Abstract: Optimization problems in chemical engineering often involve complex systems of nonlinear DAE as the model equations. The direct multiple shooting.
Copyright @ IF AC Advanced Control of Chemical Processes, Pisa, Italy, 2000
REAL-TIME OPTIMIZATION AND NONLINEAR MODEL PREDICTIVE CONTROL OF PROCESSES GOVERNED BY DIFFERENTIAL-ALGEBRAIC EQUATIONS H. Georg Bock * Moritz Diehl* Johannes P. Schloder * Frank Allgower ** Rolf Findeisen ** Zoltan N agy ***
* Interdisciplinary Center for Scientific Computing (IWR),
University of Heidelberg, Germany ** Institute for Systems Theory in Engineering, University of Stuttgart, Germany *** Faculty of Chemistry and Chemical Engineering, "Babes-Bolyai" University of Cluj, Romania
Abstract: Optimization problems in chemical engineering often involve complex systems of nonlinear DAE as the model equations. The direct multiple shooting method has been known for a while as a fast off-line method for optimization problems in ODE and later in DAE. Some factors crucial for its fast performance are briefly reviewed. Recently, this approach has been successfully adapted to the specific requirements of real-time optimization. Special strategies have been developed to effectively minimize the on-line computational effort, in which the progress of the optimization iterations is nested with the progress of the process. They use precalculated information as far as possible (e.g. Hessians, gradients and QP presolves for iterated reference trajectories) to minimize response time in case of perturbations. In typical real-time problems they have proven much faster than fast off-line strategies. Nonlinear Model Predictive Control (NMPC) may be interpreted as a special realtime optimization problem. Strategies to exploit the similarities between subsequent NMPC optimization problems are described. Numerical results for the NMPC of a high-purity distillation column subject to parameter disturbances are presented. Copyright © 2000 IFAC Keywords: Predictive Control, Nonlinear Control Systems, Differential Algebraic Equations, Numerical Methods , Optimal Control, Distillation Columns
1. INTRODUCTION
the capability to directly handle equality and inequality constraints, and the possibility to treat disturbances fast.
Real-time control based on the optimization of nonlinear dynamic process models has attracted increasing attention over the past decade, e.g. in chemical engineering (cf. Garcia et al. (1989) ; Biegler and Rawlings (1991) ; Bock et al. (2000) ; Diehl et al. (1999}). Among the advantages of this approach are the flexibility provided in formulating the objective and the process model,
One important precondition, however, is the availability of reliable and efficient numerical optimal control algorithms. Direct methods reformulate the original infinite dimensional optimization problem adaptively as a
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sequence of finite nonlinear programming (NLP) problems by a parametrization of the controls and states. Such a method is called a simultaneous solution strategy, if the NLPs are solved by an infeasible point method such as sequential quadratic programming (SQP) or generalized Gauss-Newton. Typical parametrizations are collocation, finite differences, or direct multiple shooting (cf. Bock and Plitt (1984); Leineweber (1999» , which is the basic method we treat in this paper.
to dovetail the iterative solution procedure with the process development in order to compute fast approximate dosed-loop controls. • The NMPC of a high-purity distillation column model of 164 states is treated in section 5. As a scenario, disturbances in the feed stream are considered, which result in changes of the desired operating point.
2. REAL-TIME OPTIMAL CONTROL PROBLEMS
Direct multiple shooting offers the following advantages in the context of real-time process optimization:
Throughout this paper, we consider optimal control problems of the following simplified type:
• As a simultaneous strategy, it allows to exploit solution information in controls, states and derivatives in subsequent optimization problems by suitable embedding techniques. • Efficient state-of-the-art DAE solvers are employed to calculate the function values and derivatives. • Since the integrations are decoupled on different multiple shooting intervals, the method is well suited for parallel computation. • The approach allows a natural treatment of control and path constraints as well as boundary conditions.
tf
min
u( · ),x( ·) ,p
jL(X(t) , z(t) ,U(t),P)dt to
+
(1)
E(x(tf ),p)
subject to a system of differential algebraic equations (DAE) of index one
= f(x(t), z(t), u(t) ,p)
(2)
0= g(x(t), z(t) , u(t) ,p)
(3)
B( ·)x(t)
Here, x and z denote the differential and the algebraic state vectors, respectively, u is the vector valued control function , whereas p is a vector of system parameters. Matrix B( ·) is assumed to be invertible, so that the DAE is of semi-explicit type.
The efficiency of the approach, which has been observed in many practical applications, has several reasons. One of the most important is the possibility to incorporate information about the behaviour of the state trajectory into the initial guess for the iterative solution procedure; this can damp the influence of poor initial guesses for the controls (which are usually much less known) . In the context of NMPC , where a sequence of neighbouring optimization problems is treated, solution information of the previous problem can be exploited.
Initial values for the differential states and values for the system parameters are prescribed:
= Xo
(4)
p=Po
(5)
x(to)
In addition, terminal constraints
The paper reviews results presented in Bock et al. (2000) ; Nagy et al. (2000); Bock et al. (1999) ; Diehl et al. (1999), and presents a new view on real-time optimization in NMPC. It is organized as follows :
(6)
as well as state and control inequality constraints
h(x(t) , z(t) ,u(t),p)
• In section 2, we introduce a general class of optimal control problems for NMPC that can be treated by the current implementation of the real-time direct multiple shooting method. • We sketch the direct multiple shooting method in section 3, with emphasis on the SQP method specially tailored to the solution of such highly structured NLP problems. • In section 4 we describe a boundary value embedding strategy for subsequent NMPC optimization problems. This strategy allows
~
0
(7)
have to be satisfied. Remark: The reason to introduce the parameters p as a variable subject to the equality constraint (5) has certain algorithmic advantages, as will become apparent in section 4. Solving this problem we obtain an open-loop optimal control and corresponding state trajectories, that we may implement to control a plant. However, during operation of the real process, both
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state variables and system parameters are most likely subject to disturbances, e.g. due to model plant mismatch. Hence, an optimal closed-loop or feedback control law
on the N intervals it;, t;+ll by introducing the initial values sf and si (combined : s;) of differential and algebraic states at times t; as additional optimization variables.
(8)
On each subinterval [t;, tHd we compute the trajectories x;(t) and z;(t) as the solution of an initial value problem:
would be preferable that gives us the optimal control for a sufficiently large range of time points to and initial values Xo and parameters Po . One computation ally expensive and storage consuming possibility would be to precalculate such a feedback control law off-line on a sufficently fine grid. In contrast to this, the present paper is concerned with efficient ways to calculate this feedback control in real-time for progressing to . One important variant of the optimization problem (1)-(7) arises in NMPC, where the final time t I progresses with to : (9)
The constant Tp is called the prediction horizon . In this case the closed-loop control u does no longer depend on the time.
B( ·)i;(t)
= f(x;(t), z;(t), u; , p)
(13)
0= g(x;(t) , z;(t) , u;,p) -o:;(t)g(s f,sf ,u;,p) x;(t;) = sf
(14) (15)
Here the subtrahend in (14) is deliberately introduced to allow an efficient DAE solution for initial values and controls sf, sf , u; that may violate temporarily the consistency conditions (3) . Therefore we require for the scalar damping factor 0: that o:;(t;) = 1. For more details on the relaxation of the DAE the reader is conferred, e.g. to Leineweber (1999) or Schulz et al. (1998) . Note that the trajectories x;(t) and z;(t) are functions of the initial values, controls and parameters sf , si , u;, p. Analogously, the integral part of the cost function is evaluated on each interval independently:
(10)
J
ti+l
L;(sf, sf, u;,p)
3. DIRECT MULTIPLE SHOOTING FOR OPTIMAL CONTROL
=
L(x;(t), z;(t) , u;,p) dt (16)
ti
The solution of the real-time optimal control problem is based on the direct multiple shooting method, which is reviewed briefly in this section. For a more detailed description see e.g. Leineweber (1999) .
3.2 The Problem
Structured
Nonlinear
The parametrization of problem (1)-(7) using multiple shooting and a piecewise constant control representation leads to the following structured nonlinear programming (NLP) problem :
3.1 Parametrization of the infinite optimization problem
N-l
min
L
+
L;(sf,si,u;,p)
U,8,p i==O
The parametrization of the infinite optimization problem consists of two steps. For a suitable partition of the time horizon [to, t I J into N subintervals it;, tHd with
we first discretize the control function u( ·). For simplicity, we assume here that it is parametrized as a piecewise constant vector function
= u; for t E [t;, t;+lJ,
E(sN'P) (17)
subject to the initial value and parameter constraint
(11)
u(t)
Programming
sf = Xo ,
(18)
P=Po ,
(19)
the continuity conditions
sf+l
(12)
= x;(tHd
i = 0, 1, ... N -1, (20)
and the consistency conditions i=O , I , . . . N. (21)
0= g(sf,sf,u;,p)
but every parametrization with local support could be used without changing the structure of the problem.
Control and path constraints are imposed pointwise at the multiple shooting nodes
In a second step the DAE are parametrized by multiple shooting . We decouple the DAE solution
h(sf,sf,u;,p)
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~
0
i
= 0, 1, . .. N
(22)
As a consequence, the Hessian of f. has a block diagonal structure with blocks 'V';,l;(Wi , >',J.L) . Similarly, the multiple shooting parametrization introduces a characteristic block sparse structure of the Jacobian matrices 'VG(w)T and 'VH(w)T .
as well as at the terminal point r(siv,p) { ; }
o.
(23)
It is of crucial importance for performance and numerical stability of the direct multiple shooting method that these structures of (24) are fully exploited. A number of specific algorithmic developments contribute to this purpose:
3.3 SQP for Multiple Shooting
The above NLP problem (17)-(23) is solved by a sequential quadratic programming (SQP) method tailored to the multiple shooting structure. The NLP can be summarized as minF(w) w
G(w)=O
subject to
{
H (w)~O ,
• For the exploitation of the block diagonal structure of the Hessian, three versions are recommended for different purposes: a) A numerical calculation of the exact Hessian corresponds to Newton 's method. This version is recommended if the computation of the Hessian is cheap, and in the case of neighbouring feedback control, where the Hessian can be computed and stored in advance. The use of the "exact" Hessian has excellent local convergence properties. For globalisation techniques based on trust regions are needed, since the Hessian may become indefinite far from the optimal solution. b) Partitioned high rank updates as introduced in Bock and Plitt (1984) speed up local convergence with negligible computational effort for the Hessian approximation. c) A third approach to obtain a cheap Hessian approximation - the constrained GaussNewton method - is recommended in the special case of a least squares type cost function F(w) = !"G(w)"~ . The matrix 'VufJT'VufJ is already available from the gradient computation and provides an excellent approximation of the Hessian, if the residual G(w) of the cost function is sufficiently small. • Special recursive QP solvers are used for problem (26) that exploit the block sparse structure of (24). Both active set strategies and interior point methods are available for the treatment of large systems of inequality constraints (Bock and Plitt (1984) ; Leineweber (1999) ; Steinbach (1995)) . • Leineweber (1999) developed a reduction technique for DAE systems with a large share of algebraic variables. He exploits the linearized algebraic consistency conditions for a reduction in variable space, so that only reduced gradients and Hessian blocks need to be calculated, which correspond to the differential variables only (cf. SchlOder (1988) ; Bock et al. (1988)) • The solution of the DAE initial value problems and the corresponding derivatives are computed simultaneously by specially designed integrators which use the principle of internal numerical differentiation. In particular, the integrator DAESOL,
(24)
where w contains all the multiple shooting state variables and controls as well as the model parameters. The discretized dynamic model is included in the equality constraints G(w) = O. Starting from an initial guess wo , an SQP method for the solution of (24) iterates wk+l = w k + a k Ll.wk,
k = 0, 1, . . . , (25)
where a k E ]0, 1] is a relaxation factor, and the search direction Ll.w k is the solution of the quadratic programming (QP) subproblem 'VF(wk)T Ll.w
+ -1 Ll.wT Ak Ll.w 2
(26)
subject to G(w k ) + 'VG(wk)T Ll.w=O H(w k ) + 'VH(wk)T Ll.w~O.
(27)
A k denotes an approximation of the Hessian of the Lagrangian function f,
where>. and J.L are the Lagrange multipliers. Some remarks are in order on how to exploit the multiple shooting structure in the construction of a tailored SQP method. Due to our choice of state and control parametrizations the NLP problem and the resulting QP problems have a particular structure: the Lagrangian function f is partially separable , i.e., it can be written in the form N
f(w , >. , J.L)
=
Lfi(Wi , >',J.L),
(29)
i=O
where Wi are the components of the primal variables w which are effective on interval [ti , ti+l] only. This separation is possible if we simply interpret the parameters p as piecewise constant continuous controls.
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(see e.g. Bauer et al. (1997)), which is based on the backward-differentiationformulae (BDF), was used in the numerical calculations presented in this paper. • The DAE solution and derivative generation can be performed in parallel on the different multiple shooting intervals. Numerical test show very satisfactory speedup. Important for the use of the above methods in the real-time context is their excellent local convergence behaviour. By proper strategies to select stepsizes a k or trust regions nk or both, global convergence can be theoretically proven. Reassuring as this property is, it is of lesser importance in real-time optimization, as generally no runtime bounds can be established. For a detailed description of globalisation strategies available in the latest version of direct multiple shooting (MUSCOD2) the reader is referred to Leineweber (1999).
4. REAL-TIME EMBEDDING STRATEGIES In a real-time scenario we aim at solving a sequence of optimal control problems that differ by progressing time to, and by initial values Xo which we expect to deviate from the values predicted by the model. We must also expect that some of the parameters Po, which are assumed to be constant in the model, are subject to disturbances.
In contrast to this, the authors suggest an algorithm which differs from this approach in two important aspects: First, we propose to start the solution iterations from the solution for the reference values Xo,Po instead of the deviated values. It turns out that the formulation of the constraints (18), (19) , which can be considered as an Initial Value Embedding of the optimal control problem into the manifold of perturbed problems, is crucial for the real-time performance. Not only all important components of the first quadratic program are already available, but also the decomposition and solution of this QP can be largely done without knowledge of the actual values x~, p~ . This differs significantly from the conventional approach , where all function values and derivatives need to be recalculated before even any QP can be formulated. In our approach, the first iteration is available in a negligible fraction of the time of a whole SQP iteration! Moreover, it is easy to show that the error of this first correction in the NLP variables compared to the solution of the full non linear problem is only second order in the size of the perturbation. Based on this observation, we secondly propose not to iterate the nonlinear solution procedure to convergence, but rather use the following scheme:
+ +
The time between consecutive optimization problems must be short enough to guarantee a suffienctly fast reaction to disturbances. Fortunately, we can assume that we have to solve a sequence of neighbouring optimization problems. Let us assume that a solution of the optimization problem for values to, Xo, Po is available, including function values, gradients and a Hessian approximation, but that at time to the real values of the process are the deviated values x~,p~ . How to obtain an updated value for the feedback control u(x~ , p~, tf - to)?
Apply the result of the first correction of the controls immediately to the real process during some process duration 6. During this period 6 first compute the full QP solution, and based on this iterate, compute a new linear-quadratic model expansion for the next time step, and solve the QP as far as possible to prepare the immediate feedback response for the following step.
Remark 1: Note that the prescribed procedure is not equivalent to the multiple shooting SQP method as described above: in each step or iteration a different optimal control problem is treated, with different Xo,Po in the NMPC case, and additionally different t f - to for shrinking time intervals as in batch processes.
A conventional approach would be - to start the SQP procedure as described above from the deviated values x~,p~ and to use the old control values Ui for an integration over the complete interval tf - to, and - to iterate until a given (strict) convergence criterion is satisfied.
Remark 2: In contrast to SQP methods, the solution procedures have to be modified, e.g. the quadratic programs have to be solved without knowledge of the unknown values XO,PO generating a feedback law that includes state and control inequality constraints.
Note that in the meantime the old control variable will be used, so that no response to the disturbed values x~,p~ is applied so far .
Remark 3: The time 6 required for model expansion and full QP solution depends on the complexity of the model and the optimization problem, the numerical solution algorithms involved and the available computer. If 6 is not sufficiently small, parallelization may be a remedy.
In time critical processes this may take much too long to be able to cope with the nonlinear dynamics.
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Fig. !. Closed-loop simulation result for step disturbances in XF. By measuring only Tl3 and T 27 , the disturbance is gradually detected by an EKF, as can be seen by the second line in the graph XF. The control response is shown in the graphs V and L , while the necessary CPU time for each optimization problem is plotted in the graph at the bottom. The sampling time is 5 seconds, whereas the predicted control horizon comprises 5 prediction intervals of altogether 100 seconds. The control model is of 164th order. 5. NMPC OF A HIGH-PURITY DISTILLATION COLUMN As a realistic application example we consider the control of a high purity binary distillation column with 40 trays for the separation of Methanol and n-Propanol. The column is modelled by a DAE with 42 differential states and 122 algebraic states, that is described in Nagy et al. (2000).
set point is used . These two concentrations are much more sensitive to changes in the inputs of the system than the product concentrations; if they are kept constant, the product purities are safely maintained for a large range of process conditions. As concentrations are difficult to measure, we consider instead the tray temperatures , which correspond directly to the concentrations via the Antoine equation.
5.1 The distillation column
5.2 State estimation To obtain an estimate of the 42 differential system states and of the model parameters F and XF by measurements of the two temperatures T 13 , T27 only, we have implemented a variant of an Extended Kalman Filter (EKF) .
The binary mixture is fed in the column with flow rate F and molar feed composition XF. Products are removed at the top and bottom of the column with concentrations XB and XD The column is considered in L/V configuration, i.e. the liquid flow rate L and the vapor flow rate V are the control inputs. The control problem is to maintain the specifications on the product concentrations XB and XD despite disturbances in the feed flow F and the feed concentration XF .
An EKF is based on subsequent linearizations of the system model at each current estimate; each measurement is compared with the prediction of the nonlinear model, and the estimated state is corrected according to the deviation. The weight of past measurement information is kept in a weighting matrix, which is updated according to the current system linearization.
As usual in distillation control, the product purities XB and XD at reboiler and condenser are not controlled directly - instead, an inferential control scheme which controls the deviation of the concentrations on tray 13 and 27 from a given
In addition to an ordinary EKF our estimator can incorporate additional knowledge about the
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model used 5th order 164th order
N-10 N=5 1/ avrg max I avrg 1/ max 0.08 s I 0.04 s 1/ 0.14 s 0.08 s 1.2 s 1.3 s 1 0.8 sI/ 2.4 s
N-20 avrg max 0.31 s 5.0 s
0.22 s 2.5 s
Table 1. CPU times for different horizon lengths, in seconds. The first row shows the results for a reduced control model with only 5 differential states, and the second row shows the CPU times for the detailed model with 42 differential and 122 algebraic states. N is the number of multiple shooting intervals in the prediction horizon (each of 20 sec length). All simulations were performed on a Compaq Alpha XP 1000 workstation. possible range of states and parameters in form of inequality constraints. This is especially useful as the tray concentrations need to be constrained to be in the interval [0, 1] to make a reasonable simulation possible. The performance of the estimator was such that step disturbances in the model parameters were completely detected after 120 seconds (ie. two minutes) , as can be seen in Figure 1 for an example disturbance scenario. Remark: It is interesting to observe that our estimation scheme with two measurements and two model parameters even allows to treat model plant mismatch. By interpreting the two parameters as integrators our scheme is able to find consistent steady states and parameters in the model that fit steady state measurements of the plant. This made it possible to perform closed-loop simulations with a reduced order control and estimation model with only five differential states, which is described in Nagy et al. (2000).
= IIx E(x(tf),P) = IIx -
L(x , z, u , p)
x sllb
+ lIu - u.llk
xsll~.
(33) (34)
and an terminal inequality constraint: r(x(tf ) ,p)
= 0: -
E(x(tf ), p)
(35)
Here, the weighting matrices Q, R, p. and the scalar 0: are chosen according to the theory of quasi-infinite horizon NMPC (cf. Chen and Allgower (1998)) , as specified in Nagy et al. (2000). This controller setup would ensure nominal stability for the closed-loop system in the case of perfect state information and exact solution of the optimization problems. In our numerical solution approach, the determination of the current set point for given parameters and the dynamic optimization problem are performed simultaneously. As system state and parameters are the (smoothed) result of an EKF estimation, they only slightly vary from one optimization problem to the next , so that favourable conditions for a one-iteration embedding strategy are given.
5.3 Controller Design 5.4 Numerical Results
Given an estimate of the system parameters Po (here F and XF), our controller first determines an appropriate desired steady state for states and controls x. , Z., and u •. This is done by posing a steady state constraint
Z., u.,Po) = 0 g(x. , Z. , u. ,Po) = 0 f(x.,
For a realistic test of the algorithm we have performed closed-loop simulations with the plant modelled by the 164th order model. The estimation and optimization are performed either with the same model or with a reduced model of fifth order (see the remark on model plant mismatch in subsection 5.2).
(30) (31)
In the disturbance scenario shown in Figure 1 we consider three step changes in the feed concentration XF : starting from the nominal value, x F is first reduced by 10%, later increased by 15% and at the end reduced by 5% to reach the nominal value again. In the closed-loop simulation in Figure 1 a control model of 164th order with a prediction horizon of 100 sec is used. In the first two graphs the controlled tray temperatures are shown - they are also used as the only measurements for the state estimator. It can be seen that they vary only by some tenths of centigrades
(32) where the measurement function y determines the controlled tray temperatures from the system state, so that equation (32) restricts the steady state to satisfy the inferential control aim of keeping the temperatures T 13 , T27 at fixed values T 13re j , T27ref '
Secondly, the open-loop optimal control problem is posed in the form (1)-(7) , with quadratic Lagrange and Mayer terms:
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during the whole scenario, which implicitly ensures highest purity of the product streams. At the bottom the CPU time for each optimization problem is plotted, which is well below the 5 seconds that are used as a sampling time. Please note that this CPU time does not cause any delay between new state estimate and control response as our embedding strategy delivers an immediate response requiring only one QP solution . This time is mainly needed to prepare the linearizations for the next immediate response. In Table 1 numerical results for the same scenario are shown for several simulations. In these simulations we use either the 5th order or the 164th order model for control and prediction horizon lengths of 100 sec, 200 sec and 400 sec. The predicted control steps have a length of 20 sec in all examples, so that we have N = 5, 10,20 multiple shooting intervals, respectively. As the computation times vary from one optimization problem to the next due to integrator adaptivity, we list not only the average CPU time for one optimization, but also the maximum CPU time observed for each simulation. The values from Figure 1 can be found in the first columns of the last row. Even for a detailed model of 164th order and a prediction horizon of 400 sec, the CPU time meets the requirement to be less or equal the sampling time of 5 seconds. 6. CONCLUSIONS New real-time and NMPC schemes based on the direct multiple shooting method are described. Their features are an Initial Value Embedding that leads to a negligible response time after disturbances, and the immediate application of the computational results after each iteration. An application to the NMPC of a high-purity distillation column shows excellent performance with CPU times in the range of seconds per optimization problem. Here, even the use of a 164th order model is feasible for real-time control. ACKNOWLEDGEMENTS Financial support from Deutsche Forschungsgemeinschaft in the Schwerpunktprogramm "Echtzeit-Optimierung grofter Systeme " is gratefully acknowledged. The authors are indebted to Daniel B. Leineweber for fruitful discussions on real-time problems in chemical engineering. References Bauer, 1., F. Finocchi, W.J. Duschl, H.-P. Gail and J .P. SchlOder (1997). Simulation of chemical reactions and dust destruction in protoplanetary
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