≤ 0, uT B(x)u = uT B0 + i=1

using standard numerical computation. Let g be defined by gi = uT Bi u for i = 1, . . . , m. The vector g is now a cutting plane for the starting ellipsoid. The cutting plane cuts the starting ellipsoid in two, and the half-space {z|g T (z −x) < 0} contains a feasible set, if it exists. To see this observe that, uT B(z)u = uT B(x)u − g T (z − x). Now, the minimum volume ellipsoid containing the half-space is given by, E˜ = {z | (z − a ˜)T A˜−1 (z − a ˜) ≤ 1} where A˜ and a ˜ are, Ag  a ˜ =a− , (m + 1) g T Ag

A˜ =

m2 m2 − 1



  2 T T A− A g Agg AgA . m+1

The next point in the iteration k, can be calculated as x(k+1) = x(k) −

 −1/2 (k) 1 g . A(k) g (k)T A(k) g (k) m+1

If the obtained x is infeasible, find a new u and a smaller ellipsoid and iterate until x satisfies B(x) > 0. The obtained ellipsoid contains the feasibility set of the problem, including the interior point x. To minimize the objective function cT x, define the c ˜ as the cutting plane, and find a minimum volume for the set {z | cT (z − x) < 0} ∩ E, that is the part of the latest ellipsoid where the objective function is smallest. Find

1.3 Optimization

17

a feasible point for this ellipsoid and define c as the cutting plane until optimum is reached. It can be shown that the ellipsoids volume in the feasibility part decreases exponentially in the iteration, which indicates a polynomial time convergence of the algorithm. Numerically, however, interior-point methods are more efficient. For example the LMI Control Toolbox for Matlab uses an interior-point method. Interior-point methods use so called barrier functions and analytical centers of these in the iteration along the path of center to obtain the optimum of the eigenvalue problem (5). These concepts will be described briefly here. The optimization problem, λopt = inf {cT x : B(x) > 0}, x

has the property that for each λ > λopt ,

 B(x) 0 Gλ (x) = > 0, 0 λ − cT x

(6)

for all feasible x. The barrier function,  log det G−1 λ (x) x ∈ X φλ (x) = ∞ x∈ / X, where X is the feasibility set of (6), defines the analytic center of (6) through, x∗ = arg min φλ (x). The curve x∗ (λ) for λ > λopt is called the path of center of the optimization problem. It can be shown that x∗ (λ) is a smooth function and has a limit as λ → λopt which is the optimum of the problem. The analytical center can be obtained by use of standard Newton algorithm, given an initial value of x ∈ X and a λ, as x(k+1) = x(k) − α(k) H(x(k) )g(x(k) ) where α(k) is the step-length in the iteration step, H and g are the Hessian and the gradient of φ respectively, gi = −tr(G−1 (x)Gi ),

Hij (x) = −tr(G−1 (x)Gi G−1 (x)Gj )

where tr denotes the trace of a square matrix. This corresponds to solve a total least squares problem of the same order as the original problem. Here, the problem structure, that is the data Bi can be exploited to achieve a more efficient algorithm. The step-length can be obtained using different strategies, which guarantee that the next iterate is feasible. The most efficient is a simple line search. Different ways are possible to perform the optimization over λ when a x∗ (λ) is obtained. One of the simplest is the method of center, λ(l+1) = (1 − θ)cT x(l) + θλ(l) x(l+1) = x∗ (λ(l) )

18

Chapter 1 Introduction

where θ is a fix constant with 0 < θ < 1 and x∗ (λ(l) ) is obtained as described above in the k-iterations. Other, more efficient types of method to minimize λ, are the so-called primal and dual methods. These use a dual description of the problem in the form of an LMI, see e.g. [8] and references therein.

Chapter 2

Summary of the papers 2.1

Paper I

Gain Scheduling using Linear Parameter-Varying Systems and H∞ Synthesis In this paper, linear parameter dependent (LPV) systems are investigated, especially how they can be used in a gain scheduling control design framework. The most fundamental property of a system, stability is treated in a general way, that is using standard Lyapunov theory. This leads to the problem of finding a Lyapunov function. For true linear time-invariant systems, it can be shown that quadratic functions are necessary and sufficient to prove stability of a linear system. For LPV systems, a quadratic Lyapunov function is potentially conservative. An extension of a quadratic function with a constant Lyapunov matrix in the quadratic form is to let this matrix depend on the parameters of the LPV system, in a general way. As LPV systems are (if the parameter is true exogenous) linear time-variant systems with in advance unknown parameters, the parameter dependence is a logical extension. As analysis and synthesis tool, a parameter dependent extension of the bounded real lemma is used. In the synthesis case, the resulting closed loop system is used in the lemma. If a controller is sought that satisfies the control design specification, that is the augmentation, two difficulties arise. Firstly, the bounded real lemma is not linear, but bi-linear in the decision variables (the controller matrices and the Lyapunov matrix), and secondly, the problem is parameterized. The controller matrices dependence can be eliminated, such that only a part of the Lyapunov matrix is sought in the first step. This problem is convex in the decision variables. Then a controller can be computed explicitly in the parts of the Lyapunov matrix obtained and in the plant data, or using the bounded real lemma, which now is linear in the controller. A simplified expression for a controller is presented. A non-parameterized problem can be solved that gives sufficient conditions for the

19

20

Chapter 2 Summary of the papers

parameterized one. The non-parameterized one assumes a certain structure of the parameter dependence of the solutions, namely an affine dependence. For this to make sense, it is assumed that the plant has the same dependence. In the paper, an additional reduction of the constraints for the non-parameterized problem is presented, which makes the related optimization faster to solve. As the goal is to use the synthesis method to nonlinear systems, the mapping of nonlinear system to linear parameter varying system is discussed. An indirect measure, on how well a nonlinear system will behave given an approximation on the LPV form, is presented. The synthesis method is illustrated on a 3 state Moore-Greitzer compressor model example using an analytic transformation of the nonlinear model to an affine LPV system.

2.2

Paper II

On the Computation of LPV Controllers An important special class of linear parameter varying systems is LPV systems with affine parameter dependence. This class of systems is considered in this paper. When using a parameter dependent Lyapunov matrix, a parameterized linear matrix inequality solves the existence problem of the controller. An ad hoc, but reasonable assumption is to let the parameter dependent Lyapunov matrix mimic the plant data parameter dependence that is affine in this case. The existence problem then becomes a parameterized LMI with quadratic parameter dependence. Parameterized LMIs are hard to solve numerically, but due to the quadratic dependence, sufficient conditions in terms of LMIs can be derived. It all boils down to characterize whenever a quadratic function is negative, or whenever it is sufficient to check vertices of the domain of the function for negativeness. Here, a new way of doing so is presented which combines so called multi-convexity and monotonicity of the function. The existing methods of finding a controller, given a solution to the existence problem, most often contains the time derivative of the parameters. This is however not desirable, since the time derivative of the parameter is not a measurable quantity. The practical use of such a controller is therefore limited. A way to avoid this problem is to put further constraint to the solution of the existence problem. Here, less conservative constraints are introduced in comparison to earlier work. The presented approaches are applied to a missile example.

2.3 Paper III

2.3

21

Paper III

LPV-based Gain Scheduling Technique applied to a Turbo Fan Engine Model One of the largest shortcomings of using linear parameter varying systems in gain scheduling control design is the lack of good methods of rewriting/approximating nonlinear systems to LPV systems. This is especially true when the nonlinear model is of high order and when the right hand side of the model contains look-up tables and implicit functions. Most often a set of linearizations is mapped into an LPV model using interpolation or least square fitting to a simple function. However, there are no results that relate such a mapping to a nonlinear plant. In fact, closeness to stationarity is essential, and only local validity is obtained. In this paper, the use of so-called velocity based linearization is investigated as a tool to achieve an LPV model. The LPV model is used in a parameter dependent state feedback control design for the nonlinear system. An existence condition for a parameter dependent state feedback controller is derived, in terms of LMIs. Also, an explicit feedback control law is derived, expressed in the LPV model and in the solution to the existence problem. Augmentation considerations are discussed in this framework, for example compensation for the velocity form. The design method is illustrated by use of a 7 state FORTRAN code turbo fan jet engine model.

22

Chapter 2 Summary of the papers

Chapter 3

Concluding remarks In this thesis, gain scheduling using linear parameter varying (LPV) systems and a H∞ like synthesis for this type of systems, have been investigated. The synthesis method relies on that a good description of a nonlinear process to be controlled can be found in the form of an LPV system. This is a non-trivial assumption, and few systematic tools exist. Given an LPV description of the process, the synthesis method is casted as an optimization problem in the form of parameterized linear matrix inequalities. These can be relaxed into a set of linear matrix inequalities, for which there exist very efficient numerical tools to obtain an optimal solution. The controller is parameter dependent, and has a guarantee of performance in the user defined operating range. The performance specifications are translated into augmentation filters. The analogy with linear time invariant augmentation filters in H∞ control design, becomes increasingly weaker when the operating conditions of the process varies. This fact makes the design process harder than in the traditional linear time invariant case of robust controller design. As the controller is parameter varying and therefore more complex than a linear robust controller, additional caution must be taken in the use of such a controller. The presented method is motivated to be used whenever, • The plant is strongly nonlinear, such that it is motivated with a nonlinear controller. • The plant can be expressed as a linear parameter-varying system with sufficiently good accuracy. • The operating conditions change in the same time frame as the states of the plant, such that traditional ad hoc mappings of gain scheduling schemes fails.

23

24

3.1

Chapter 3 Concluding remarks

Future work

Since the synthesis method in this thesis relies on good LPV descriptions, a systematic tool to derive LPV models from a general nonlinear process model would increase the practical usefulness of the synthesis method. Also, an explicit measure on how well the LPV description is, would reveal whenever such a controller design framework is motivated. Comparing to the linear time invariant case, discretization of the controller becomes more critical for parameter varying controllers. Since the parameter may vary in the same time frame as the closed loop system, a piece-wise constant argument becomes less motivated. The effects of the controller discretization on the closed loop system have not been investigated in depth. Due to the negative definiteness of the LMIs in the synthesis process, some robustness to this problem is offered, but not in a quantitative measure, see [3].

Bibliography [1] OBIDOCTE, 1997. BRITE/EURAM,BE 97-4077. [2] K. J. ˚ Astr¨om and B. Wittenmark. Adaptive Control. Electrical Engineering: Control Engineering. Addison Wesley, Readings, Massachusetts, second edition, 1995. [3] P. Apkarian. On the discretization of LMI-synthesized linear parameter-varying controllers. Automatica, 33(4):665–661, 1997. [4] P. Apkarian and H. D Tuan. Parameterized LMIs in control theory. SIAM Journal of Control & Optimization, 38(4):1241–1264, 2000. [5] T. Azuma, R. Watanabe, and K. Uchida. An approach to solving parameterdependent LMI conditions based on finite numbers of LMI conditions. In American Control Conference, pages 510–514, 1997. [6] T. Azuma, R. Watanabe, K. Uchida, and Fujita M. A new LMI approach to analysis of linear systems depending on scheduling parameter in polynomial form. Automatiserungstechnik, 48, 2000. [7] G. Becker and A. Packard. Robust performance of linear parametrically varying systems using parametrically-dependent linear feedback. System & Control Letters, 23:205–215, 1994. [8] S. Boyd, L. Ghaoui, E. Feron, and V. Balakrishnan. Linear Matrix Inequalities in System and Control Theory. SIAM, 1994. [9] F. Bruzelius. On the computation of LPV controllers. Technical report, Dep. of EE Enginering, Chalmers University of Technology, 2002. [10] F. Bruzelius and C. Breitholtz. Gain scheduling via affine linear parametervarying systems and H∞ synthesis. In Conference on Decision & Control, pages 2386–2391, 2001. [11] J. C. Doyle. Analysis of feedback systems with structured uncertainties. IEE Proceedings, 129:242–250, 1982. [12] P. Gahinet. Explicit controller formulas for LMI-based H∞ synthesis. Automatica, 32(7), 1996. [13] P. Gahinet and P. Apkarian. A linear matrix inequality approach to H∞ control. International Journal of Robust and Nonlinear Control, 4:421–448, 1994. 25

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[14] P. Gahinet, A. Nemirovski, A. Laub, and M. Chilali. LMI Control Toolbox for Matlab. The Mathworks Inc., 1995. [15] M. H¨arefors. A Study in Jet Engine Control. PhD thesis, Dept of Signals and System, Chalmers University of Technology, 1999. [16] A. Helmerson. Methods for Robust Gain Scheduling. PhD thesis, Dept of Electrical Engineering, Link¨oping University, 1995. [17] I. Horowitz. Fundamental theory of linear feedback control systems. Trans. IRE on Autom. Control, 1959. [18] I. Horowitz. Quantitative Feedback Design Theory (QFT), volume 1. QFT Publications, Boulder, Colorado, 1993. [19] T. Iwasaki and R. Skelton. All controllers for the general H∞ control problem: LMI existence conditions and state-space formulas. Automatica, 30, 1994. [20] J. Kelly and Evers J. An interpolation strateqy for dynamic compensators. In The AIAA Guidance, Navigation, and Control Conference, pages 1682–1690, 1997. [21] H Khalil. Nonlinear Systems. Prentice-Hall, Inc, Upper Saddle River, second edition, 1996. [22] D. Leith and Leithead W. On formulating nonlinear dynamics in LPV form. In Conference On Decision and Control, pages 3526–3527, 2000. [23] R. Nichols, R. Reichert, and W. Rugh. Gain scheduling for h-infinity controllers: a flight control example. IEEE Transactions on Control Systems Technology, 1(2):69–79, 1993. [24] A. Packard, K. Zhou, P. Pandey, and G. Becker. A collection of robust control problems leading to LMI’s. In Conference on Decision and Control, pages 1245–1250, 1991. [25] W. Rugh and J. Shamma. Research on gain scheuling. 36(10):1401–1425, October 2000.

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[26] M. Safonov. Stability margins of diagonally perturbed multivariables feedback systems. IEE Proceedings, 129:251–256, 1982. [27] J. Shamma and M. Athans. Analysis of nonlinear gain-scheduled control systems. IEEE Transaction on Automatic Control, 35:898–907, 1990. [28] J. Shamma and M. Athans. Guaranteed properties of gain sheduled control of linear parameter-varying plants. Automatica, 27:559–564, 1991. [29] N. Z. Shor. Minimization Methods for Non-differentiable Functions. Springers Series on Computational Mathematics. Springer-Verlag, Berlin, 1985.

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[30] D. Stilwell and W. Rugh. Stability preserving interpolation methods for the synthesis of gain scheduled controllers. Automatica, 36, 2000. [31] H. Tuan and P. Apkarian. Relaxations of parameterized LMIs with control applications. In Conference on Decision & Control, pages 1747–1752, 1998. [32] H. Tuan and P. Apkarian. Monotonic relaxation for robust control: New characterization. In American Control Conference, pages 1914–1918, 2000. [33] L. Vandenberghe and S. Boyd. sp: Software for Semidefinite Programming. User’s Guide, Beta Version. K.U. Leuven and Stanford University, October 1994. [34] F. Wu, X. H. Yang, A. Packard, and Becker G. Induced L2 norm controller for LPV systems with bounded parameter variation rates. International Journal of Robust and Nonlinear Control, 6:983–988, 1996. [35] V. A. Yakubovich. Solutions of certain matrix inequalities encountered in automatic control theory. Soviet Math. Dokl., 3:620–623, 1962. [36] G. Zames. Feedback an optimal sensitivity: Model reference transformations, multiplicative seminorms, and approximate inverses. IEEE Transactions on Automatic Control, 2(26):301–320, apr 1981. [37] K. Zhou, J. Doyle, and Glover K. Robust and Optimal Control. Prentice-Hall, Inc, 1995.

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Paper I GAIN SCHEDULING USING LINEAR PARAMETER-VARYING SYSTEMS AND H∞ SYNTHESIS F. Bruzelius Technical report R003/2001, Chalmers University of Technology, April 2001.

Comments: Also partly presented at the 4th Swedish-Russian Control Conference, Moscow 2001 and at the 40th Conference on Decision and Control 2001, Orlando, Florida.

29

30

Paper I

Gain Scheduling using Linear Parameter-Varying Systems and H∞ Synthesis F. Bruzelius Department of Signals and Systems, Control and Automation Laboratory Chalmers University of Technology, SE-412 96 Gteborg, Sweden

Abstract The method of Gain Scheduling is widely spread in industry. One of the main hazards with this way of designing controllers however is that in general neither stability nor performance can be globally guaranteed in advance. These properties must be checked by extensive simulations, or by some other post-analysis. In this report, an alternative method for Gain Scheduling is presented where stability and performance can be guaranteed, given a special class of system, namely Linear Parameter Varying systems (LPV). The method is an H∞ -like design method for this type of systems, which offers robustness to disturbances and model errors. Some extensions of existing results are presented, where a parameter dependent Lyapunov function and a constraint called multi-convexity are used to get a tractable optimization problem. Application to non-LPV systems, where the nonlinear model is rewritten as an LPV system, is also treated in this report. The method is illustrated by application to a 3-state Moore-Greitzer compressor model, including nonlinear simulations.

Keywords: gain scheduling, linear parameter-varying control, parameterized linear matrix inequalities, H∞ -control, robust nonlinear control.

Introduction The method of Gain Scheduling is widely spread in applications e.g. in reactor tanks [18] or in jet engine control [13]. We may define gain scheduling, see [14], as a linear parameter varying feedback regulator whose parameters are changed as functions of operating conditions. Traditional gain scheduling methods try to combine linear design techniques and linear thinking in a sub-problem divide-andconquer manner. In practice, the concept is to linearize the plant to be controlled in 31

32

Paper I

a number of stationary points (also called operating points), and design control laws for each linearization using a linear design method. The next step is to connect the controllers such that a global controller is achieved that covers the whole pre-defined operating range. This can be done in numerous ways. The main strategies or groups of methods are switching/hybrid or interpolation schemes. The switching scheme, where the linear controllers are switched between, depending on which operating point that are closest to the current condition, has drawbacks. First there are implementation aspects. To be able to keep the desired goal of the linear designs, the design points must be relatively close together, since the linearization of the plant is only valid in a neighborhood of the design point. This means that a large number of linear controllers must be implemented and in addition a scheme to prevent inactive controllers to drift away, called bump-less transfer. Secondly, the switching surfaces need to be checked for stability in a post analysis, for instance by simulation. Third, the linear design must be “more” robust since they must cover an area of validity in the operating range. An example of the switching scheme is given in [13]. The main drawback of the interpolation scheme, where the linear controllers are interpolated into one nonlinear controller, is that stability in general can not be guaranteed in advance, see e.g. [26]. In some approaches of interpolation, stability can not be guaranteed even in a neighborhood of a design point. Different interpolation strategies have been studied in literature such as in [28], where the resulting interpolated controller stabilizes the closed loop for all fixed operating conditions. However this is an ad hoc method, and the behavior of the resulting feedback system may be different than expected from the linear design. Another approach to the Gain Scheduling problem is, to from a linearization (dependent of the operating condition), design one controller dependent of the operating condition and then make a post compensation for the first order approximation done in the linearization step, see e.g. [18]. In this paper, yet a different approach to the gain scheduling problem is taken, namely by use of a linear parameter varying system solve a parameter dependent H∞ problem, an LPV-H∞ problem. The divide-and-conquer strategy is lost and the resulting controller is scheduled directly, which guarantees stability and L2 performance of the closed loop system. The control synthesis is carried out by optimizing a performance index over a set of linear matrix inequalities (LMIs). The potential hazard of this method, in addition to the nontrivial step of describing the nonlinear plant as an LPV system in a good way (if at all possible), is the absence of solutions in some cases. A scheme to increase the solvability is presented here, using a parameter dependent Lyapunov function. The use of parameter dependent Lyapunov functions usually results in a set of parameterized LMIs that can not be checked exactly. Different approaches have been presented to get around this problem, such as griding of the parameter space and evaluation of the parameterized constraints on this grid. By doing so the number of constraints will explode as the dimension of the parameter space increases. The scheme presented here results in a set of LMIs

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that will not grow essentially, in comparison to parameter independent Lyapunov approaches. In contrast to griding approaches, the parameterized LMIs are here tractable to solve, using necessary conditions. The resulting set of relaxed LMIs will however have a smaller (or equal) solution set than the parameterized LMIs, i.e. the possibility of finding a solution might be reduced. An advantage over the switching and the stability preserving interpolation scheme is that the presented framework is unified, i.e. the design is performed in one step instead of many linear designs and interpolation/switching schemes. A relevant problem, that arises when using LPV-H∞ control design, namely the mapping/approximation from a nonlinear model to an LPV system is also treated here briefly. In contrast to other gain scheduling strategies, where a linearization is most often enough, we here want to describe the nonlinearities globally (or at least in some region) as a linear parameter varying system. The rest of the paper is divided into sections as follows. First a section on basic properties of LPV systems, such as stability is presented. Next a section on synthesis of a gain scheduled H∞ , with optimization constraints and controller formulas is given. The mapping from a nonlinear model to an LPV system and some of the consequences such mapping has to the closed loop system, follows. Last a small numerical example on a Moore-Greitzer compressor model is given, where a control design using LPV-H∞ technique is performed.

1

LPV systems

If we have gain scheduling control design in mind, a natural model description is, x(t) ˙ = A(ρ(t))x(t) + B1 (ρ)w(t) + B2 (ρ)u(t) z(t) = C1 (ρ(t))x(t) + D11 (ρ)w(t) + D12 (ρ)u(t) y(t) = C2 (ρ(t))x(t) + D21 (ρ)w(t)

(1.1)

where x(t) ∈ Rnx is the state vector of the model, u(t) ∈ Rmu is the input, w(t) ∈ Rmw is an immeasurable disturbance and ρ(t) ∈ Rp is the scheduling variable (also called parameter vector) of the system that describes the operating condition. The performance output is z(t) ∈ Rnz and y(t) ∈ Rny is the measured output available to the controller. The parameter vector is assumed to be a priori unknown, but measured on-line and used as an additional gain scheduling “feedback” to the controller. A system like (1.1) is called linear parameter varying system (LPV) if the parameter vector ρ(t) is an exogenous signal, i.e. does not depend on the states or the outputs and quasi-LPV if ρ(t) depends on any output or state vector component. Observe that even in the case where the parameter vector is exogenous it makes no sense to use a frequency transform of (1.1), since we are assuming ρ(t) to be varying in time. An LPV system like (1.1) might be looked upon as a generalization of a linear time-varying system (LTV), as they coincide for a given trajectory of the parameter

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vector ρ(t) ≡ ρ0 (t), and coincide with a linear time invariant system (LTI) for a fixed parameter vector ρ(t) ≡ ρ0 . In applications, the range of operation is often known in advance. This can be addressed directly in (1.1) by setting bounds on the parameter vector, i.e. ρ(t) ∈ Γ ⊂ Rp where Γ is a compact set. From an application point of view, it also often makes sense to say that the parameter vector is piece-wise continuous or even continuous in t. Further on in the report, we will call the function space of piece-wise continuous functions bounded by Γ ⊂ Rp for F(Γ). If in addition a function f (t) has bounded derivative, and these bounds are within√a subset ∆ ⊂ Rp , this will be denoted as f ∈ F(Γ, ∆). The Euclidean norm, i.e. xT x, is denoted x and the L2 -norm as xL2 .

1.1

Stability of LPV systems

For stability observations we consider an autonomous LPV system, x(t) ˙ = A(ρ)x(t),

x(t0 ) = x0 ∈ Rnx ,

ρ ∈ F(Γ, ∆).

(1.2)

The first to establish for stability of (1.2) is that all entries in the A(·) matrix are bounded, or equivalently that A(·) is a smooth function matrix and that the parameter rate is bounded, i.e. there exists a small enough ε > 0 such that ρ(t) ˙ < ε. Secondly, if the eigenvalues of A(ρ(t)) have negative real parts for all values of ρ(t), then the system is stable. A proof of this can be found in e.g. [22]. This result relates the slow-varying parameter changes to LTI stability. There are results that relates slow-varying parameter systems with LTI instability, but generally LTI stability for frozen parameters (or all possible) does not imply LPV stability. Also LPV stability does not imply LTI stability for frozen parameters. This is illustrated by two examples taken from [2]. Example 1.1 The LPV system, 
−1 + aρ21 1 + aρ1 ρ2 x, x(t) ˙ = −1 + aρ1 ρ2 −1 + aρ22

(1.3)

is LTI stable for all ρ2 ≤ 1 and a < 2. However if ρ1 = cos t and ρ2 = sin t i.e. ρ = 1, then the solution of (1.3) goes to infinity when t increases. Example 1.2 The LPV system,
 −1 − 5ρ1 ρ2 1 − 5ρ21 x(t) ˙ = x, −1 + 5ρ22 −1 + 5ρ1 ρ2

(1.4)

has a solution that goes to zero  for ρ1 = cos t and ρ2 = sin t, but the eigenvalues for this system is λ1,2 = −1 ± 5ρ2 − 1, which indicates that the underlying LTI system is unstable for ρ2 > 25 .

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The two preceding examples show that LPV stability is closely connected to LTV stability. It is therefore convenient to define stability of LPV systems as for LTV systems, i.e. using the transition matrix, Definition 1.3 A system like (1.2) is said to be asymptotically stable if for any t0 ≥ 0, x0 ∈ Rnx and ρ(t) ∈ F(Γ, ∆), where Γ, ∆ are compact sets, if lim x(t) = 0,

t→∞

or equivalently, lim Φρ (t, t0 ) = 0,

t→∞

where Φρ denotes the transition matrix for (1.2) with a given trajectory of ρ. This definition will coincide with standard Lyapunov stability. If there exists a symmetric positive definite upper bounded matrix Q(ρ) (Q < k) such that, ˙ A(ρ)T Q(ρ) + Q(ρ)A(ρ) + Q(ρ) < 0, for all ρ ∈ F(Γ, ∆) then (1.2) is asymptotically stable, see e.g. [23] for a proof. A more conservative way of verifying stability is to use a constant Lyapunov matrix i.e. V (x) = xT Qx. Since this reduces the complexity, it is commonly used. Asymptotic stability is also referred to as quadratic stability to point out the use of a quadratic Lyapunov function, V (x) = xT Qx, for analysis. A non-autonomous system like (1.1) is said to be asymptotically stable if the system is asymptotically stable for input u(t) ≡ 0.

1.2

Analysis of LPV systems

In standard H∞ control, a lemma called Bounded Real Lemma (BRL), plays an important role for both analysis and synthesis. Theorem 1.4 The linear time invariant (LTI) system, x(t) ˙ = Ax(t) + Bu(t) y(t) = Cx(t) + Du(t) is asymptotically stable and D + C(sI − A)−1 B∞ < γ, if there exist a γ > 0 and a positive definite symmetric matrix P such that,  T  A P + P A P B CT  BT P −γI DT  < 0 (1.5) C D −γI

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A proof can be found in any literature on H∞ or robust control, e.g. [35]. An extension of this lemma to be valid for linear time varying systems can be found in [21]. The extension to LPV systems like (1.1), using a constant matrix P can be found in e.g. [6]. The use of one single quadratic Lyapunov function, i.e. V (x) = xT P x, is potentially conservative and in [34] a parameter dependent Lyapunov function, V (x) = xT P (ρ)x, is used. The parameter must then have known bounds of the rate of change, i.e. ρ ∈ F(Γ, ∆). The (1, 1)-block-element in (1.5) will then get an additional term, the derivative of the Lyapunov matrix, P˙ (ρ). The parameter dependent Lyapunov matrix will destroy the convex property in the synthesis problem, discussed in the next section. The following modified BRL theorem found in e.g. [34] is included here with a proof for completeness. Theorem 1.5 Given the LPV system, x(t) ˙ = A(ρ(t))x(t) + B(ρ(t))u(t) y(t) = C(ρ(t))x(t) + D(ρ(t))u(t) assume that there exists a positive definite symmetric matrix W (ρ) and constants γ > 0, such that   T ˙ (ρ) W (ρ)B(ρ) C T (ρ) A (ρ)W (ρ) + W (ρ)A(ρ) + W  (1.6) −γI DT (ρ)  < 0 B T (ρ)W (ρ) C(ρ) D(ρ) −γI for all ρ(t) ∈ F(Γ, ∆). Then the system is asymptotically stable and the induced L2 -norm is upper bounded by γ, that is ||y(t)||L2 ≤ γ||u(t)||L2 + β(||x0 ||). The function β(x) ∈ K∞ i.e. β(0) = 0, strictly increasing and β(r) → ∞, r → ∞. Proof. To prove the norm bound, we use the same method as in [6], and first proof the case when D(ρ) = 0. The general case with D(ρ) = 0 is transformed to the case D(ρ) = 0 via a state transformation. Observe that (1.6) implies that the (1, 1)-element is negative definite, which gives the asymptotical stability from Lyapunov stability theory. Consider the case D(ρ) = 0. Let V (x) = xT W x implying that the derivative of V along the state trajectories becomes ˙ x. V˙ = xT (AT W + W A)x + 2uT B T W x + xT W where the ρ dependence has been left out for notation sake. By using standard completion of squares, V˙ = xT (AT W + W A)x − γ||u − γ −1 B T W x||2 . . . ˙x +γ||u||2 + γ −1 xT W BB T W x + xT W

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Next, taking the Schur complement of (1.6), see e.g. [8], gives ˙ + γ −1 (W BB T W + C T C) < 0 AT W + W A + W Hence,

V˙ < γ||u||2 − γ −1 ||y||2 .

Integration of V˙ gives  V ≤

t

(γ||u||2 − γ −1 ||y||2 )dσ + V (0)

0

The fact that V ≥ 0 ∀t = 0 completes the proof of L2 gain for the case when D(ρ) = 0. The general case D(ρ) = 0, is handled by using the unitary parameter dependent input-output transformation, [6], 
−1

−1 T  ˜1 D γ y(t) u(t) γ D = ˜ 2 −γ −1 D u˜(t) γ −1 y˜(t) D ˜ 2 = (I − γ −2 DDT )1/2 . The state transformation ˜ 1 = (I − γ −2 DT D)1/2 and D where D inverse is guaranteed since the lower 2 × 2 block matrix of (1.6) is negative definite, and will transform the problem to, 


ˆ x x˙ Aˆ B = ˆ u˜ y˜ C 0 which completes the proof. Observe that when ρ contains part of the state vector, the proof is valid for only a region of the state space enclosed by parts of (Γ, ∆), that is the region Λ explicitly defined by Λ = {x|(ρ(x, θ), ρ(x, ˙ θ)) ∈ (Γ, ∆)}, where θ is a true exogenous input. A system with the property of upper bounded induced L2 -norm of γ is also called a γ-dissipative system, see e.g. [15].

2

Synthesis

The synthesis of Gain Scheduled H∞ controllers is most often based on the bounded real lemma, see previous section, where the lemma is used on the closed loop. This problem is neither linear nor convex in the controller and Lyapunov matrices, which makes it hard to solve. In addition, it is parameterized by ρ and ρ˙ which makes it even more complicated to solve. Techniques involving projection is used to eliminate

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the controller dependence from the corresponding parameterized LMI (1.5). A new set of symmetric positive definite matrices will then give the solution to the new parameterized LMI problem. This problem and the control finding problem form an equivalence to the original (BRL) problem, that is convex in the Lyapunov matrix and the controller. Yet the parameterization of the LMIs remains. To be able to solve this as an LMI problem without griding the parameter space [34], we restrict ourselves to treat affinely dependence on the parameters, e.g. 


x x˙ A0 + ρj Aj B0 + ρj Bj , (2.1) = C0 + ρj Cj D0 + ρj Dj u y where ρj denotes the jth component of ρ. Also, to get a convex optimization problem, we assume that the controller has the same order as the (augmented) plant. Since the parameter vector is also assumed to be bounded in a hyper-cube, it is equivalent to check LMIs in the 2p corners of this hyper-cube if a quadratic Lyapunov function is considered. This is with these assumptions a convex feasibility problem, or if we minimize over upper bound of the L2 gain γ a convex optimization problem on LMI form. Such problems can be solved numerically using e.g. LMITOOL with the SP package, see [31], or LMI Control Toolbox for MATLAB, [11]. For further information on LMIs in control theory see [8]. Another form of parameter dependence frequently treated in literature is Linear Fraction Transform (LFT), which means that the LPV system can be described as a linear time-invariant system with a parameter feedback, e.g. an autonomous system x(t) ˙ = A(ρ(t))x(t) will have the following form, x(t) ˙ = Ax(t) + Bq q(t), v(t) = Cv x(t) + Dvq q(t) q(t) = diag(ρ1 (t)Is1 , . . . , ρp Isp )v(t).

(2.2)

This means that the entries of A(·) will be real-valued rational functions of the parameter ρ(t). Further on in this report we will focus on affine parameter dependent LPV systems. On gain scheduling using LPV systems with LFT dependence see e.g. [19, 14, 3, 16, 24]. A third type of parameter dependence is what could be called general LPV system, where only the boundedness of the entries in the state space realization is assumed. Synthesis of these types of LPV-systems is generally hard and, as mentioned earlier, a griding approach of the parameter space is the only way to get a numerically solvable synthesis problem.

2.1

LMI constraints

Like in linear H∞ the bounded real lemma (1.5), or some modified version, is used on the closed loop system. The controller, x˙ K (t) = AK (ρ(t))xK (t) + BK (ρ(t))y(t) u(t) = CK (ρ(t))xK (t) + DK (ρ(t))y(t),

(2.3)

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is not assumed to have the same structure as the plant, i.e. affine parameter dependence, at this stage. The closed loop, i.e. the LPV system (1.1) with the feedback (2.3) becomes, x˙ cl (t) = Acl (ρ(t))xcl (t) + Bcl (ρ(t))w(t) (2.4) ycl (t) = Ccl (ρ(t))xcl (t) + Dcl (ρ(t))w(t), where xTcl = [xT , xTK ], and 
A(ρ) + B2 (ρ)DK (ρ)C2 (ρ) B2 (ρ)CK (ρ) Acl (ρ) = , AK (ρ) BK (ρ)C2 (ρ) 
B1 (ρ) + B2 (ρ)DK (ρ)D21 (ρ) Bcl (ρ) = , BK (ρ)D21 (ρ) 

Ccl (ρ) = C1 (ρ) + D12 (ρ)DK (ρ)C2 (ρ) D12 (ρ)CK (ρ) ,

 Dcl (ρ) = D11 (ρ) + D12 (ρ)DK (ρ)D21 (ρ) . which are the matrices used in (1.5) or (1.6). The BRL is not convex in the decision variables, i.e. the controller matrices and the Lyapunov matrix, and is not suitable as a constraint when optimizing over the performance level γ. Therefore the controller dependence in this matrix inequality is eliminated. The BRL is then split up in a set of parameterized LMIs for the Lyapunov matrix, discussed in this section, and a set of parameterized LMIs for the controller, discussed in the next section. The most frequently used way, e.g. in [10, 3, 6, 34, 8], of eliminating the controller from the Bounded Real Lemma’s LMI formulation is by projection. The following projection lemma is then used, where a proof can be found in [10]. Lemma 2.1 Consider the LMI, Ψ + U T K T V + V T KU > 0

(2.5)

where Ψ = ΨT , K is the independent matrix variable and U , V has full rank. Denote V⊥ as the orthogonal complement to V and U⊥ to U respectively. The LMI (2.5) is feasible if and only if (2.6) V⊥T ΨV⊥ > 0, U⊥T ΨU⊥ > 0

Setting,






  

 A 0 B1 0 I Ap = , Cp = C1 0 , C = , , Bp = 0 0 0 C2 0 



0 B2 0 , B= , D12 = 0 D12 , D21 = I 0 D21

and the controller,


K=

AK BK CK DK

 ,

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Bcl = Bp + BKD21 Dcl = D11 + D21 KD21

Applying Lemma 2.1 to the BRL with V = [BT W, 0, D12 ] and U = [C, D21 , 0] and eliminate “zero” rows and columns, and partition the Lyapunov matrix W in BRL Theorem 1.5, will give the following formulation, found in e.g. [10, 3], Theorem 2.2 Assume that the system (1.1) has bounded ρ dependence and the maT (ρ)] have full row-rank uniformly trices R1 (ρ) = [C2 (ρ), D21 (ρ)], R2 (ρ) = [B2T (ρ), D12 in ρ ∈ Γ. Then there exists an LPV controller satisfying Theorem 1.5 if and only if there exist matrices X(ρ) = X T (ρ), Y (ρ) = Y T (ρ) and a scalar γ > 0 satisfying the following LMIs  

T X˙ + XA + AT X XB1 C1T  0 N NX 0 X T T  0 satisfying,  ¯  T ˜k + X ˜i X ˜iB ˜1i C˜1i ˜ i A˜i + A˜Ti X
T  X δ  k k ˜ T T ˜ 1i 0 < 0 ˜1 X ˜i ˜ 11i  − σ R1i  R B −γI D i 0 ˜ 11i −γI C˜1i D   T ˜1i
T  B − k δ k Y˜k + Y˜i A˜Ti + A˜i Y˜i Y˜i C˜1i  ˜ 2i R ˜ 2i 0 < 0 ˜ ˜ ˜   R C1i Yi −γI D11i − σ 0 T T ˜1i ˜ 11i −γI B D
 Xj Aj + ATj Xj Xj B1j T − σR1j R1j ≥ 0 T B1j Xj 0
 T Yj ATj + Aj Yj Yj C1j T R2j ≥ 0 − σR2j C1j Yj 0 
˜i I X >0 I Y˜i

(2.11)

(2.12)

(2.13) (2.14) (2.15)

for i = 1, . . . , 2p , j = 1, . . . , p, where X(ρ) = X0 +

p j=1

p

ρj Xj =

2

˜i, αi X

Y (ρ) = Y0 +

i=1

p

p

ρj Y j =

j=1

2

αi Y˜i .

i=1

Before proving this theorem, we need a lemma and a definition. Definition 2.5 A scalar function f : Rn → R is said to be Multi-convex if, ∂2 f (x) ≥ 0, ∂x2i

i = 1, . . . , n

Lemma 2.6 Consider a scalar quadratic function f : Ω ⊂ Rn → R where Ω is bounded by a hyper-cube, Ω = {x ∈ Rn |xi ≤ xi ≤ x¯i },

i = 1, . . . , n.

Denote the 2n vertex of this set by Ωi , i = 1, . . . , 2n . If f is multi-convex, then f (x) < 0 ∀x ∈ Ω if and only if it is negative on the vertices of Ω, i.e. f (Ωi ) < 0 for i = 1 . . . , 2n A proof can be found in e.g. [2]. We are now ready to proof Theorem 2.4. Proof. Applying Finsler’s Lemma to Theorem 2.2, gives parameter dependent versions of the two LMIs (2.11), (2.12) and (2.15). Pre and post multiply (2.11), (2.12) by a non-zero arbitrary vector of proper dimension. We have now two quadratic functions in ρ. Taking the second derivative with respect to ρi implies that these quadratic functions are multi-convex if these derivatives are negative, or equivalently if (2.13) and (2.14) hold. Then using Lemma 2.6 on the two functions gives

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the sufficient condition of negative definiteness in the vertex of Γ, i.e. (2.11) and (2.12). Last, the LMIs (2.15) gives a necessary and sufficient condition for positive definiteness of (2.9) due to the affine structure. For more complete proof of this part see [4]. What remains to be proven is that it is necessary and sufficient to check the upper and lower bounds respectively in the space of the derivative of the ˜ i > 0 and Y˜i > 0 for i = 1, . . . , 2p which parameter, α. ˙ LMI (2.15) implies that X ˜ ˜ ˙ gives that the upper bound on X(ρ) and −Y˙ (ρ) are i δ¯i X i and − i δ i Yi respectively. Taking the Schur complement on the corresponding α-dependent ˙ versions of (2.11) and again pre and post multiply this LMI with an arbitrary nonzero vector z of proper dimension, p

g(α, ˙ α) =

2

˜1, X ˜ 2 , . . .)z < 0 ˜ i z + z T L(α, γ, X α˙ i z T X

(2.16)

i=1

˜ 2 , . . .) denotes the α˙ independent parts of the expression. Since ˜1, X where L(α, γ, X the parameter is only assumed to be bounded and with a bounded rate of change, (2.16) must hold for all values of α and α˙ within the specified hyper-cubes Γ and ∆ respectively, independent of each other. Let us denote the set of feasible solu˙ M2 . Pick an tions to (2.16) as M1 and the corresponding solution to maxα˙ g(α, α) ˙ which implies arbitrary solution S ∈ M1 , trivially this must satisfy maxα˙ g(α, α) that M1 ⊆ M2 since ∆ is compact. Then pick an arbitrary solution S  ∈ M2 which gives a negative value of maxα˙ g(α, α) ˙ for all values of α ∈ Γ. Since g(α, α) ˙  is a monotonically increasing function of α˙ the solution S is also a solution of M1 which gives that M1 = M2 . Using the same argument on (2.12) proves that the worse cases in ∆ are equivalent to check the hole ∆ space.

A. Conservatism Observe that the two matrices X(ρ), Y (ρ) are assumed to be of affine structure as the plant data (A(ρ), B(ρ) etc.). This is a reasonable ad hoc assumption, to be able to solve the problem without griding the problem in the parameter-space. Though this will introduce some conservatism to the solution, since the general solution not necessary will depend affinely on the parameter. Also observe that ¯δ ∈ δ, / ∆. However, from practical experience, the conservatism introduced has small significance. The consequence of Theorem 2.4 is that the number of LMIs becomes 3 · 2p + 2p in comparison to the constant Lyapunov matrix approach 2 · 2p + 1. In contrast to other results as in [4, 34] where the number of LMIs is at least 2 · 22p + 2p + 2p this approach will not explode in size. For example consider a scheduling problem with four scheduling variables, i.e p = 4, the number of LMI constraints will be 56 and 536 respectively. The numerical computation effort is still much higher than in the constant Lyapunov matrix approach, since the cost of more decision variables is higher in general than the number of constraints. The use of the multi-convexity property may in some cases be conservative. In con-

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trast to the griding approach of the parameter space, the number of LMIs is fixed to the dimension of the parameter space. The multi-convexity property that is a directional property and a mixture of griding and multi-convexity in different directions to keep the number of LMIs down and get a less conservative solution, is proposed in [4]. Another way of reducing the conservatism introduced by the multi-convexity is to relax the two parameterized LMIs with non-negative multipliers, e.g. set corresponding parameterized LMI in the left hand side of (2.11) to − (λ0 + k λk ρ2k ) I instead of zero, also in [4]. By doing so the multi-convexity demand is relaxed and the left hand side of e.g. (2.13) will be −λj I instead of zero. Other relaxations of parameterized LMIs, [30], are to exclude extreme-points of the function in the parameter, i.e. it is sufficient to check negativeness in the lower part of the hyper-cube if the derivative is negative or upper part if the derivative is positive. This is a monotonic relaxation.

xT L(Xi , α)x6

Multi-convexity 

Monotonic relaxation 

α

Figure 1: Different types of relaxations. Dots indicates points to be checked. Necessary and sufficient conditions for existence of a H∞ controller in the LTI case are that the plant is stabilizable and detectable, see e.g. [35]. For the LPV case of constant scheduling parameter (e.g. ρ˙ = 0) this holds too, but for varying parameters the derivative term in (2.11) and(2.12) mixes the matrix variables. The extra conservatism introduced by the multi-convexity of the problem gives additional conditions for existence of solution to the LMI problem not further investigated here.

2.2

Controller formulas

Since the controller was eliminated from the formulation of the BRL, we need to find one controller realization, given a plant description (1.1) and a solution (X(ρ), Y (ρ), γ) of the synthesis LMIs, e.g. (2.11)-(2.15). Observe that with a solution given, the existence of a parameter dependent controller that satisfies the γ-performance problem is guaranteed. As mentioned in the previous section, the controller is not assumed to be affine in the parameters. There might be a restriction in saying that the controller is affine in the parameters, since no such assumption was made deriving Theorem 2.4. It would however be preferable due to implementation aspects of the controller.

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In the case of constant Lyapunov matrix, and the assumption of constant R1 and R2 matrices in Theorem 2.2, the affine controller assumption is more motivated, due to affinity in the BRL. The controller finding problem is to find a state space realization that satisfies the BRL for all parameter vectors within Γ. It is straight forward to solve a feasibility problem in the vertex of Γ, and the resulting controller is then a convex interpolation of the solutions of the feasibility problem. If there are additional bounds on the controller realization, it is always possible to solve these under the BRL constraint using a conventional optimization tool. Different approaches has been suggested in the literature to get a specific balanced controller realization, some of them are presented further on in this section. In the parameter dependent Lyapunov case, the problem in finding a controller is harder. The BRL in this case is obviously not affine in the parameters. In fact, if an affine parameter dependent controller is assumed and exists, the BRL parameterized LMI is to the worst of power 4 in the parameter. Therefore the reasoning on vertexes does not apply here. A controller can of course be computed for every parameter and parameter derivative combination (Γ, ∆), and mapped to a controller that covers the complete (Γ, ∆) space. This approach is computationally demanding, and is much like the griding approach for the Lyapunov finding/performance optimization problem not exact. In addition there are no guarantees that the controller will have an affine structure in the parameter, and may not even be continuous. This speaks against intuition, since the plant is assumed to have the affine structure. Another solution to the problem, presented in e.g. [34] and in this report, is to find an algebraic expression containing the plant data and the solutions (X(ρ), Y (ρ), γ). These solutions are not affine either, and some additional assumptions must be made on the plant data. Here we will present some algorithms to solve the problem for fixed values of (α, α) ˙ and an algebraic expression. In addition, a brief discussion is carried out on how to find a controller that does not explicitly depend on the derivative of the parameter vector α = T ρ. Other schemes, such as [34, 4], give a ρ˙ dependent controller, which may not be desired due to e.g. noisy measures of parameters etc. The first step in finding the controller is to reconstruct a Lyapunov matrix, W (ρ) in Theorem 1.5. This is needed in both an algebraic solution and a numerical solution for a fixed parameter value. One way of reconstructing the Lyapunov matrix is,  

I X Y I , (2.17) = W 0 NT MT 0 where W is the Lyapunov matrix in (1.6) for the closed loop system. M, N ∈ Rnx ×nx are any matrices such that M N T = I − Y X uniformly in ρ. From (2.10) we get the identities, X(ρ)Y (ρ) + N (ρ)M T (ρ) = I, N T (ρ)Y (ρ) + Z(ρ)M T (ρ) = 0 ∀ρ ∈ Γ,

(2.18) (2.19)

where Z(ρ) ∈ Rnx ×nx is the (2, 2)-block matrix element of W . We are searching for elements of W , i.e. N and Z. Observing that Z must be positive definite, we can

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solve for M T in (2.19), i.e. M T = −Z −1 N T Y and use this in (2.18). We get after pre-multiplying by Y −1 the identity, N (ρ)Z −1 (ρ)N (ρ)T = X(ρ) − Y (ρ)−1 ,

(2.20)

whose left hand side is guaranteed positive definiteness from (2.15). Evaluating the left hand side of the identity (2.20) for different ρ ∈ Γ, we can use e.g. singular value decomposition (SVD) to compute N and Z for fixed values of ρ. Using an argument similar to the one in the proof of Theorem 2.4, the LMI (1.6) is ˙ is replaced by its largest eigenvalue times the identity matrix. To see fulfilled if W this we can use the Schur argument of (1.6) and the well known inequality, T ¯ λ(A)xT x ≤ xT Ax ≤ λ(A)x x,

¯ where λ(A) and λ(A) denotes the the smallest and largest eigenvalues (parameter ¯ W ˙ ), but this dependent in our case) of A respectively. We can compute maxρ˙ λ( will however introduce conservatism to the solution and a controller that satisfies the upper bound parameter rate does not necessarily exist. Another approach to eliminate the ρ˙ dependence in the controller is, extensively discussed in [1], is to restrict either X or Y to be constant, that is independent of the parameter. Two algorithms to solve the constant (or fixed value of (ρ, ρ)) ˙ controller finding problem are given below. Recall that the LMI problem in finding a controller, given a solution (X, Y, γ), may be partitioned as in Lemma 2.1, that is Z + P T KQ + QT KP < 0

(2.21)

˙ . K is the controller where Z, P, Q depend only on the system data, W and W partitioned as
 AK BK K= . CK DK An algorithm to solve (2.21) was proposed in [17], Algorithm 2.7 i. Compute a symmetric matrix Σ > 0 such that Z − P T ΣP < 0. ii. Set K = −ΣP ΘQT (QΘQT )−1 where Θ = (Z − P T ΣP )−1 . For badly scaled problems this algorithm may give large norm solutions, which is not suitable for implementation. Another algorithm that gives a numerically more tractable solution was presented in [9]. The algorithm is presented here in detail, slightly modified to fit Theorem 1.5, i.e. the use of a parameter dependent W . For completeness we also give a solution to a single sided LMI used in this algorithm.

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Lemma 2.8 [9] Consider the LMI ˜ +K ˜TP < 0 Z + PTK

(2.22)

˜ is unknown. Let where Z = Z T , P are given and the matrix K   T   Σ 0  V1 P = U1 U2 0 0 V2T be a SVD of P . Then a solution of (2.22) is given by, ˜ = − 1 αU1 V1T − (U1 Σ−1 V1T )ZV2 V2T K 2 where α > max(0, λmax (Σ−1/2 V1T ZU1 Σ−1/2 ). Algorithm 2.9 Evaluate the plant data for the value of (ρ, ρ). ˙ † † D12 )D0 (D21 D21 ), where D0 is any matrix such that ||Dcl || < γ i. Set DK = (D12 † and A denotes the Moore-Penrose pseudo-inverse of A.

ii. Compute least-squares solutions to,       C2 0 D21 0 ΘB1 T   D21 = − −γI DclT  B1T X  0 Dcl −γI C1 + D12 DK C2     T   0 D12 B2T 0 T T  0 −γI DclT  ΘC1 = −  B1T + D21 DK B2T   D12 Dcl −γI C1 Y where  denotes matrices without interest here. † † iii. Set π12 = I − D12 D12 , π21 = I − D21 D21 . Compute ΘB2 and ΘC2 as the solution of the two LMIs

Ψ + C2T π21 ΘB2 + ΘTB2 π21 C2 < 0 Π + B2 π12 ΘC2 + ΘTC2 π12 B2T < 0 or ΘB2 = 0, ΘC2 = 0 if π21 C2 = 0 and π12 B2T = 0 respectively. Ψ and Π are given by, Ψ = AT X + XA + X˙ + ΘTB1 C2 + C2T ΘB1 + T    T −1  T T T ΘTB1 ΘTB1 γI −DclT B1 X + D21 B1 X + D21 + C1 + D12 DK C2 −Dcl γIγ C1 + D12 DK C2 Π = AY + Y AT − Y˙ + B2 ΘC1 + ΘTC2 B2T + T    T −1  T T T T T γI −DclT B1 + D21 B1 + D21 DK B2T DK B2T + C1 Y + D21 ΘC1 −Dcl γIγ C1 Y + D21 ΘC1 where M, N is given by M N T = I − Y X, e.g. a SVD.

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iv. Set ΘC = ΘC1 +π12 ΘC2 and ΘB = ΘB1 +π21 ΘB2 . The controller (AK , BK , CK ) is then given by, N BK = −XB2 DK + ΘTB CK M T = −DK C2 Y + ΘC ˙ +N˙ M T + −N AK M T = XB2 ΘC +ΘTB C2 Y +X(A−B2 DK C2 )Y +(A+B2 DK D21 )T +XY  +

T ΘTB B1T X + D21 C1 + D12 DK C2

T 

γI −DclT −Dcl γIγ

−1 

T T DK B2T B1T + D21 C1 Y + D21 ΘC



If some additional assumptions are made on the plant data, the following closed form expressions can be derived. Corollary 2.10 Given a solution X(ρ), Y (ρ), γ to Theorem 2.4. Assuming that one of the following statements are fulfilled: † † = I and D21 D21 = I ∀ρ ∈ Γ. • D12 D12 † † • D11 = 0, (I − D21 D21 )C2 = 0 and (I − D12 D12 )B2 = 0 ∀ρ ∈ Γ.

Then the closed loop stability and γ-dissipativity is guaranteed by the controller, † † D11 D21 DK (ρ) = −D12  − DK C2 Y )M −T CK (ρ) = (C

T ) BK (ρ) = N −1 (−XB2 DK + B ˙ + N˙ M T + X(A − B2 DK C2 )Y + AK (ρ) = −N −1 (XY  + A)M  −T  T C2 Y + XB2 C +B

where, † T  = γ(D12 C D12 )−1 B2T − D12 C1 Y T −1 T † T  = γ(D21 D21 B ) C2 − (D21 ) B1 X, T −1 T T  = (A + B2 DK C2 ) + γ (XB1 + BD  21 )(B1T + D21 A DK B2T ) +  γ −1 (C1T + C2T DK D21 )(C1 Y + D21 C)

Proof. A direct application of Algorithm 2.9 and the matrix inversion lemma, [35].

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u w

- Wu

- zu

- WT

- zT

- ?- WS

- zS

- P

-y

Figure 2: Augmented plant with weighting filters.

2.3

Design

As in LTI H∞ design the loop-shaping or weighting filters like in figure 2 is also used in LPV H∞ design. The main thing to keep in mind from a synthesis point of view when selecting filters is that the fragile structure of the problem must be kept, i.e. the affine parameter dependence. This means that weighting filters must be LTI systems. We might be able to use the multi-convexity property for nonquadratic functions to allow parameter dependence, but this will always introduce new conservatism to the problem, when optimizing. Observe that some parameter dependence of the weighting filters will keep the affinity, e.g. by letting Wu in figure 2 be parameter dependent. Then LPV based H∞ may be viewed as a unified framework of interpolating controllers such that the parameter dependent controller stabilizes the system for all parameters ρ ∈ F(Γ), since the design weights may be “different” in different operating points (vertex of Γ). For further discussion on loop shaping of nonlinear systems see [15]. The controller will as mentioned earlier, in the framework presented here, always have the same order as the plant plus the weighting filters. Order reduction of the controller is a non-trivial procedure, and LTI methods may not be applicable directly since the dynamics is different in different regions of Γ. A process model order reduction before the design would probably be preferable. The closed loop behavior in general, when the parameter is varying arbitrary within the predefined bounds (Γ, ∆), will not be known in other respects than the stability and the γ-dissipativity.

3

Application to nonlinear systems

In the general case the nonlinear model, x˙ = f (x, u, w, θ) z = g(x, u, w, θ) y = h(x, w, θ)

(3.1)

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where x, y, z, u is defined as above and θ a true exogenous online measured signal, the nonlinear H∞ approach of solving a partial differential inequality (namely a Hamilton-Jacobi-Isaacs inequality) fails or is hard to solve numerically if the model is of high order, i.e. many states. In general, these types of problems are impossible to solve analytically, and the resulting non-convex numerical problem is NP-hard to solve (there is no algorithm that solves the problem in polynomial time). The LPV approach on the other hand, resulting in a convex optimization problem, is relatively easy to solve using conventional optimization tools. On the other hand the LPV-H∞ control synthesis is only applicable to models of a special structure. The nonlinear model must be rewritten into an LPV model. An intuitive way of transforming a model like (3.1) to an LPV system, is to make an interpolation/approximation of a set of linearizations in different operating points. That is ∂ ∂ ∂ x + ρi ∂u f (xi , ui , 0, θi )˜ u + ρi ∂w f (xi , ui , 0, θi )w x˜˙ = i ρi ∂x f (xi , ui , 0, θi )˜ z˜ = y˜ =

i

∂ ∂ ∂ ρi ∂x g(xi , 0, θi )˜ u + ρi ∂u g(xi , ui , 0, θi )˜ u + ρi ∂w g(xi , ui , 0, θi )w

i

∂ ∂ ρi ∂x h(xi , 0, θi )˜ u + ρi ∂w h(xi , 0, θi )w



(3.2)

where x˜ = x − xss , u˜ = u − uss , y˜ = y − yss and uss , xss , yss is given by 0 = f (xss , uss , w, θ), θi are parameter values given in the operating range and ρi = ρi (x, y, θ) are basis functions that interpolate/approximate the linearizations at a finite set of operating points i. The basis functions are usually polynomial functions or linear interpolations of the operating points, for the sake of simplicity. This strategy has been used successfully in many applications for example in [29, 32, 27, 7]. In the framework of LPV-H∞ (and in quasi-LPV) control synthesis this will in general introduce some additional conservatism since the basis function, e.g. power function, is not independent, and the parameter space Γ is not a hyperrectangle. A strategy to reduce this conservatism is presented in [5], based on a refined polyhedron search in geometric space spanned by the power functions. Since the step of rewriting a system like (3.1) to an LPV system in general will be an approximation of the original nonlinear model, a highly relevant question arise: When is this transformation possible or more precisely, when is it motivated to use the LPV framework for control of nonlinear problems? In the general case it is hard to answer this question, since it is highly dependent of the structure of the nonlinearity of the process. Special cases such as input affine systems of special structures may in sometimes be written as true quasi-LPV systems, see e.g. [20]. Other types of nonlinearities that will be written as quasi-LPV’s are e.g. saturations of the input. These can be modeled using the following technique, presented in [33]. Assume a scalar amplitude input saturation,  u ≥ umax  umax , , umin < u < umax (3.3) usat (u) = u  umin , u ≤ umin By introducing the scheduling variable θ(u) = usat (u)/u, it will give information about saturation degree. By multiplying θ to the input matrices B2 and D12 the

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plant model will reflect the status of the saturation. In addition we can include the limited rate of change of the actuators by seeking a controller satisfying the derivative of θ. A natural measurement of how good the approximation is compared to the nonlinear model, i.e. the error introduced in the step of linearization and interpolation/approximation would be the norm of the error/residuals between the LPV system and the original model, ||f (x, u, w, θ) − A(ρ(x, θ))x − B(ρ(x, θ))u||∞ < . Here we present a different measurement of the approximation. For simplicity lets consider control and disturbance affine systems, that is x˙ = f (x, θ) + G1 (x, θ)w + G2 (x, θ)u z = h1 (x, θ) + N11 (x, θ)w + N12 (x, θ)u y = h2 (x, θ) + N21 (x, θ))w,

(3.4)

where the variables are the same as above and the involved mappings are Lipschitz continuous and Gi , Nij bounded such that a solution is well defined. Let us also, for simpler algebraic manipulations, consider the closed loop of an input disturbance affine nonlinear system, and give an estimate for what effect the approximation will have. Theorem 3.1 Consider the input affine nonlinear system, x˙ = f (x, θ) + G(x, θ)w y = h(x, θ).

(3.5)

Assume that the parameter affine LPV system, x˙ = A(ρ)x + B(ρ)w y = C(ρ)x,

(3.6)

approximates (3.5), using some function ρ = ρ(x, θ). Assume that (3.6) satisfies Theorem 1.5, with a W (ρ) and a γ. Then (3.5) is γn -dissipative if there exist a finite γε > 0 such that GT W x2 + h2 ≤ γn . T −1 (B T W x2 + Cx2 ) θ,x=0 2x W (Ax − f ) + γ

γε = sup

(3.7)

for all (ρ, ρ) ˙ ∈ (Γ, ∆). Proof. It is not hard to see that (3.5) is γn -dissipative (see e.g. proof of Theorem 1.5) if ˙ x + γn−1 (xT W GGT W x + hT h) < 0. H = f T W x + xT W f + xT W def

(3.8)

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52 From the assumption on the LPV system (3.6) we know that, ˙ + γ −1 (W BB T W + C T C))x < 0 xT (AT W + W A + W adding and subtracting (3.8) to the above equation gives 2xT W (Ax − f ) + γ −1 xT W (BB T −

γ γ GGT )W x + γ −1 (xT C T Cx − hT h) + H < 0 γn γn (3.9)

Denote, p(γh ) = γ −1 xT W (BB T −

γ γ GGT )W x + γ −1 (xT C T Cx − hT h) + 2xT W (Ax − f ). γn γn

From (3.9) sufficient condition for H < 0 for some γh is p(γh ) ≥ 0 or equivalently (3.7) holds. In addition (3.5) is internally stable if (3.5) is z-detectable, that is if u, w ∈ L2 implies that x → 0 as t → ∞, see [15]. There will in general be a trade-off in the step of describing a nonlinear system as an LPV system between accuracy of the approximated LPV system and complexity of the LPV system that will propagate to the controller. Theorem 3.1 gives, despite the conservatism in the proof, a picture of how well the approximation must be to preserve the desired properties (stability, γ-performance etc.) of the closed loop system. A simple structure of the parameter-vector is preferable since the controller will have the same structure dependence as assumed in the previous section.

4

Numerical example

Here we give a small novel numerical example to the LPV-H∞ control design, applied to a jet engine compressor model. This model is taken from [12], where a nonlinear H2 controller is computed for a compressor model. The numerical computations here were performed using LMI Control Toolbox, which is a user friendly interface package for solving LMI feasibility problems and optimization problems with LMI constraints. Consider the 3 state Moore-Greitzer compressor model, Φ˙ = −Ψ + Ψc − 3ΦR √ ˙ = 12 (Φ + 1 − v Ψ) Ψ β R˙ = σR(1 − Φ2 − R), R(0) > 0,

(4.1)

where Φ is the annulus averaged mass flow coefficient, Ψ is the plenum pressure rise, R is the squared amplitude of circumferential flow asymmetry, Ψc is the compressor characteristic relating pressure rise in the plenum to the mass flow, v the control

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which is proportional to the throttle area and β, σ is system dependent constants. The compressor characteristic is most often approximated by a cubic function in Φ, Ψc (Φ) = 1.1469 + 1.5Φ − 0.5Φ3 .

(4.2)

This is a simplification of the true phenomena, since Φ < 1 is unstable and therefore not physically observable. However the approximation (4.2) will give the same peak operating point of the compressor as the true system ( at Φ = 1). The model (4.1) is derived to reflect nonlinearities in a compressor system. Two important physical phenomena in a compressor that correspond to instabilities of (4.1) are surge, which is oscillations of the annulus average flow of the compressor system, and rotating stall which are regions of reduced or reversed flow that rotates around the annulus of the compressor. The non-stall stable equilibria of (4.1) for Φ > 1 is given by,     Φ0 Φ  Ψ  =  Ψc (Φ0 )  . 0 R 0

(4.3)

These are however not controllable by throttle feedback in the region Φ ≤ 1 without full state information. There also exists an unstable set of equilibria,     Φ0 Φ  Ψ  =  Ψc (Φ0 ) − 3(1 − Φ20 )Φ0  , 1 − Φ20 R 0 which is stabilizable in the region that corresponds to stall. To be able to describe the compressor model as an LPV system, a state transformation is carried out that moves the non-stall equilibria to the origin. For simplicity √ assume that Φ and Ψ are known such that the transformation u = β12 (1 − v Ψ − Φ) makes sense. Further, the signal u is used as the control input. For an arbitrary non-stall equilibria (4.3) with Φ0 > 1 the model (4.1) can then be rewritten as, r˙ = σ(1 − Φ20 )r − σr(2Φ0 φ + φ2 ) − σr2 φ˙ = −ψ + 32 (1 − Φ20 )φ − 3Φ0 r − 12 φ(φ2 + 3Φ0 φ + 6r) ψ˙ = −u,

(4.4)

where φ = Φ − Φ0 and ψ = Ψ − Ψc (Φ0 ) and r = R. From (4.4) one can see that both r and φ (and φ2 ) are needed to be able to describe (4.4) as an affine quasi-LPV exactly. Now, assuming that R and Φ are measured, one LPV system that describes (4.1) is,     0 0 0 f1 (ρ) x˙ =  −3Φ0 f2 (ρ) −1  x +  0  u (4.5) 0 0 0 −1 where ρ = [r, φ, φ2 ]T , x = [r, φ, ψ]T and, ˜ 0 − σρ1 − 2σΦ0 ρ2 − σρ3 f1 (ρ) = σ Φ ˜ 0 − 3ρ1 − 3 Φ0 ρ2 − 1 ρ3 f2 (ρ) = 32 Φ 2 2 ˜ 0 = 1 − Φ20 Φ

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The square amplitude of circumferential flow asymmetry R is of course positive, which implies that it is pointless to search for a stabilizing solution for negative values of R. Further, the third scheduling parameter is also always positive. Letting R be in the interval [0, 2] and R˙ in [−0.5, 0.5] and φ, φ˙ be in [−0.1, 5], [−1, 1] respectively. This implies that the parameter is in, ρ ∈ Γ = {x|0 ≤ x1 ≤ 2, −0.1 ≤ x2 ≤ 5, 0 ≤ x3 ≤ 25} and ρ˙ ∈ ∆ = {x| − .5 ≤ x1 ≤ 0.5, −1 ≤ x2 ≤ 1, −10 ≤ x3 ≤ 10} The main objectives of the control design is to keep the state variables inside a neighborhood of the equilibrium (the origin) and to keep the derivative of the control input u at a reasonable level. This can be translated to set penalty to the control k2 s and to the outputs i.e. WT = diag(c1 , c2 ). signal and its derivative i.e. Wu = k1 + s+ν This implies that the augmented plant will be,     0 0 0 a11 (ρ) 0  −3.9 a22 (ρ) −1 0    x +  0 u x˙ =   0  −1  0 0 0  0 0 0 −100 −10     0.001 0 0 0 0 z =  0 0.1 0 0  x +  0  u (4.6) 0 0 0 1 0.11

 1 0 0 0 1 0 y = x+ w 0 1 0 0 0 1 a11 (ρ) = −2.76 − 4ρ1 − 10.4ρ2 − 4ρ3 a22 (ρ) = −1.035 − 3ρ1 − 1.95ρ2 − 0.5ρ3 where σ = 4 and Φ0 = 1.3, which corresponds to an equilibrium pressure Ψc (1.3) = 1.9984 or 94% of the peek value Φc (1) = 2.1496, and with the weight constants c1 = 10−3 , c2 = 0.1, k1 = 0.01, k2 = 0.1 and the PI cut-off frequency ν = 100. Applying Theorem 2.4 on (4.6) using LMI Control Toolbox and optimizing over the performance index γ will result in the optimum γopt = 0.7626. The LPV system (4.6) does not satisfy the conditions of Corollary 2.10 for explicit controller formulas. Though, since D11 = 0 and π21 C2 = 0, π22 B2 = 0 in Algorithm 2.9 uniformly in ρ for this problem, we may use the explicit controller formulas in Corollary 2.10 except for DK = 0. A simulation of the closed loop with the nonlinear compressor model and the derived controller, see figure 3, reveals that the system is stable as long as we stay in the specified bounds (Γ, ∆). Observe that the steady state error in φ is due to the fact that no integral states have been introduced. An alternative to this design would be to ignore the flow asymmetry state R, since this is stable in the region Φ ≥ 1, and just use Φ and Φ2 as scheduling variables, to get a smaller problem. This was however not done here.

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1

r(t)

0.5 0 0

1

2

3 time

4

5

6

7

1

2

3 time

4

5

6

7

1

2

3 time

4

5

6

7

φ(t)

0.5 0

−0.5 −10 10

u(t)

5 0

−5 0

Figure 3: Nonlinear simulations of closed loop performance, initial values r(0) = 1 and φ(0) = −0.1 and a step disturbance of magnitude 0.3 to the control input u(t) at t = 5.

5

Conclusion

An alternative way of gain scheduling has been presented, using Linear Parameter Varying systems using a parameterized H∞ framework in the LMI form. The use of a parameter dependent Lyapunov function, which enables an explicit exploitation of the parameter rate variation, is treated. To avoid griding of the parameter space and the parameter derivative space, an ad hoc fixed solution structure, to the cost of potential conservatism, has been used. Though, this results in a set of parameterized LMIs, which are NP-hard to solve. Here a so called multi-convex relaxation is used to get a set of LMIs to the cost of increased numbers of LMIs and some conservatism of the problem. An extension of existing results is presented, that shows that in an affine parameter dependent case, it is necessary and sufficient to check a worst case scenario of the parameter derivatives. This result reduces the number of LMIs in the problem, without causing additional conservatism. The controller finding problem in the parameter dependent Lyapunov function case is a parameterized LMI feasibility problem. Some existing controller formulas are briefly discussed, which solve the problem for a fixed value of the scheduling parameter and its derivative. A strategy to eliminate the explicit parameter derivative dependence is presented here. With some assumptions on the (augmented) plant, there exist explicit closed form solutions to the controller finding problem, one presented here. These are however not affine in the parameter, but rather complicated

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expressions. A brief discussion on how to approximate nonlinear systems to affine (quasi-) LPV system and the effect this approximation will have on the closed loop system is also given. In this framework it is of greater importance to get a good global (or in a region) description of the system, than in the traditional framework of gain scheduling where a set of linearizations most often is sufficient. A small numerical example is given. Here the method is applied to a 3-state MooreGreitzer compressor model, including all steps of the design. Despite some simplifications and assumptions on the plant model, this example is illustrative for such a design. A feasible solution to the LPV-H∞ problem was found, which gives a stabilizing solution since no approximations were performed when rewriting the nonlinear model to a quasi-LPV model.

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[33] F. Wu and K. M. Grigoriadis. LPV-based control of systems with amplitude and rate actuator saturation constriants. In American Control Conference, pages 3191–3195, 1999. [34] F. Wu, X. H. Yang, A. Packard, and Becker G. Induced L2 norm controller for LPV systems with bounded parameter variation rates. International Journal of Robust and Nonlinear Control, 6:983–988, 1996. [35] K. Zhou, J. Doyle, and Glover K. Robust and Optimal Control. Prentice-Hall, Inc, 1995.

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Paper II ON THE COMPUTATION OF LPV CONTROLLERS F. Bruzelius Technical Report R002/2002, Department of Signals and Systems, Chalmers University of Technology, G¨ oteborg, Sweden, January 2002.

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On the computation of LPV controllers F. Bruzelius Department of Signals and Systems, Control and Automation Laboratory Chalmers University of Technology, SE-412 96 G¨ oteborg, Sweden

Abstract In this paper, a new characterization of parameterized linear matrix inequalities (LMI) as a finite set of LMIs is proposed. The method is computationally more demanding, and may be viewed as a combination of existing methods. It is illustrated in the framework of robust gain scheduling for linear parameter varying (LPV) systems, with bounded parameters and parameter derivatives. In this framework, a new way of finding a controller that is independent of the derivative of the parameter is proposed. This controller is potentially less conservative than controllers designed by use of existing methods. Both results are illustrated by a numerical example.

1

Introduction

The use of gain scheduling is widely spread, in many different types of applications. This intuitive way of designing nonlinear controllers has been one of the the most frequently used methods in practical applications. However, in the most naive form, there are no theoretical results that guarantees system stability when such a method has been used to design the controller. This is due to the fact that one has overlooked the rate of change of the parameter that determines the operating conditions, see e.g. [13]. The classical method of gain scheduling involves a set of linear time invariant controllers, designed for different operating conditions. A mapping strategy (switching or interpolating) forms the nonlinear feedback for the operating range. The linear design is most often based on linearizations of the nonlinear plant, which implies that local stability can be guaranteed. However, interpolation schemes may destroy this property. Different interpolation schemes have been proposed, such as in [14], that preserve the stability. Recently, in [12], the notion of linear parameter varying (LPV) systems was introduced. This class of systems, which depends causally of an on-line measured 63

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parameter vector, is a natural way of describing dynamic changes in a process, and therefore also natural in the framework of gain scheduling. The pioneering work by [8], which is a H∞ synthesis method formed as a set of linear matrix inequalities (LMI), has a natural extension to the class of LPV systems [5]. Together with the development of efficient optimization software, [9], for LMIs, this has opened a promising way for systematic design of gain scheduled controllers. The synthesis may be based on a single quadratic Lyapunov function [5] or a quadratic parameter dependent Lyapunov function [16]. There is also a parallel synthesis method based on linear fraction transform (LFT) [2] and [10] which uses µ like methods. We will in this paper focus on the Lyapunov approach. Using one single Lyapunov function is, as pointed out in [16], potentially conservative, though. However, using information on the bounds of the rate of change and a parameter dependent Lyapunov function less conservatism can be obtained. Using a parameter dependent Lyapunov function results in parameterized LMIs. Additionally, the controller will in general, depend on the time derivative of the parameters. This paper deals with these two difficulties, within the standard setup. A parameterized LMI represents one LMI per value of the parameters, and are therefore hard to solve. Two ways to solve these problems have been proposed in the literature. Either by a griding of the parameter space, that is, evaluating the parameterized LMI on a lattice, or analytically finding a necessary set of LMIs. The first method, nonetheless, will become numerically intractable even for problems with parameter vectors of low dimensions. Here we present a new way of characterizing the parameterized LMIs to a finite set of LMIs, using the vertices of the parameter space. It boils down to characterize whenever a quadratic function has its maximum in a vertex of its domain. Since the parameters are only known on-line and not a priori, the parameter vector and its time derivatives can be treated as two independent quantities. We will here exploit this to derive a controller that is independent of the time derivative of the parameter. The remainder of the paper is organized as follows. Notation, problem formulation and supporting lemmas are introduced in section 2. Necessary conditions for the existence of a parameter dependent controller formed as a set of LMIs is treated in section 3, and a method to find a parameter dependent controller in section 4. Finally a numerical example in section 5 is given.

2

Preliminaries

Before presenting the main results of this paper, we give the notation, the problem formulation and some supporting lemmas on LMIs. For block matrices a
denotes the transpose of the corresponding block. Symmetry is denoted as S(B) = B + B T , for brevity. The ρ parameter and time dependence

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are dropped whenever it is obvious, for compactness. For time derivatives we use the dot notation, that is f˙ for the time derivative of f . An index indicates the corresponding coefficient in an affine form, for example D11i . For hyper-cubes H, the set of vertices (or corner points) is denoted as vertH. The Moore-Penrose pseudo-inverse of a matrix M is denoted M † . Consider the system LPV system, x(t) ˙ = A(ρ)x(t) + B1 (ρ)w(t) + B2 (ρ)u(t) z(t) = C1 (ρ)x(t) + D11 (ρ)w(t) + D12 (ρ)u(t) y(t) = C2 (ρ)x(t) + D21 (ρ)w(t)

(2.1)

where x ∈ Rn , y ∈ Rny and z ∈ Rnz are the state vector, the measured output and the auxiliary performance outputs respectively. The signals u ∈ Rnu , w ∈ Rnw and ρ ∈ Rp are the control input, the disturbance input and a scheduling parameter that is assumed to be measured on-line but unknown a priory. Especially our attention will be focused on LPV systems with affine parameter dependence, or shorter affine LPV systems, that is, A(ρ) = A0 +

p


ρk Ak ,

B1 (ρ) = B10 +

k=1

p


ρk B1k

k=1

We also assume that the parameter ρ is bounded in a hyper-cube, that is ρ(t) ∈ H = {ρ ∈ Rp |ai ≤ ρ ≤ bi },

∀t ≥ 0

and ρ˙ is bounded in the same manner, ρ(t) ˙ ∈ Hd = {ρ˙ ∈ Rp |ci ≤ ρ˙ ≤ di },

∀t ≥ 0.

We can without loose of generality make −ci = di via an affine transformation of ρ. The class of systems is uniquely determined by the value of A(ρ), B1 (ρ) etc. in the vertices of H, also known as polytopic systems. We will further on use the notation affine LPV system for the system (2.1) and include the bound H and the symmetric bound Hd . The problem formulation is to find a parameter dependent controller that gives closed loop stability and an induced L2 -norm less then some positive constant γ from the disturbance input w to performance output z, that is,   2 T 2 zL2 = z(t) z(t)dt ≤ γ w(t)T w(t)dt. This problem is solved by the following extension of the well known bounded real lemma, applied to the closed loop system. Lemma 2.1 Consider the LPV system, x(t) ˙ = E(ρ)x(t) + F (ρ)w(t) z(t) = G(ρ)x(t) + H(ρ)w(t),

ρ∈D

(2.2)

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where D is a compact set, and ρ˙ ∈ Dd compact set. If there exists a matrix W (ρ) > 0 for all (ρ, ρ) ˙ ∈ D × Dd and a scalar γ > 0 such that  T  ˙ (ρ) W (ρ)F (ρ) GT (ρ) E (ρ)W (ρ) + W (ρ)E(ρ) + W  (2.3)
−γI H T (ρ)  < 0

−γI Then system (2.2) is asymptotically stable and has an induced L2 -norm from w to z less then γ. In [16] a complete proof of this lemma can be found. We will in this paper frequently encounter so called parameterized LMIs. They correspond to one LMI per single value of a parameter vector. Constraints like this are inherently hard to solve, and a relaxation that reduces the set of infinitely many LMIs to a finite set is in practice necessary for finding a solution. One alternative is to grid the parameter space in a lattice and evaluate the parameterized LMI on this. However, as the parameter space grows in dimension, the griding points grows exponentially and results in an intractable numerical problem. We propose a new way of relaxing parameterized LMIs into a finite set of LMIs supported by the following lemma. Lemma 2.2 Let f : D → R be a quadratic function, where D = {x ∈ Rn : ai ≤ xi ≤ bi } is a hyper-cube. Then f (x) < 0 ∀x ∈ D if f (vi ) < 0,

∀vi ∈ vertD

(2.4)

and if there exist semi-definite functions αj : D → R such that ∂f ∂2f (x) ≤ , αj (x) ∂xj ∂x2j

∀x ∈ D,

j = 1, . . . , n

(2.5)

Proof. What we have to prove is that f (x) will have its maximum in some vertex of D. Let y ∗ = [y1∗ , . . . , yn∗ ] be the global maximizer of f over D. If y ∗ is not a vertex of D, we have ai < yi∗ < bi for some i. Consider the function, ∗ ∗ g(yi ) = f (y1∗ , . . . , yi−1 , yi , yi+1 , . . . , yn∗ )

If the right hand side of (2.5), i.e. the second derivative of f is negative, then g is a monotonic function. On the other hand if right hand side of (2.5) is positive or equal to zero, g is a convex function. Hence, the maximum of g is obtained at either ai or bi . We have, g(yi∗ ) ≤ max(g(ai ), g(bi )) Since y ∗ is the global maximizer of g we have, g(yi∗ ) ≥ max(g(ai ), g(bi ))

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That is,

g(yi∗ ) = max(g(ai ), g(bi ))

Repeating this argumentation for all i, gives that the function f will obtain its maximum in some vertex of D, which concludes the result. We now give two standard tools for LMIs Lemma 2.3 The following conditions are equivalent,  1. 2.

Q S ST R

 >0

R > 0, Q − SR−1 S T > 0

The lower condition is most often referred to as the Schur complement of the first. This lemma can be found in [6], or in any other standard text on LMIs. Next a tool to eliminate matrix variables is given by, Lemma 2.4 The LMI, G + U XV T + V X T U T > 0, where X is the independent matrix variable, is feasible whenever there exists a scalar σ such that G − σU U T > 0, G − σV V T > 0

This lemma is most often called Finslers lemma. In [3] a parameter dependent version of this lemma is presented. It is proven that one can choose a large enough scalar σ independently of the parameter.

3

Controller existence

If we apply lemma 2.1 to the closed loop system, the matrix inequality (2.3) is not only parameter dependent, but also nonlinear in the decision variables W and in the controller. What has become a standard practice is to eliminate the controller dependence in (2.3), by using for example lemma 2.4. The problem is then split up in first finding a W and then construct a controller satisfying (2.3). In this section we focus on the first problem. This is a partial differential inequality in parts of W . An ad hoc solution, suggested in [16], is to mimic the parameter dependence of the plant. The problem is then reduced to a set of parameterized LMIs, that can be relaxed to a finite set of LMIs. Using lemma 2.2 results in the following controller existence theorem.

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Theorem 3.1 Consider the affine dependent LPV psystem (2.1). If there exist p symmetric positive definite matrices X(ρ) = X0 + i=1 Xi ρi , Y (ρ) = Y0 + i=1 Yi ρi and real scalars σ, γ, αi and βi such that 

X˙ + S(XA) 

 ˙ −Y + S(AY ) 



  T C2 C2 XB1 C1T T −σ
−γI D11

−γI   T B2 B2T Y C1 B1 −γI D11  − σ 


−γI

 C2T D21 0 T D D21 21 0  < 0
0  T B2 D12 0 T 0  0, I Y

(3.1)

(3.2)

(3.3)

for all (ρ, ρ) ˙ ∈ vertH × vertHd ,  T αi S(Xi A + XAi ) − 2S(Xi Ai ) αi (Xi B1 + XB1i ) − 2Xi B1i αi C1i T  
0 αi D11i

0   T T T T T αi S(C2i C2 ) − 2C2i C2i αi (C2i D21 + C2 D21i ) − 2C2i D21i 0 T D ) − 2D T D
αi S(D21i 0≤0 −σ  21 21i 21i

0   T T T βi S(Ai Y + AYi ) − 2S(Ai Yi ) βi (Yi C1 + Y C1i ) − 2Yi C1i βi B1i 
0 βi D11i 

0   T T T T T βi S(B2i B2 ) − 2B2i B2i βi (B2i D12 + B2 D12i ) − 2B2i D12i 0 T ) − 2D D T
βi S(D12i D12 0≤0 −σ  12i 12i

0 

(3.4)

(3.5)

for all ρ ∈ vertH and i = 1 . . . p, then there exists a stabilizing LPV controller such that zL2 ≤ γwL2 , for initial condition x(0) = 0 and for all (ρ, ρ) ˙ ∈ H × Hd Proof. We will briefly show how the parameterized LMIs (3.1), (3.2) and (3.3) are sufficient for the existence of a parameter dependent controller. In [7] and [3] complete proofs of this result can be found. Partition the Lyapunov matrix W in lemma 2.1 as follows,  W =

X N N T X2



 W

−1

=

Y M M T Y2

Define the full rank matrices,  ΠX =

X I NT 0



 ΠY =

I Y 0 MT





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The congruence transformation T = diag(ΠY , I, I), applied to the bounded real lemma (Lemma 2.1) for the cloosed loop system, results in  ˙  + AT + C T DT B T XB1 + BD  21 C T + C T DT DT  2) A X + S(XA + BC 2 1 2 K 2 K 12   B1 + B2 DK D21  21
−Y˙ + S(AY + B2 C) Y C1T + CD  T + DT DT D 

−γI D11 21 K 12


−γI

   < 0,  (3.6)

 B,  C  are matrix variables depending on the controller (AK , BK , CK , DK ) where A, ˙ as and on the Lyapunov matrix W and its time derivative W  2 Y − XB2 C  = N AK M T − X Y˙ + N M˙ T + X(A − B2 DK C2 )Y + BC  A  = N BK + XB2 DK B  = CK M T + DK C2 Y C

(3.7) (3.8) (3.9)

Applying a parameter dependent version of lemma 2.4 to the matrix inequality above with respect to the matrix variable, 

K K + A + B2 DK C2 B A K C DK will end up in (3.1) and (3.2). Positiveness of W is guaranteed by (3.3). What remains to be proven is that (3.4) and (3.5) in vertH gives sufficient conditions for negativeness of (3.1) and (3.2) in (ρ, ρ) ˙ ∈ H × Hd if (3.1) and (3.2) are negative in vertH × vertHd . Since ρ˙ is linear in (3.1) and (3.2), it is necessary and sufficient to check the vertices of Hd . Multiply (3.1) with an arbitrary nonzero vector x, f (ρ) = xT M(ρ)x. Applying lemma 2.2 and we get, αi

p
∂f ∂ 2f ˜ j ρj + M ˜ 0) − M ˜ i )x ≤ 0, M − 2 = xT (αi ( ∂ρi ∂ρ j=1

(3.10)

where Mi are matrices depending on the plant data and (X, Y, γ). The constraint (3.10) is equivalent to (3.4). Since (3.4) is affine in ρ it is sufficient to check the vertices of H. The treatment of the parameterized LMI (3.2) is analogous. Even though the number of LMIs is growing exponentially, the computational complexity using for example an interior point algorithm such as [9] grows at worst linearly in the number of constraints and cubically in the number of decision variables. However, viewing αi and βi in theorem 3.1 as decision variables, it is more critical that these do not enter linearly in the inequalities. The most interesting values of these multipliers are though, zero which corresponds to multi-convexity [3], and large absolute values which corresponds to monotonic relaxation [15]. Thus, multi-convexity and monotonic relaxation are special cases of lemma 2.2. Since each set of multiplier values corresponds to one optimization/feasibility problem,

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the practical use of theorem 3.1 is potentially limited. Whenever the best possible performance level is of higher priority than computational time, nonetheless, the proposed method is a good alternative to a brute force griding approach. Often multi-convexity and monotonicity is too conservative properties to demand on the parameterized LMI directly. An upper bound of the parameterized LMI can easily be obtained using a positive definite matrix function with quadratic dependence with these properties is then a less conservative solution. This is also illustrated in the numerical example in section 5.

4

Controller Formulas

In this section we will focus on how to find a parameter dependent controller that in a closed loop with the LPV system (2.1) satisfies the γ norm bound for all possible trajectories of the scheduling variable specified by the bounds H and Hd . This is achieved if we can find a controller realization, that satisfies lemma 2.1. In the preceding section necessary components of W in (2.3) were found. The goal is to find a controller given for a given solution to theorem 3.1, (X, Y, γ). Of particular interest is a controller that is independent of the time derivative of the parameter. The practical use of a controller that depends on the time derivative of the parameter, ρ, ˙ is of course limited. However given this solution (X, Y, γ), there do not exist methods to find a ρ˙ independent controller. A conservative method was proposed in [4] and extensively discussed in [1]. The idea is to let one of X and Y be constant. Here we propose a different method, based on that we can find a worst case ρ˙ in Hd . To be able to find such a ρ˙ we must let the linear coefficients of affine forms X and Y be semi-definite. This will introduce conservatism to the solution of (X, Y, γ) though, however it is not clear that this will result in a more conservative solution than letting either X or Y to be constant. In fact in the numerical example the proposed method is less conservative. The above discussion can be formalized in the following theorem. Theorem 4.1 Given a solution to theorem 3.1 where Xi and −Yi i = 1, . . . , p are either positive or negative semi-definite matrices. Assuming that one of the following statements are fulfilled: † † = I and D21 D21 = I ∀ρ ∈ Γ. • D12 D12 † † )C2 = 0 and (I − D12 D12 )B2 = 0 ∀ρ ∈ Γ. • D11 = 0, (I − D21 D21

Then the closed loop stability and a L2 -norm upper bounded by γ guaranteed by the controller, x˙ K (t) = AK (ρ)xK (t) + BK (ρ)y(t) u(t) = CK (ρ)xK (t) + DK (ρ)y(t)

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where, † † DK (ρ) = −D12 D11 D21  − DK C2 Y CK (ρ) = C

T ) BK (ρ) = V (−XB2 DK + B p
Xi αi Y + X(A − B2 DK C2 )Y + AK (ρ) = −V ( i=1

 + A)   T C2 Y + XB2 C +B and † T  = γ(D12 C D12 )−1 B2T − D12 C1 Y T −1 T † T  = γ(D21 D21 B ) C2 − (D21 ) B1 X, T −1 T T  = (A + B2 DK C2 ) + γ (XB1 + BD  21 )(B1T + D21 A DK B2T ) +  γ −1 (C1T + C2T DK D21 )(C1 Y + D21 C)

V = (I − XY )−1  −di if Xi ≤ 0, αi = di if Xi ≥ 0,

Xi =  0 Xi =  0

Proof. First, observing that the congruence transformation T = diag(ΠX , I, I) applied on the bounded real lemma matrix inequality (2.3), resulting in (3.6), is not dependent on ρ. ˙ Since N and M T can be chosen with full rank, see [8], we can let  in (3.7) be independent of ρ. A ˙ Using a Schur argument on (3.6) we get,   p
Xi 0 < L(ρ), (4.1) ρ˙ i 0 −Yi i=1

where the right hand side is independent of ρ. ˙ Clearly this inequality holds for all ˙ = α such that, ρ˙ ∈ Hd , especially for ρ(t)   p
Xi 0 ≥ 0 ∀t, (αi − ρ˙ i (t)) 0 −Yi i=1

due to the fact that the matrices are semi-definite and that Hd is symmetric around zero. We have found a worst case ρ˙ in Hd for the specific solution (X, Y, γ). Solving (3.7)-(3.9) for AK , BK , CK and making a controller-state transformation xK = M T z, result in the controller expression given in the theorem. In [7] a proof can be found  B,  C)  guarantees negative definiteness of (2.3), that the givens specific choice of (A, which completes the proof. The assumption on the plant data D11 etc. in theorem 4.1 is a sufficient condition for a proper closed loop, that is with a Dcl = 0. This assumption can however be eliminated using the technique in [7].

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Observe that in a practical situation the inverse (I − X(ρ)Y (ρ))−1 does not need to be computed on-line. To see this we observe from (3.3) that the spectral radius of XY always is greater than one. We can write, (I − X(ρ)Y (ρ))−1 =

∞


(X(ρ)Y (ρ))−k =

k=1

n


Z(ρ)k

k=1

where we have used the matrix version of a geometric sum in the first inequality and Cayley-Hamiltons identity in the last equality. There are no guarantees that Z(ρ) is a fraction of polynomials in ρ of finite degree, though. In fact, in the general case it will have infinite degree. However, we can always scale the parameter ρ such that it makes sense to approximate by ignoring higher order terms of ρ.

5

Numerical example

For numerical illustration we give an example originating from [11] and modified in [3] and [15]. We use the same notation and specification as in [3] and [15] to be able to compare. The LMI Control Toolbox, [9], for Matlab is used for numerical optimization. An LPV model describing the longitudinal dynamics of a missile is given by, 

 

α˙ q˙



=

wθ1 wθ2 ηz q





 =



 =

−0.89 1 −142.6 0 θ1 0 0 θ2



−1.52 0 0 1



α q



−1 0 1 0 

α q



+ 

α q

0 −0.89 178.25 0



wθ1 wθ2





+

−0.119 −130.8



δfin ,

 ,

(5.1)

 .

The states α and q describe the angle of attack and the pitch rate of the missile, the input δfin is the fin deflection and the output ηz is the measurement of the vertical acceleration. The two time varying parameters θ1 and θ2 reflect the resulting changes in the missile aerodynamics for angles of attack from zero up to 20 degrees. The performance specifications are given by a settling time of 0.2 seconds with a minimal overshoot and zero steady-state error for a step command to the vertical acceleration. The controller must suppress high-frequency noise and the control action δf in should be less than 2. To meet the specifications the missile LPV model (5.1) is augmented as in figure 1, where the filter is chosen as We = 0.8,

Wu =

0.01s3 + 0.03s2 + 0.3s + 1 . 10−5 s3 + 3 · 10−2 s2 + 30s + 104

In addition an integrating function is introduced to the controller as in figure 1, N=

2 + 0.06s s

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ηc

z2 6 Wu

z1 6 We -

6

N

6

-

ηz

6 K

δfin

θ1 0  0 θ2 -

G

q

Figure 1: Augmented plant in closed loop. Method Achieved performance level γ Gridding (36 points) 0.1265 Multi-convexity in [3] 0.1293 Monotonicity in [15] 0.1270 Proposed method 0.1265 Table 1: Numerical results with different relaxation methods First we try to find a controller that meets the specifications for constant parameters within |θ1 | ≤ 1 and |θ2 | ≤ 1. As directional convexity and monotonicity are conservative properties, a upper bound of the parameterized LMI with these properties is sought. An easy way to find this upper bound is to add a positive definite matrix function, see [15], to the parameterized LMI, for example,

ajk ρj ρk I + bl ρl I + cI. (5.2) j,k

l

To ensure positiveness of (5.2) a change of parameter coordinates such that ρi ∈ [0, 1] and letting,   a11 . . . a1p   A =  ... . . .  ≥ 0, bi ≥ 0, c ≥ 0 ap1 app Lemma 2.2 applied to the sum of (5.2) and the synthesis parameterized LMIs forms the set of LMI in this numerical example. Table 1 shows the numerical result. The proposed method is used as a combination of the multi-convexity and monotonicity in different directions with corresponds to αi and βi to be zero and large absolute values. As a second numerical example we try to find a controller that is practically valid ˙ The parameter for time varying θ. That is, a controller that is independent of θ. ˙ ˙ trajectories are bounded by the rates |θ1 | ≤ 1 and |θ2 | ≤ 1. Table 2 shows the numerical results for the missile example, compared to earlier work. The first column ˙ is the achieved γ with X and Y free, that is a solution with a θ-dependent controller, that is a practically invalid controller. The two middle columns are if X or Y is constant, as proposed in [4]. The right most column shows the here proposed

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γ

X,Y free 0.1293

X constant Y constant X,Y semi-definite 0.1505 0.1405 0.1360

Table 2: Numerical results of practical valid LPV synthesis method, namely by letting Xi and −Yi i = 1, . . . , p be semi-definite with the same sign. This is an indication that the proposed method is potentially less conservative, than the one in [4].

6

Conclusions

In this paper a new characterization of parameterized linear matrix inequalities with quadratic parameter dependence, and known bounds in the form of a hyper-cube is proposed. The result is computationally more demanding than earlier work relying on multi-convexity or monotonicity. However, both these are special cases of the proposed method, since it allows both multi-convexity and monotonicity in each direction. The computationally complexity lies in the fact that all vertices of the hyper-cube, as well as the characterizing linear matrix inequalities are parameter dependent. Additionally several optimizations must be performed for different values of the multipliers. The method is applied to the problem of finding a parameter dependent controller for a parameter dependent linear system with a L2 gain performance specification. From the numerical example no drastic gain is accomplished, but an insight of the problem structure. If a parameter dependent Lyapunov function is used for deriving the controller, the controller typically depends on the time derivative of the parameters. Here we present a potentially less conservative solution than existing methods, resulting in a controller that is independent of the parameter derivatives. Since the assumption in linear parameter dependent systems is that the parameter trajectories are bounded in value and rate, but not known a priori, one can treat parameters and its derivatives as two separately independent quantities. The proposed method exploits this fact and finds a worst case for the derivatives. The numerical example also indicates that the proposed method is less conservative than existing methods.

References [1] P. Apkarian and R Adams. Advanced gain-scheduling techniques for uncertain systems. IEEE Transactions on Control Systems Technology, 6(1), 1998. [2] P. Apkarian and P. Gahinet. A convex characterization of gain-scheduled H∞ controllers. IEEE Transaction on Automatic Control, 40(5), 1995. [3] P. Apkarian and H. D Tuan. Parameterized LMIs in control theory. SIAM Journal of Control & Optimization, 38(4):1241–1264, 2000.

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[4] G. Becker. Additional results on parameter-dependent controllers for LPV systems. In IFAC World Congress, volume G, pages 351–356, 1996. [5] G. Becker and A. Packard. Robust performance of linear parametrically varying systems using parametrically-dependent linear feedback. System & Control Letters, 23:205–215, 1994. [6] S. Boyd, L. Ghaoui, E. Feron, and V. Balakrishnan. Linear Matrix Inequalities in System and Control Theory. SIAM, 1994. [7] P. Gahinet. Explicit controller formulas for LMI-based H∞ synthesis. Automatica, 32(7), 1996. [8] P. Gahinet and P. Apkarian. A linear matrix inequality approach to H∞ control. International Journal of Robust and Nonlinear Control, 4:421–448, 1994. [9] P. Gahinet, A. Nemirovski, A. Laub, and M. Chilali. LMI Control Toolbox for Matlab. The Mathworks Inc., 1995. [10] A. Helmerson. Methods for Robust Gain Scheduling. PhD thesis, Dept of Electrical Engineering, Link¨oping University, 1995. [11] R. T. Reichert. Robust autopilot design using µ-synthesis. In American Control Conference, pages 2368–2373, 1990. [12] J. Shamma and M. Athans. Analysis of nonlinear gain-scheduled control systems. IEEE Transaction on Automatic Control, 35:898–907, 1990. [13] J. Shamma and M. Athans. Guaranteed properties of gain sheduled control of linear parameter-varying plants. Automatica, 27:559–564, 1991. [14] D. Stilwell and W. Rugh. Stability preserving interpolation methods for the synthesis of gain scheduled controllers. Automatica, 36, 2000. [15] H. Tuan and P. Apkarian. Monotonic relaxation for robust control: New characterization. In American Control Conference, pages 1914–1918, 2000. [16] F. Wu, X. H. Yang, A. Packard, and Becker G. Induced L2 norm controller for LPV systems with bounded parameter variation rates. International Journal of Robust and Nonlinear Control, 6:983–988, 1996.

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Paper III LPV-BASED GAIN SCHEDULING TECHNIQUE APPLIED TO A TURBO FAN ENGINE MODEL F. Bruzelius Tech. Report R003/2002, Department of Signals and Systems, Chalmers University of Technology, G¨ oteborg, Sweden, January 2002.

Comments: Also submitted to the IEEE Control Systems Society Conference on Control Applications 2002.

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LPV-based Gain Scheduling Technique applied to a Turbo Fan Engine Model F. Bruzelius Department of Signals and Systems, Control and Automation Laboratory Chalmers University of Technology, SE-412 96 G¨ oteborg, Sweden

Abstract The Gain Scheduling controller design technique is one of the most popular approaches in industrial applications, whenever a nonlinear plant is to be controlled and the operating conditions change substantially. However, the shortcomings of such approaches are that stability and performance cannot be guaranteed in so called off-line design points, and for rapid changes of the operating conditions. Here, a method based on Linear Parameter Varying (LPV) systems is applied to a turbo fan engine model. Such a design involves a transformation of the nonlinear model to an LPV system, augmentation of the LPV plant not unlike traditional H∞ control design, and optimization regimes to produce an optimal controller. For transformation of the nonlinear model to an LPV system, the method of velocity based linearization is investigated. Synthesis for a parameter dependent state feedback controller is derived. A controller is designed based on this LPV system.

Keywords: gain scheduling, linear parameter varying systems, velocity based linearization, parameterized linear matrix inequalities, turbo fan engine H∞ control

1

Introduction

In many control design applications, where the object to be controlled changes its dynamical behavior substantially, linear controllers may be insufficient to meet the performance specifications. Nonlinear approaches are most often hard to use if the object is large, that is has many dynamic states. A popular method in hand is then Gain Scheduling technique, which tries to combine linear design thinking with controller design that meets the specifications in a non-local way. Despite many successful implementations, Gain Scheduling in its most naive form has no guarantee for performance and stability in off-design points, and for moving operating 79

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conditions. This is due to the fact that linear control design techniques based on linearizations, do not consider changes in the operating conditions and that the technique not necessarily follows the mapping of the linear controllers in off-design points conditions. Stability preserving methods such as in [16] and methods that guarantee stability for sufficiently slow changes such as in [15], have been presented. More recently, in e.g. [5], research in Gain Scheduling has come to focus on parameterized linear control methods, where the nonlinear model is expressed as a linear parameter varying system (LPV system) or quasi-LPV system and a linear control method is applied to this system for every parameter value. The change of the parameters is directly treated in the design such that, in contrast to traditional gain scheduling, stability and performance are guaranteed in the entire operating window. The operating window is the region in which the parameters are allowed to vary within, and is predefined in the control design process. The design methods most frequently treated in the literature are H∞ and H2 and hybrids of these, since they correspond to an optimization problem, that can be treated in the parameterized situation. The difficulty in applying such methods is the lack of good tools to convert or approximate nonlinear models to LPV models, [12]. In this work, the method of LPV control design is investigated, with a turbo fan jet engine model as an example. The most frequently used method for turning nonlinear models into LPV models, is to map linearizations together. There is in general, however, no obvious connection between this mapped model and the nonlinear model, see e.g. [12]. Also a variety of analytical methods are treated in the literature such as in [9] and [19]. These are however not applicable in a situation when the model is of high order, and/or contains look-up tables and implicit expressions. Here, velocity based linearization is investigated as a tool for turning nonlinear systems into an LPV form. Velocity based linearization is, given correct initial condition, an exact description of the nonlinear system. The input and output of a velocity based linearized system are the time derivatives of the corresponding nonlinear system. Problems associated with this in the control design process are discussed here. With some assumptions, an affine LPV system is mapped to the velocity based linearized system. The controller design, including the LPV transformation, is applied to a FORTRAN code turbo fan engine model, [1], with 7 states and the fuel flow as the single input. Since the jet engine model is intended to be used as an observer, parameter dependent state feedback is considered here. Sufficient conditions for the existence of such a controller are derived in terms of parameterized linear matrix inequalities, and formulas for a specific controller are derived. The design instrument is to pick augmentation filters, with auxiliary inputs and outputs. The task of the controller is to keep a small induced L2 -norm gain from the auxiliary input to the auxiliary outputs, as in the Linear Time Invariant (LTI) case of H∞ control design. When dealing with linear parameter varying systems, and especially ones with affine parameter dependence, the information about the operating condition (or the parameter) is not essential for stabilizing the system [6]. That is, if we can make an

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affine parameter dependent system stable through a parameter varying state space feedback matrix, then it is also possible to make the system stable with a constant matrix. However, one can achieve better performance by use of the parameter variations when designing the feedback law compared to a a linear gain feedback, since scheduling parameter independent feedback only is a special case. The notation in this paper is rather standard. For transpose components in block matrices, a star
is used. Moore-Penrose pseudo inverse of a matrix M is denoted by M † . Time and parameter dependences are left out whenever it is obvious. The report is divided into sections as follows. First a brief discussion on methods for turning nonlinear systems into an LPV system is given. A method called velocity based linearization is then discussed more in detail. After that, the design method used is presented, which is an state feedback H∞ -like LPV control design technique. Finally, the developed methods are applied to a turbo fan engine model, including numerical transformation of the nonlinear system into an LPV system, picking weighting filters and finding a controller. The result is indicated by simulations on the closed loop system.

2

LPV Modeling

Consider a general nonlinear model, x˙ = F (x, u, θ) y = G(x, u, θ)

(2.1)

where x denotes the state vector, y the measured output, u the control input and θ an on-line measured input quantity. To be able to apply the LPV control design framework to the system (2.1), a transformation of the system (2.1) is necessary. The goal is to find a set of differential equations on the LPV form, x˙ = A(ρ)x + B(ρ)u y = C(ρ)x + D(ρ)u

(2.2)

where ρ = ρ(x, u, θ) is the parameter or scheduling vector variable of the system, such that the solution of (2.1) is as close to the solution of (2.2) as possible in a predefined operating region of ρ. The scheduling variables or parameters may contain endogenous signals of the system and the exogenous signals that define the operating conditions. The operating region is defined by a compact set Γ, where ρ(t) ∈ Γ. For components of the parameter which correspond to exogenous signals, it is often possible to put bounds on the time derivative of that part of the parameter vector in the same fashion, that is ρ(t) ˙ ∈ ∆. A standard method to linearize nonlinear systems such as (2.1) is by truncating the Taylor series expansion of the nonlinear system, and include the Jacobian first order terms. To end up in a linear system, the series expansion is performed around a

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stationary point or a stationary trajectory of the nonlinear system, that is f (x, u) = f (x0 , u0 ) +

∂f (x0 , u0 ) ∂f (x0 , u0 ) (x − x0 ) + (u − u0 ) ∂x ∂u 2

(2.3)

2

+O(x − x0  , u − u0  , x − x0  · u − u0 ) where f (x0 , u0 ) = 0 in stationary. This will capture the local behavior in the (x0 , u0 ) surrounding. There are well known stability results relating the linearized system (2.3) to the corresponding nonlinear one. The closeness to (x0 , u0 ) is essential, however. Parameterized Jacobians to achieve an LPV system with larger region validity may not be sufficient for the application of LPV control design, see e.g. [12]. To see this, consider the following illustrative example, taken from [13], Example 2.1 Consider the nonlinear system, x˙ = f (x − 10u) where f (0) = 0 is an unique zero of the function and f is twice differentiable. By linearization, the resulting system becomes, δ x˙ =

∂f (0) ∂f (0) δx + δu. ∂x ∂u

Observe that this linearization is valid for all inputs whenever δx and δu are sufficiently small. Moreover, the linearization is only valid point-wise for all x and u satisfying x − 10u = 0 and may not be treated as an approximation of the nonlinear system. 3

Other techniques to achieve a linearized (and parameter dependent) model to be valid globally, or at least in a region, most often require the nonlinear model left hand side to be analytic or at least continuously differentiable a number of times. Most often larger models contain look-up tables that may not be expressed so easily by functions. The turbo fan engine is neither analytic nor on closed form, but we may assume with reasonable errors that the model is at least once differentiable. To be able to get around the problem of closeness to a stationary point/trajectory it is possible to differentiate in time instead of x and u etc, as suggested in [13]. We illustrate this with the following nonlinear model, x˙ = Ax + Bu + f (ρ) y = Cx + Du + g(ρ)

(2.4)

where ρ = ρ(x, u) is a vector valued function depending on x and u, with less or equal dimension to the sum of the state and the input dimensions. The constant matrix quadruple (A, B, C, D) represents the linear part of the system. An equivalent description of (2.4), that is a system with the same solution given correct initial

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values is given by differentiating the left hand sides of (2.4) with respect to time, where x˙ is denoted by ξ, ∂ρ ∂ρ ξ+ u˙ ∂x ∂u ∂f (ρ) ∂ρ ∂f (ρ) ∂ρ ξ˙ = (A + )ξ + (B + )u˙ ∂ρ ∂x ∂u ∂u

ρ˙ =

y˙ = (C +

(2.5)

∂g(ρ) ∂ρ ∂g(ρ) ∂ρ )ξ + (D + )u, ˙ ∂ρ ∂x ∂ρ ∂u

∂ρ ∂ρ and ∂u are assumed to depend only on ρ, the system If the two partial derivatives ∂x (2.5) is clearly on the LPV form (2.2) with u˙ as input and [ξ T , y T ]T as the state vector. A trivial example of such a ρ is ρ = [xT , uT ]T . There is a couple of obvious disadvantages with this re-formulation. The input is a derivative of the input to the original system. However, the model only intended to be used in a control design process. The output of the designed controller is then integrated to get the desired control u. Another problem with (2.5) is that the state vector is now of a higher order (the dimension of the state-vector plus the output-vector). A general problem in robust control design is that the design of low order controllers often leads to nonconvex optimization problems. For this work, it is assumed that the state vector (or the time derivative of it) is measured, such that a state (derivative) feedback is possible. By assuming this, the state vector has the same size in the velocity based linearization as that of the nonlinear model.

A different approach to linearize in off-equilibrium points to achieve an LPV model, is to map Taylor series expansions performed in non-stationarity, see e.g.[11]. This will however lead to a system with an affine right hand side. This can, under some circumstances, be turned into a non-affine model using an input transformation. However, closeness to stationarity issues like in Example 2.1 remains unsolved.

3

Design Method

In this section the control design method is briefly described. We begin by showing how a specification can be met using augmentation filters and how these specifications can be expressed in the form of parameterized LMIs. In the next section it will be described how to turn the control problem into a convex optimization problem, that can be solved with standard optimization code, such as LMI Lab for Matlab, [10]. The results in this section is based on fundamental properties of LMIs. For a detailed description of LMIs and how these types of problems can be solved see [7]. A standard setup in robust control is to minimize a measurement of the gain from a disturbance input to some auxiliary output of the plant, e.g. [22]. By selecting the auxiliary output one can design the controller to meet specified performance specifications. This can be done in numerous ways, see e.g. [22]. A standard way is to choose augmentation weights as in Figure 1. By selecting the linear filter

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- Wu

- zu

- WT

- zT

- ?- WS

- zS

- P

u w

-y

Figure 1: Augmented plant with weighting filters. weights (WS , WT , Wu ), for a frozen value of ρ, one can keep the interpretations of LTI systems such as frequency and loop shaping. For example one can set upper constraints on the sensitivity function S, the complementary sensitivity function T and the process sensitivity function KS. A more detailed discussion on how these filters can be chosen in the velocity based linearization case is given in the next section. The control design objective is to find a controller such that zL2 ≤ γ, wL2 (ρ,ρ)∈(Γ,∆) ˙ sup

w=0,

is obtained, where dL2 denotes the induced L2 -norm, that is,
 ∞  12 dT (t)d(t)dt . dL2 = 0

The augmented system then has the form, x˙ = A(ρ)x + B1 (ρ)w + B2 (ρ)u z = C1 (ρ)x + D11 (ρ)w + D12 (ρ)u y = C2 (ρ)x + D21 (ρ)w + D22 (ρ)u

(3.1)

where z denotes the auxiliary output of the system and w the disturbance. Observe that x now denotes the states of the plant plus the states of the augmentation filters. Without loss of generality it can be assumed that D22 = 0, see e.g. [22]. Here we will only treat the full state information case, meaning that the full state vector x is known for example by measurements, and the third row in (3.1) is of less importance. This implies that we can use state feedback (u = Kx) to achieve the disturbance attenuation. However, the states in (3.1) will contain the states of the augmentation filters, implying that the resulting control system will include these states as well. The LPV extension to the well known Bounded Real Lemma, see e.g. [21], can be directly used for the closed loop to find a feedback that fulfills the L2 -norm, but will result in a bi-linear matrix inequality problem in the feedback K and the

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Lyapunov matrix P . Such problems are numerically hard to solve. However, using transformations this inequality can be converted to an equivalent linear (affine) one. The following theorem for the state feedback problem guarantees existence of a controller. It can be used to optimize the performance measure γ, that is with respect to the matrix functions Q and R, with some additional relaxations discussed in the next section. Theorem 3.1 Consider the LPV system (3.1). A memoryless state feedback controller u = Kx achieving the disturbance rejection, zL2 ≤γ wL2 (ρ,ρ)∈(Γ,∆) ˙ sup

w=0

(3.2)

is given by K = RQ−1 where Q = QT > 0 and R are matrix-valued functions satisfying the following matrix inequality,   T QAT + QA − Q˙ + RT B2T + B2 R B1 QC1T + RT D12 T  0 satisfying   T T (A + K T B2T )P + P (A + B2 K) + P˙ P B1 C1T + K T D12 T  0 ∀ρ ∈ vertΓ,

R i ρi

A controller matrix valued function K(ρ) is then given by K(ρ) = R(ρ)Q−1 (ρ). Proof. The theorem is obtained from, assuming an affine parameter dependence on Q and R, a direct application of multi-convexity of the axial parallel parameter box Γ , see e.g. [3], on Theorem 3.1 with λi as positive multipliers and a switch of the second and third row and column. Theorem 4.1, gives a sufficient condition which indicates that it is conservative. The assumptions made are on the structure of the ρ dependence in Q and R and that the parameterized LMI is multi-convex in each axial direction of the ρ-space. The multi-convexity property can be used to the parameter box in Figure 2. However, this will result in a larger number of constraints which are potentially even more conservative. Other relaxations such as monotonicity, [18], or combinations such as [8] exist in the literature, and the conservatism of these methods are problem specific. No further investigation on differences between the different relaxation methods has been performed to this problem.

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For the augmentation, as mentioned before, only LTI filters are used. To get a hint on how to pick these linear filters, a frequency interpretation approach is applied with frozen parameters. In order to keep the controller order low, only first order augmentation filters will be considered. In a more realistic case the order of the augmentation filters are higher (typically of order 3). This will lead to a controller of high order. To obtain a lower order controller, an order reduction must be performed, which corresponds to a non-convex optimization problem. The filter Wu that sets constraints on the process sensitivity function, that is KS(s) = K(I − KP (s))−1 in the linear case, may be parameterized as ˜ u (s) = s + ωbc /Mu . W εs + ωbc where ωbc corresponds to an upper bound on the controller bandwidth and the noise frequencies, Mu an upper bound on the process sensitivity function and ε is a small positive parameter introduced to get a proper filter. It must be taken into account that the model is on velocity form (2.5) having the derivative of the original ˜ u (s) input u as its input. To compensate for this an additional integration in W could be included. A pure integration would lead to a possible loss of stabilizability and detectability, i.e. loss of a solution to our controller finding problem, and an approximation of the integration is necessary, ˆ u (s) = W

1 ˜ Wu (s) s + ε1

˜ u (s) can now be where ε1 is a small positive constant. The additional pole in W eliminated, since the filter is proper with the approximation of the integration. The filter can now be parameterized as, Wu (s) =

s + ωbc /Mu . ωbc (s + ε1 )

(4.9)

The other augmentation filter used here, WS , is used to set constraints on the sensitivity function, S(s) = (I − KP (s))−1 A parameterization of WS is, ˜ S (s) = s/MS + ωb , W s + ωb ε2

(4.10)

where MS is an upper bound on the sensitivity function, ωb is a lower bound on the bandwidth and ε2 is a small parameter to avoid a pure integration. The output of the LPV system is the derivative of the EP R , and using the same argument as above to compensate for this, the following parameterization is obtained, WS (s) =

1 ˜ WS (s) s + ε3

(4.11)

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for some small positive ε3 . Observe that using this augmentation structure, the LMIs in Theorem 4.1 can be reduced. For example D11a = 0 etc. Also worth observing is that P (ρ) will only depend on ρe , that is, only true exogenous quantities. This implies that stability can be guaranteed in the specified region of the last component of the scheduling variable, and for arbitrary fast changes internally. Reasonable values of the design were obtained with the following set of parameters: Ms = 1, ωb = 1, ωbc = 0.1 and Mu = 100. Allowing the true exogenous parameter to vary within the parameter box, ρ˙ e (t) ∈ ∆ = {p| − 0.1 ≤ pi ≤ 0.1, i = 1, 2}, resulted in an optimal γ = 1.6034 when minimizing γ over the LMIs in Theorem 4.1.

4.3

Nonlinear Simulations

Here we will present some closed loop simulations with the parameter dependent state feedback and the nonlinear engine model. The derived controller is a state (derivative) feedback controller. However, the state vector which was used in the control design includes the augmented states. These augmented states must be constructed in the controller, and feed back by the state feedback matrix function K(ρ), see Figure 5. As the sensitivity augmentation filter WS was connected to the derivative of the output EP R , and an approximative integration was included in the filter, see (4.11), one can cancel the approximative ˜ S (4.10) needs to be constructed in integration for the derivative such that only W the controller. This implies, assuming the state derivative of the engine is measured, that the controller will have three continuous states, or nc = n u + n a − ny = 1 + 3 − 1 = 3 where nc , ny and na are the orders of the controller, the output of the plant (the jet engine) and the augmentation filters, respectively. Observe that the resulting controller has the form,  u = K(ρ(τ ))xa (τ )dτ,

(4.12)

with xa (t) = [x˙ T (t), xTu (t), xTS (t)]T . One may argue that the state derivative of the jet engine, x, ˙ is an unknown quantity, not available for feedback. An alternative implementation to (4.12) is to use the estimated derivatives,    F (x(τ ), u(τ ), θ(τ ))

 dτ. Kj (ρ(τ )) Ku (ρ(τ )) KS (ρ(τ ))  xu (τ ) u= xS (t)

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-



u -

Engine

− EP R -

+ 

EPref R

x˙  u˙

K(ρ)



-

xS



˜ S (s)  W

xu

Wu (s)

Figure 5: The engine in closed loop. where K = [Kj , Ku , KS ] and xu and xs are the states from the augmentation filters, see Figure 5. This will however introduce the model uncertainties into the control law. Using partial integration on (4.12),   u(t) = Kj (ρ(τ )x(τ ) +

−

dKj (ρ(τ )) dτ



 x(τ ) Ku (ρ(τ ))) KS (ρ(τ ))  xu (τ )  dτ, xS (t) 

implying that another possibility to eliminate the x(t) ˙ dependence in the controller would be to assume that the parameter changes slow, i.e. to say that ρ(t) ˙ ≈ 0. Then the synthesis method used here would reduce to the case of standard parameter dependent state feedback controller based on a mapping of linearizations, see e.g. [20]. In the implementation phase, a discretization of the controller is necessary. To reduce the computation in a real-time situation, the controller dynamics (parameter updates) does not need to be updated for every sample interval. This is due to the negativeness of the synthesis LMIs. For a more detailed discussion on such matters see [2]. A take-off situation of the closed loop system has been simulated, using the nonlinear turbofan engine model, [1]. The surroundings of the simulation were defined by an assumed intake Mach number with linear increase from zero to 0.8 in 30 seconds and a linear increase of the altitude from 0 to 10688 meters also in 30 seconds, see Figure 6. This is of course a non-realistic take-off situation, e.g. a vertical mean

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value velocity of 356 m/s. However, the parameters are within range of the control design and the situation illustrates the potential of the controller. One model was used as an observer and one as a model of the turbo fan engine. Actuators and burner dynamics are modeled as first order linear filters with relatively fast responses and therefore not considered in the design step. In Figure 6 in the 1600

1.5

1500

1.4

1400 1.3

T41

EP R

1300 1200

1.2

1100 1.1

1000 900 0

10

20 time

30

1 0

40

1.4

4500

1.2

4000

1

20 time

30

40

10

20 time

30

40

XN L

3500

WF E

0.8

3000

0.6

2500

0.4

2000

0.2 0 0

10

10

20 time

30

40

1500 0

Figure 6: Nonlinear simulation with state derivative feedback. upper right graph, one can observe that the tracking of the reference EPref R (dashed) is good. After one second the idle speed engine is given a thrust command step (EPref R ), and after 10 seconds the altitude and the intake Mach number start to increase. The controller is able to compensate for such changes of the surrounding. The wear of the compressor is closely related to the temperature T41 and its time derivative. A fast change of T41 is not desirable. In a more realistic manoeuvre the change of T41 would be slower than in this simulation.

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97

Concluding remarks

A procedure on how to make a control design using a parameter dependent feedback applied to a nonlinear jet engine model has been presented. An L2 -norm performance measure, not unlike H∞ control design, was developed for state feedback controllers, using a velocity based linearization of the nonlinear model. Augmentation considerations are discussed, especially problems related to the fact that the velocity model has a differentiation on the inputs and the outputs. Connected to this, a brief discussion on the implementation of the state derivative controller is carried out. Only simple (low order) linear time invariant augmentation filters was used in the control design. The frequency interpretation of these filters becomes weaker and weaker as the operating conditions change. Numerical methods to perform an approximation of the nonlinear model to an affine structured linear parameter varying system, was investigated, using different norms and different sets of scheduling parameters. Simulations of the closed loop system, using one model as an observer, shows a good tracking performance. As all control design methods based on linear parameter varying (LPV) systems, have the weak link of mapping nonlinear models into LPV models. The velocity based linearization method has a more solid theoretical foundation than, what has become a standard method, mapped stationary Taylor linearizations. However, the difference in the two methods of turning nonlinear models to LPV models is only an integration and numerically, the sensitive numerical linearization has to be used to obtain Jacobian matrices and the mapping procedure is no different.

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Appendix Selected variables of importance. Fan speed, State variable XN L Compressor speed, State variable XHL TM 3B Metal temperature at HPC, State variable TM 3C Metal temperature at HPC, State variable TM 4B Metal temperature at Combust. chamber, State variable TM 42B Metal temperature at HPT, State variable TM 42C Metal temperature at HPT, State variable Gas temperature at HPT inlet cooling bleed recovery T41 Pressure ration, Output EP R ZXM Intake Mach number, exogenous input Altitude , exogenous input ZALT Fuel flow, control input WF E Pressure at induct P2 Pressure at LPT P5 where HPC denotes the High Pressure Compressor, HPT the corresponding turbine, and LPT the Low Pressure Turbine.

P2

TM 3C TM 3B

TM 4B

T41

P5 TM 42C TM 42B

References [1] OBIDOCTE, 1997. BRITE/EURAM,BE 97-4077. [2] P. Apkarian. On the discretization of LMI-synthesized linear parameter-varying controllers. Automatica, 33(4):665–661, 1997. [3] P. Apkarian and H. D Tuan. Parameterized LMIs in control theory. SIAM Journal of Control & Optimization, 38(4):1241–1264, 2000.

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[4] G. Bara and J. Daafouz. Parameter-dependent control with γ-performance for affine LPV systems. In Conference on Decision & Control, pages 2378–2379, 2001. [5] G. Becker and A. Packard. Robust performance of linear parametrically varying systems using parametrically-dependent linear feedback. System & Control Letters, 23:205–215, 1994. [6] F. Blanchini. The gain scheduling and the robust state feedback stablization problems. IEEE Transactions on Automatic Control, 11:2061–2070, 2000. [7] S. Boyd, L. Ghaoui, E. Feron, and V. Balakrishnan. Linear Matrix Inequalities in System and Control Theory. SIAM, 1994. [8] F. Bruzelius. On the computation of LPV controllers. Technical report, Dep. of Signals and Systems, Chalmers University of Technology, 2002. [9] F. Bruzelius and C. Breitholtz. Gain scheduling via affine linear parametervarying systems and H∞ synthesis. In Conference on Decision & Control, pages 2386–2391, 2001. [10] P. Gahinet, A. Nemirovski, A. Laub, and M. Chilali. LMI Control Toolbox for Matlab. The Mathworks Inc., 1995. [11] T. A. Johansen, K. Hunt, P. J. Gawthrop, and H. Fritz. Off-equilibrium linearisation and design of gain scheduled control with application to vehicle speed control. Control Engineering Practice, 6:167–180, 1998. [12] D. Leith and Leithead W. On formulating nonlinear dynamics in LPV form. In Conference On Decision and Control, pages 3526–3527, 2000. [13] D. J. Leith and W. E. Leithead. Comments on the prevalence of linear parameter varying systems. Technical report, Dep. of EE Enginering, University of Strathclyde, Glasgow UK, 1999. [14] E. Nordgren, Z. Gastineau, S. Adibhatla, G.S. Grewal, and O.D.I. Nwokah. Robust multivariable turbofan engine control: A case study. In Conference On Decision & Control, pages 1086–1097, 1994. [15] J. Shamma and M. Athans. Analysis of nonlinear gain-scheduled control systems. IEEE Transaction on Automatic Control, 35:898–907, 1990. [16] D. Stilwell and W. Rugh. Stability preserving interpolation methods for the synthesis of gain scheduled controllers. Automatica, 36, 2000. [17] X. Sun and I. Postlethwaite. Affine LPV modeling and its use in gain scheduled helicopter control. In UKACC International Conference on Control, 1998. [18] H. Tuan and P. Apkarian. Monotonic relaxation for robust control: New characterization. In American Control Conference, pages 1914–1918, 2000.

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