Automatica 36 (2000) 527}540
Quasi-Min-Max MPC algorithms for LPV systemsq Yaohui Lu, Yaman Arkun* School of Chemical Engineering, Georgia Institute of Technology, Atlanta, GA 30332-0100, USA Received 12 February 1999; received in revised form 9 August 1999
A scheduling model predictive controller is presented for polytopic linear parameter varying systems with input and output constraints. It is shown that the receding horizon implementation of the feasible solutions guarantees closed-loop stability.
Abstract In this paper a new model predictive controller (MPC) is developed for polytopic linear parameter varying (LPV) systems. We adopt the paradigm used in gain scheduling and assume that the time-varying parameters are measured on-line, but their future behavior is uncertain and contained in a given polytope. At each sampling time optimal control action is computed by minimizing the upper bound on the `quasi-worst-casea value of an in"nite horizon quadratic objective function subject to constraints on inputs and outputs. The MPC algorithm is called `quasia because the "rst stage cost can be computed without any uncertainty. This allows the inclusion of the "rst move u(kDk) separately from the rest of the control moves governed by a feedback law and is shown to reduce conservatism and improve feasibility characteristics with respect to input and output constraints. Proposed optimization problems are solved by semi-de"nite programming involving linear matrix inequalities. It is shown that closed-loop stability is guaranteed by the feasibility of the linear matrix inequalities. A numerical example demonstrates the unique features of the MPC design. ( 2000 Elsevier Science Ltd. All rights reserved. Keywords: Model predictive control; Linear parameter varying systems; Scheduling; Closed-loop stability; Linear matrix inequality
1. Introduction Model Predictive Control (MPC), also known as moving or receding horizon control, has originated in industry as a real-time computer control algorithm to solve linear multivariable problems that have constraints and time delays (Cutler & Ramaker, 1980; Richalet, 1980). Today most process industries use MPC in one form or another as their advanced control technology. Basically MPC solves on-line an open-loop constrained optimization problem at each time instant and implements only the "rst element of the optimal control pro"le. The optimization is repeated at the next sampling time by updating the initial condition with the new state. It is well known that this receding horizon implementation of the q This paper was not presented at any IFAC meeting. This paper was recommended for publication in revised form by Associate Editor A. Tits under the direction of Editor T. Basar. * Corresponding author. Dean of Engineering, Koc University, Istanbul, Turkey. Tel.:#212-2296674; fax:#212-2293006. E-mail address:
[email protected] (Y. Arkun)
open-loop optimal control pro"le gives rise to a stationary feedback control law (Bitmead, Gevers & Wertz, 1990). In the past most industrial applications have used the "nite horizon implementation of MPC. However, despite many reported successful applications, the "nite horizon MPC algorithms are di$cult to analyze theoretically since closed-loop asymptotic stability depends on many tuning parameters in an unnecessarily complicated way and no guarantees are provided (Bitmead et al., 1990). Realizing this drawback several researchers have recently revisited MPC and studied it as a constrained in"nite horizon LQR problem, which led to useful stability results. For example for linear plants with input and output constraints reference Rawlings and Muske (1993) is able to perform the optimization over an in"nite prediction horizon by using only the "rst N control moves and setting the remaining (in"nitely many) moves to zero. It is shown that feasibility of the resulting quadratic program guarantees stability. Instead of setting the control inputs to zero after a certain horizon, Scokaert and Rawlings (1998) and Chmielewski and Manousiouthakis (1996) use a "xed feedback control law to obtain a "nite
0005-1098/00/$ - see front matter ( 2000 Elsevier Science Ltd. All rights reserved. PII: S 0 0 0 5 - 1 0 9 8 ( 9 9 ) 0 0 1 7 6 - 4
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parameterization of the inputs over an in"nite prediction horizon. For nonlinear plants similar in"nite horizon MPC algorithms have been developed to guarantee closedloop stability. For example Michalska and Mayne (1993) suggests a switching dual-mode horizon controller in which a local linear feedback control law is applied once the states enter an invariant terminal region and a "nite horizon predictive controller is applied outside the terminal region. In Chen and AllgoK wer (1998) the in"nite horizon MPC cost function is bounded by a "nite horizon stage cost with a terminal penalty term so that the resulting nonlinear optimization problem can be solved numerically. A local state feedback law is used to compute the terminal penalty term o!-line. Local stability is proven by the existence of a feasible solution to the optimization problem. In this paper we consider the class of linear parameter-varying or LPV systems whose state-space matrices are a$ne functions of some time-varying parameter vector p(k). In recent years there has been a renewed interest in LPV systems, especially to provide useful theoretical foundations for gain scheduling control (see Shamma & Athans, 1991; Apkarian, Gahinet & Becker 1995; Wu & Packard, 1995). The common theme in these works is to make the controller parameter dependent so that when time-varying parameters are measured in real-time, the controller becomes self-scheduling and o!ers potential performance improvements over a "xed robust controller. The purpose of this paper is to develop in"nite horizon scheduling MPC algorithms for LPV plants. In doing so MPC is extended to apply to an important class of systems. Both input and output constraints are addressed and stability guarantees are provided. The rest of the paper is structured as follows. Section 2 de"nes the basic system of interest as a polytopic Linear Parameter Varying (LPV) model and introduces the necessary notation. In Section 3, Model Predictive Control (MPC) algorithm `Quasi-Min-Maxa is formulated for the unconstrained case "rst, and then modi"ed to include both input and output constraints. Stability proofs and comparison with some of the existing MPC algorithms are also given. In Section 4, simulations on a numerical example illustrate the application of the developed algorithms. Finally, Section 5 concludes the paper.
2. Problem statement We consider discrete polytopic LPV systems whose system matrices are a$ne functions of a parameter vector p(k): x(k#1)"A(p(k))x(k)#B(p(k))u(k)
(1)
where l A(p(k))" + p (k)A , j j j/1
l B(p(k))" + p (k)B j j j/1
(2)
with x3Rnx denoting the state, u3Rnu the control and p"[p , p ,2, p ]3Rl the parameter vector. The 1 2 l time-varying parameter vector p(k) belongs to a convex polytope P, i.e. l + p (k)"1, j j/1
04p (k)41. j
(3)
Therefore, when p "1 and p "0 for j"1, 2,2, l and i j jOi, the LPV model (1)}(2) reduces to its ith linear time-invariant `locala model, i.e. [A(p(k)), (p(k))]" [A , B ]. Then it is clear that as p(k) varies inside its i i polytope P, the LPV system matrices vary inside a corresponding polytope ) whose vertices consist of l local system matrices. [A(p(k)), B(p(k))]3)"C M[A , B ],[A , B ],2,[A B ]N o 1 1 2 2 l l (4) where C denotes the convex hull. In the above deo scription p (k) can represent a measured physical varij able after it is normalized or can be interpreted as a time-varying weight assigned to the jth local model. Our ultimate goal is to control nonlinear processes based on their approximate LPV models. In fact the technique of global linearization (Liu, 1968; Boyd, Gharui, Feron & Balakrishnan, 1994) suggests that such an approximation is possible, although it can be conservative. Motivated by this result we have applied LPV modeling to chemical processes (Banerjee, Arkun, Ogunnaike & Pearson, 1997) where we "rst obtain a family of linear local models at di!erent operating points of the non-linear plant, and then construct a global LPV model by interpolating the local models using the parameter vector p(k). Both the state and the parameter vector of the LPV model are estimated on-line to match the nonlinear plant dynamics. Throughout this paper we will assume that the plant to be controlled is given by the polytopic LPV model (1)}(2) and no mismatch exists between the model and the plant. Both the parameter vector p(k) and the state x(k) are assumed to be available in real-time. Thus at every sampling time k the current system [A(p(k)), B(p(k))] is known exactly but the future systems [A(p(k#i)), B(p(k#i)), i51] are uncertain and vary inside a prescribed polytope. With this prelude we now embark on the formulation of the MPC problem we want to solve.
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3. MPC problem
529
Following an approach given in Kothare, Balakrishnan and Morari (1996), it is easy to derive an upper bound on the worst value of J=. First de"ne a quadratic func1 tion (k)TBT QI (k)Q0.5 >(k)TR0.5 j j A QI (k)#B >(k) QI (k) 0 0 j j 50, Q0.5QI (k) 0 cI 0 QI (k)
R0.5>(k)
B
[A(p(k))x(kDk)#B(p(k))u(kDk)]T x(kDk)TQ0.5 u(kDk)TR0.5
1
0
0
where the feedback gain is given by F(k)">(k)QI ~1(k).
cI
j"1, 2,2, l
(14)
(15)
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Proof. See Appendix A. h Remark 1. In the above linear matrix inequalities, p(k) and x(kDk) are the measured parameter and state, respectively. Optimization determines the "rst move u(kDk) that gets implemented, the terminal weight P(k), and the feedback gain F(k) which guarantees that the future states evolve inside an invariant ellipsoid. Remark 2. When parameter-dependent feedback law (9) is used, Theorem 1 holds if all the local models have the same B matrices, i.e. B "B for all j"1, 2,2, l. This is j j not a serious restriction as shown in Apkarian et al (1995). The condition will be met if the models include actuator dynamics. If not, then this condition can be satis"ed by "ltering the input. In this case, >(k) is replaced by > (k) and the local feedback gains are given by j F (k)"> (k)QI ~1(k), and they are scheduled according to j j the parameter-dependent global feedback law (9). Theorem 2. Unconstrained Quasi-Min-Max algorithm (P1) is stable when its feasible solutions are implemented in a receding horizon fashion. Proof. See Appendix D. h Remark 3. Note that Kothare et al. (1996) has presented an MPC design technique (Min-Max) to control linear time-varying systems that have a polytopic uncertainty description. However, in Kothare et al. (1996) the parameters are not available in real-time; therefore, the controller is robust but not scheduling. Since we assume that the current model [A(p(k)), B(p(k))] and the state x(kDk) are available, x(k#1Dk) can be predicted exactly and inequality (10) will guarantee that the remaining uncertain future states Mx(k#1#iDk), i51N belong to an invariant ellipsoid (see Lemma B.1 in Appendix B). In Kothare et al. (1996) the authors also establish an invariant ellipsoid, but in their case x(k#1Dk) is uncertain and belongs to the ellipsoid along with the rest of the future states Mx(k#1#iDk), i51N. As a result the upper bound is derived for J=, while ours is given on J= after subtract0 0
ing the "rst stage cost, i.e. we bound J=. Because of this 1 di!erence in problem formulation we keep the "rst move u(kDk) as a free decision variable (see (14)) and impose feedback on the remaining inputs ;=. In Kothare et al. 1 (1996) both u(kDk) and ;= are given by a feedback law. 1 The two techniques are compared in Fig. 1. 3.2. Problem 2 (P2). Constrained Quasi-Min-Max MPC Input and output constraints can be expressed as linear matrix inequalities and therefore included in the MPC problem. Consider the peak bounds on inputs: Du (k)D4u , k50, j"1, 2,2, n . j j,.!9 u
(16)
Inputs ;= can be split into two parts Mu(kDk), ;=N, and 1 0 since u(kDk) is a free decision variable, constraint on u(kDk) can be imposed directly: Du (kDk)D4u , j"1, 2,2, n . j j,.!9 u
(17)
For the remaining inputs ;=(k), when the control law is 1 given by (8), the existence of a symmetric matrix X such that
A
B
X
>
>T
QI
, j"1, 2,2, n 50, with X 4u2 j, .!9 u jj
(18)
guarantees that Du (k#iDk)D4u , i51, j"1,2,2, n j j, .!9 u (see Appendix B). For parameter-dependent control law (9), local models must have a common B, and >(k) in the above LMIs should be replaced by > (k) for j"1, 2,2, l j (see Remark 2). For output constraints, we consider outputs of the form y(k#iDk)"Cx(k#iDk), i51 at sampling time k and divide them into two parts as well. One is y(k#1Dk) which is the next immediate output; the second set is the remaining future outputs >=(k)"My(k#iDk), i52N. 2 Considering the Euclidean norm constraint, for the immediate output y(k#1Dk), one has DDC[A(p(k))x(kDk)#B(p(k))u(kDk)]DD 4y . 2 .!9
Fig. 1. Comparison of Quasi-Min-Max and Min-Max (Kothare et al., 1996).
(19)
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For the remaining outputs, one can show that (see Appendix C) if
A
B
QI
[A QI #B >]TCT j j 50, C(A QI #B >) y2 .!9 j j
j"1, 2,2, l (20)
then max DDy(k#1#i D k)DD 4y , i51. (21) 2 .!9 *A(p(k`i)), B(p(k`i))+|) For the parameter-dependent control law (9), >(k) in the equations above should be replaced by > (k) for j j"1, 2,2, l (see Remark 2). Similarly, peak bounds on individual outputs can be translated to su$cient LMI constraints as well. Constrained Quasi-Min-Max algorithm (P2) with control law given by (8) can now be summarized as an optimization problem (13) subject to LMIs (14)}(15); constraints on inputs (17) and (18); constraints on outputs (19) and (20). When parameter-dependent feedback law (9) is used, proper changes should be made in the LMIs as outlined in Remark 2. Theorem 3. Constrained Quasi-Min-Max algorithm (P2) is stable when its feasible solutions are implemented in a receding horizon fashion. Proof. See Appendix D. h
531
others and bounds it explicitly as in (17) so that the implemented input can reach its bounds and the feasibility characteristics are improved as it will be demonstrated by the simulations. Similar arguments in favor of the new algorithm can be made in the case of output constraints since y(k#1Dk) is separated from the rest of the outputs and not subjected to the LMIs (20) resulting again from the invariant ellipsoid. 3.3. Problem 3 (P3). Modixed Quasi-Min-Max MPC In (P2) although conservatism is improved by not subjecting the "rst input and output to invariant ellipsoid constraints, Quasi-Min-Max algorithm still includes constraints on all the future inputs and outputs. As shown in Appendix D this guarantees closed-loop stability when the inputs are implemented in a receding horizon fashion. However this is not the only way to guarantee stability. In the modi"ed Quasi-Min-Max MPC algorithm (P3) proposed here, we relax the future constraints (18) and (20) and bound only the implemented input u(kDk) and the resulting immediate output y(k#1Dk). To guarantee stability we require that the objective function monotonically decreases (as in the proof of Theorem 3) by explicitly including this constraint in the modi"ed algorithm after the initialization step k"0. Consider the objective function of algorithm (P2): '(k)"x(kDk)TQx(kDk)#u(kDk)TRu(kDk)
Remark 4. In Arkun, Banerjee and Lakshmanan (1998) a di!erent type of constrained Min-Max MPC algorithm is given. The objective function J= is not split into two 0 parts and the upper bound is computed for the worstcase value of J= and not of J= as done in Quasi-Min1 0 Max. As a result in Arkun et al. (1998) u(kDk) is not a free optimization variable but is governed by the feedback law. Therefore the algorithm in Arkun et al. (1998) bounds all the inputs Mu(k#iDk), i50N (including u(kDk)) based on (18) which leads to conservatism. By (18), bounds on inputs are cast as suzcient LMIs on the feedback gains using the fact that all the predicted states belong to an invariant ellipsoid (see Lemma 3.1 in Appendix B). Because of this, the feedback gain can be unnecessarily small due to the size of the invariant ellipsoid, and this results in conservatism by which the inputs may never reach its bounds. The new algorithm, QuasiMin-Max (P2), separates the "rst input u(kDk) from the
#x(k#1Dk)TP(k)x(k#1Dk) and include the `monotonically decreasinga condition as an explicit constraint (Lyapunov constraint) in the optimization problem: '(k)('(k!1).
(22)
Modi"ed Quasi-Min-Max algorithm (P3) is de"ned as an optimization problem (12) subject to LPV model (1)}(2), inequality (10), the control law (8) or (9), and constraints on inputs (17), constraints on outputs (19), and Lyapunov constraint (22). Theorem 4. The optimization problem (P3) with the control law given by (8) can be solved by the following semi-dexnite programming: min c c,u(k@k), QK (k),Y(k)
(23)
subject to
A
c
B
[A(p(k))x(kDk)#B(p(k))u(kDk)]T x(kDk)TQ0.5 u(kDk)TR0.5
A(p(k))x(kDk)#B(p(k))u(kDk)
QK (k)
0
0
Q0.5x(kDk)
0
I
0
R0.5u(kDk)
0
0
I
50
(24)
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A A
'(k!1)
B
[A(p(k))x(kDk)#B(p(k))u(kDk)]T x(kDk)TQ0.5 u(kDk)TR0.5
A(p(k))x(kDk)#B(p(k))u(kDk)
QK (k)
0
0
Q0.5x(kDk)
0
I
0
R0.5u(kDk)
0
0
I
(25)
'0
B
QK (k)AT#>TBT QK (k)Q0.5 >TR0.5 j j A QK (k)#B > QK (k) 0 0 j j 50, j"1, 2,2, l Q0.5QK (k) 0 I 0 QK (k)
R0.5>
0
I
0
and constraints on input (17) and on output (19):
A B A
(26)
0.2300
u (kDk)!u i i,.!9 40, i"1, 2,2, n (27) u !u !u (kDk) i,.!9 i C [A(p(k))x(kDk)#B(p(k))u(kDk)]!y i i,.!9 40, !y !C [A(p(k))x(kDk)#B(p(k))u(kDk)] i..!9 i i"1, 2,2, n . (28) y The feedback gain is given by F(k)">(k)QK (k)~1. When scheduling feedback control law (9) is used, >(k) is replaced by > (k); LMIs are satisxed for j"1, 2,2, l, and local j models are required to have common B's (see Remark 2).
0.2601 B " , 1 0.1213
Proof. See Appendix E. h
A is stable, with eigenvalues of !0.5982; !0.1160 and 1 0.6297$0.4254i, and A is unstable with eigenvalues of 2 !1.5901; 1.7151; 1.4980; 0.0798. The parameter vector p(k) is available on-line and its measured values are plotted in Fig. 2. The control objective is to regulate the state variables from x(0)"[!0.3964, 0.4377, !1.0905, 1.1137]T to the origin. Constraints on input u(k) and a particular output y(k)"x (k) will be considered. 2 Unique features of the proposed MPC algorithms are studied next, and their feasibility and performance characteristics are compared.
A A
B
B
To intialize the algorithm, in the "rst step k"0, Lyapunov constraint is disregarded temporarily and the optimization solves (23) subject to (24), (26), input constraint (27) and output constraint (28). Theorem 5. Modixed Quasi-Min-Max algorithm (P3) is stable when its feasible solutions are implemented in a receding horizon fashion. Proof. See Appendix F. h
4. Example We consider an LPV system which belongs to a polytope formed by two local discrete models:
A
0.2093 !0.1981 !0.2394
0.2717 A " 2 !0.4700
0.4598
0.5602
B
0.5671
1.3782
0.6700 !0.8600 !1.2400
0.3456 !0.6312 !1.4594
B "B . 2 1
,
1.8936
4.1. Study of **Quasi++ feature
Remark 5. From the proofs of Theorems 5 and 3, we can see that whenever algorithm (P2) is feasible, Optimization (P3) can "nd a feasible solution. However feasibility of (P3) does not necessarily imply feasibility of (P2).
0.2730
1.3452
B
0.0660 0.3021 !0.5012
0.2717 0.4416 0.5602 !0.7123 A " , 1 0.3051 !0.7865 0.7651 0.3121 0.7962 !0.1452 0.5231 !0.9345
`Quasia is the principal new feature of the algorithms proposed in this paper. The objective function J= is 0 divided into two parts: the "rst stage cost x(kDk)TQx(kDk)# u(kDk)TRu(kDk) and the remaining cost J=. An upper 1 bound is constructed on J=, and on-line optimization 1 minimizes the "rst step cost plus the upper bound on J= (`Quasi-worst-casea). As a result the "rst control 1 move u(k D k) is separated from the future inputs given by a feedback law and is considered as a free optimization variable. In Arkun et al. (1998) a di!erent type of scheduling Min-Max MPC algorithm is given where u(k D k) is not a free optimization variable but is governed by the feedback law (9). In the presence of constraints this results in conservatism and loss of performance (see Remark 4). This is shown in Fig. 3.
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Fig. 2. The LPV model behavior.
Fig. 3. Performance comparison of Quasi-Min-Max with (9) on ;= and Min-Max (Arkun et al., 1998). 1
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Fig. 4. Performance comparison of Quasi-Min-Max (P2) and modi"ed Quasi-Min-Max (P3).
Table 1 E!ect of `quasia on feasibility Algorithm
No output constraint Input feasibility range
u "1.91 .!9 Output feasibility range
Min-Max (Arkun et al., 1998) Quasi-Min-Max with (9) on ;= 1
u 51.91 .!9 u 51.77 .!9
y 51.36 .!9 y 50.89 .!9
It is observed that Quasi-Min-Max improves the performance with both states and input reaching the origin in shorter time and giving lower objective function values '(k). Furthermore, we can see that the input is not able to reach the saturation point for the algorithm in Arkun et al. (1998) due to conservatism discussed earlier. It is also shown that the optimal output can reach its bound in Quasi-Min-Max (see Fig. 4) while this is not true for Min-Max (see Fig. 3). The ranges of the values that input and output bounds can take when the algorithms can "nd a feasible solution are given in Table 1. From Table 1, it is clear that the feasibility range is increased for Quasi-Min-Max. 4.2. Study of **Scheduling++ feature Quasi-Min-Max MPC is called `schedulinga because the "rst control input u(kDk) depends on measured p(k)
and feedback gain F(k) for the future inputs can be chosen to be parameter dependent as well (see (9)). When both of these features are absent, (e.g. p(k) is not available in real-time, so u(kDk) cannot be separated and F(k) is not parameter-dependent), Quasi-Min-Max MPC reduces to the robust MPC algorithm given in Kothare et al. (1996). Table 2 compares the two methods and shows that when robust MPC is infeasible and if p(k) is available, one can implement Quasi-Min-Max MPC and "nd a feasible stabilizing solution. The same table also shows that for the same problem Quasi-Min-Max will not "nd a feasible solution unless the feedback gain F(k) is parameter dependent as given in (9). This is due the fact that it is not possible to "nd one feedback gain to stabilize both [A , B ] and [A , B ] (i.e. LMI (15) is not feasible). The 1 1 2 2 parameter-dependent feedback law (9) introduces two di!erent gains, one for each model, and LMI (15) is now feasible due to this additional degree of freedom. It
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535
Table 2 E!ect of `schedulinga on feasibility Algorithm
No output constraint Input feasibility range
u "1.91 .!9 Output feasibility range
Robust Min-Max (Kothare et al., 1996) Quasi-Min-Max with (8) on ;= 1 Quasi-Min-Max with (9) on ;= 1
Infeasible Infeasible u 51.77 .!9
Infeasible Infeasible y 50.89 .!9
Algorithm
No output constraint Input feasibility range
u "1.91 .!9 Output feasibility range
Quasi-Min-Max with (9) on ;= 1 Modi"ed Quasi-Min-Max with (9) on ;= 1
u 51.77 .!9 u 51.46 .!9
y 50.89 .!9 y 50.77 max
Table 3 E!ect of `Lyapunov constrainta on feasibility
should be noted that the feedback gains which are computed for the future inputs Mu(k#i D k), i51N are never implemented. Therefore, the future values of the parameters which appear in the feedback law (9) are not needed by the algorithm. The reason that the future feedback gains are computed is to guarantee that the sequence of implemented inputs Mu(kDk)N is stabilizing. Clearly this sequence is scheduled by the measured value of p(k) since LMI (14) depends on p(k). 4.3. Study of **Lyapunov constraint++ feature `Modi"eda Quasi-Min-Max (P3) constrains only the implemented input u(kDk) and the resulting output y(k#1Dk), and adds Lyapunov constraint explicitly in the optimization. From the proof of Theorem 3, it is easy to show that these constraints will be satis"ed when the constraints in Quasi-Min-Max (P2) are satis"ed. Fig. 4 shows that when (9) on ;= is used in both 1 algorithms, they are all feasible and their performances are quite similar with modi"ed algorithm (P3) performing slightly better. The feasibility ranges of the two algorithms are shown Table 3. It is shown that the feasibility feature of algorithm (P3) is improved compared with algorithm (P2) (see Remark 5). Comparing all the results it is also seen that modi"ed Quasi-Min-Max with (9) on ;= performs best among all the MPC algorithms 1 discussed here.
5. Conclusions In this paper, in"nite horizon Quasi-Min-Max MPC algorithms have been developed for discrete, polytopic linear parameter varying systems. The on-line optimization problems include input and output constraints and can be solved by semi-de"nite programming. It is shown
that receding horizon implementation of the feasible solutions guarantees closed-loop stability. The implemented control moves are dependent on parameters which are assumed to be available (measured) in real-time; therefore, the controllers are also termed `schedulinga. The new MPC algorithms are shown to o!er decreased conservatism and improved feasibility characteristics. A numerical example demonstrates the salient features of the proposed MPC algorithms and their superior performance over other approaches. Throughout the paper it is assumed that both the states and parameters are measured, and there is no plantmodel mismatch. Future work should consider relaxing some of these assumptions.
Acknowledgements The authors gratefully acknowledge the "nancial support of E. I. duPont de Nemours and Co., Inc. and the National Science Foundation through Grant CTS9522564.
Appendix A. Proof of Theorem 1 Minimization of x(k D k)TQx(k D k)#u(k D k)TRu(k D k)# x(k#1Dk)TP(k)x(k#1Dk) with P(k)'0 is equivalent to min c u(k@k), P(k),c subject to
(A.1)
x(kDk)TQx(kDk)#u(kDk)TRu(kDk) #[A(p(k))x(kDk)#B(p(k))u(kDk)]T ]P(k)[A(p(k))x(kDk)#B(p(k))u(kDk)]4c.
(A.2)
536
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The feedback gain matrix is given by F(k)">(k)QI ~1. This results in (15). When scheduling feedback control law (9) is used, if all the local models have a common B, i.e. B "B for j
By using Schur complements and de"ning cQI ~1"P(k), this is equivalent to min c u(k@k),QI ,c
(A.3)
subject to
A
B
[A(p(k))x(kDk)#B(p(k))u(kDk)]T x(kDk)TQ0.5 u(kDk)TR0.5
1 A(p(k))x(kDk)#B(p(k))u(kDk)
QI (k)
0
0
Q0.5x(kDk)
0
cI
0
R0.5u(kDk)
0
0
cI
which proves (14). Substituting u(k#iDk)"F(k)x(k#iDk), i51 and the state space equation (1)}(2) into (10), one can get
(A.4)
50
j"1, 2,2, l, the above proof holds after de"ning F (k)"> (k)QI ~1 and replacing > and B by > and B, j j j j respectively.
x(k#iDk)TM[A(p(k#i))#B(p(k#i))F(k)]T ]P(k)[A(p(k#i))#B(p(k#i))F(k)]
Appendix B. Casting input constraints into LMIs (A.5)
!P(k)#F(k)TRF(k)#QNx(k#iDk)40.
We make use of the following lemma.
That is satis"ed for all i51 if
Lemma B.1 (Invariant ellipsoid). Consider the system (1)}(2) and let ) be a polytope. At time k, it is supposed that (15) holds for some QI '0, c and >"FQI . Also suppose u(k#iDk)"F(k)x(k#iDk), i51. If
[A(p(k#i))#B(p(k#i))F(k)]TP(k)[A(p(k#i)) #B(p(k#i))F(k)]!P(k)#F(k)TRF(k)#Q40 (A.6) which can be expressed as LMI by substituting P(k)" cQI ~1, QI '0 and >(k)"F(k)QI :
A
QI (k)
x(kDk)TQx(kDk)#u(kDk)TRu(kDk)#x(k#1Dk)T ]P(k)x(k#1Dk)4c
QI (k)A(p(k#i))T#>(k)TB(p(k#i))T QI (k)Q0.5
(B.1) >(k)TR0.5
A(p(k#i))QI (k)#B(p(k#i))>(k)
QI (k)
0
0
Q0.5QI (k)
0
cI
0
R0.5>(k)
0
0
cI
i"1,2,2,R.
Since the inequality is a$ne in [A(p(k#i)), B(p(k#i))], it is satis"ed for all
max x(k#jDk)TP(k)x(k#jDk)4c, j51. *A(p(k`j))B(p(k`j))+|) (B.2) Thus, Z"MzDzTPz4cN is an invariant ellipsoid for the predicted states x(k#jDk), j51 of the uncertain system.
B
QI (k)AT#>(k)TBT QI (k)Q0.5 >(k)TR0.5 j j A QI (k)#B >(k) QI (k) 0 0 j j 50, Q0.5QI (k) 0 cI 0 QI (k)
R0.5>(k)
0
0
(A.7)
it follows that
[A(p(k#i)), B(p(k#i))]3C M[A B ], o 1 1 [A B ],2,[A B ]N (A.8) 2 2 l l if and only if there exist QI '0, >(k)"F(k)QI and c such that
A
B
50,
cI
j"1,2,2, l.
(A.9)
Y. Lu, Y. Arkun / Automatica 36 (2000) 527}540
537
Proof. If there exist QI '0, c and >"FQI such that (15) holds and if
Appendix C. Casting output constraints into LMIs
x(kDk)TQx(kDk)#u(kDk)TRu(kDk)#x(k#1Dk)T
It is assumed that Lemma B.1 is satis"ed and Z is an invariant ellipsoid for the predicted states. For the outputs >=(k)"[y(k#iDk), i52], we apply the derivation 2 in Kothare et al. (1996) to this invariant ellipsoid. We have
]P(k)x(k#1Dk)4c
(B.3)
then x(k#1Dk)TP(k)x(k#1Dk)4c
(B.4)
since Q'0, R50. From Appendix A, we know that (B.6) is equivalent to (B.1). Therefore x(k#1#iDk)TP(k)x(k#1#iDk) !x(k#iDk)TP(k)x(k#i) 4![x(k#iDk)TQx(k#i)#u(k#i)TRu(k#i)], i51.
(B.5)
That is x(k#1#iDk)TP(k)x(k#1#iDk)4x(k#iDk)T ]P(k)x(k#iDk), i51
(B.6)
and thus Z"MzDzTPz4cN is an invariant ellipsoid for the predicted states of the uncertain system. This completes the proof. h Once Lemma B.1 is satis"ed, the constraints on ;=(k)"Mu(k#iDk), i51N can be cast into LMIs 1
A
max DDy(k#1#iDk)DD 2 iz1 "max DDC[A(p(k#i))#B(p(k#i))F(k)]x(k#iDk)DD 2 iz1 4max DDC[A(p(k#i))#B(p(k#i))F(k)]zDD , i51 2 z|Z "p6MC[A(p(k#i))#B(p(k#i))F(k)]QI 0.5N, i51. Therefore DDy(k#1#iDk)DD 4y , i51 2 .!9 [A(p(k#i))B(p(k#i))]3), i51 if
for
any
p6MC[A(p(k#i))#B(p(k#i))F(k)]QI 0.5N4y , i51 .!9 (C.1) or QI 0.5[A(p(k#i))#B(p(k#i))F(k)]TCTC[A(p(k#i)) #B(p(k#i))F(k)]QI 0.54y2 I, i51 .!9 which is equivalent to
B
QI
[A(p(k#i))#B(p(k#i))F(k)]TCT
C[A(p(k#i))#B(p(k#i))F(k)]
y2 I .!9
50, i51.
(C.2)
(C.3)
following Kothare et al. (1996). Consider the peak bounds:
This is satis"ed for all [A(p(k#i)), B(p(k#i))]3), i51, if and only if
Du (k#iDk)D4u , i51, j"1, 2,2, n . j j,.!9 u
A
(B.7)
It holds that
A
B
>
>T
QI
50, with X 4u2 j,.!9 jj
with j"1, 2,2, n , then it is guaranteed u Du (k#iDk)D4u , i51, j"1,2,2, n . j j,.!9 u
(C.4)
which is the same as (20).
Appendix D. Proof of Theorems 2 and 3 (B.8)
Thus if there exists a symmetric matrix X such that X
B
[A QI #B >]TCT j j 50, C(A QI #B >) y2 I j j .!9 j"1,2,2, ¸
max Du (k#iDk)D2"max D(>QI ~1x(k#iDk)) D2 j j iz1 iz1 4max D(>QI ~1z) D2 j z|Z 4DD(>QI ~0.5) DD2 j 2 "(>QI ~1>T) . jj
QI
(B.9) that
First we will establish that the optimal solution at time k is a feasible solution to the problem at time k#1. Therefore the decision variables at k#1 are set equal to the optimal values computed at time k i.e. u(k#iDk#1)"u(k#iDk)"F(k)x(k#iDk) i51 and P(k#1)"P(k). Now it has to be shown that this solution satis"es the LMIs at k#1. Speci"cally (14), (17), (19) need to be checked since these are the ones that change from one sampling time to another as they explicitly
538
Y. Lu, Y. Arkun / Automatica 36 (2000) 527}540
depend on the measured parameter and state and implemented input. When (18) is satis"ed at time k, u(k#1Dk) is feasible. Then it is clear that (17) will be satis"ed at time k#1 with u(k#1Dk#1)"u(k#1Dk). As for the output constraint (19), since (20) is satis"ed for any [A B]3) at time k, y(k#2Dk) is feasible. Then it follows that (19) is feasible at k#1 when u(k#1Dk#1)"u(k#1Dk) is implemented at time k#1. Now we need to check (14). After submitting u(k#1Dk#1)"F(k)x(k#1Dk) and P(k#1)"P(k), (14) is equivalent to
Combining with Eq. (D.3), we have x(k#1Dk#1)TQx(k#1Dk#1) #u(k#1Dk#1)TRu(k#1Dk#1) # x(k#2Dk#1)TP(k#1)x(k#2Dk#1) 4x(k#1Dk)P(k)x(k#1Dk) (x(kDk)TQx(kDk)#u(kDk)TRu(kDk) (D.5)
#x(k#1Dk)TP(k)x(k#1Dk) for x(kDk)O0, u(kDk)O0. We de"ne a candidate Lyapunov function
x(k#1Dk)TM[A(p(k#1))#B(p(k#1))F(k)]T ]P(k)[A(p(k#1))#B(p(k#1))F(k)]
'(k)"x(kDk)TQx(kDk)#u(kDk)TRu(kDk) (D.1)
!P(k)#F(k)TRF(k)#QNx(k#1Dk)40. This is true if
#x(k#1Dk)TP(k)x(k#1Dk). By checking Lyapunov stability conditions '(k)50,
[A(p(k#1))#B(p(k#1))F(k)]TP(k)[A(p(k#1)) #B(p(k#1))F(k)]!P(k)#F(k)TRF(k)#Q40. (D.2) The above inequality holds since (15) is satis"ed at time k and [A(p(k#1)), B(p(k#1))]3). Thus the feasible solution of the optimization at time k is also feasible at time k#1. Now, we are ready to prove the stability. From (10), when i"1
(D.6)
'(k#1)!'(k)(0,
we have the conclusion that the controller guarantees asymptotic stability, i.e. both x(kDk) and u(kDk) converge to zero. When scheduling feedback control law (9) is used, the proof holds with the appropriate changes discussed at the end of Appendix A.
Appendix E. Proof of Theorem 4
x(k#1Dk)TQx(k#1Dk)#u(k#1Dk)TRu(k#1Dk) #x(k#2Dk)TP(k)x(k#2Dk) (D.3)
4x(k#1Dk)TP(k)x(k#1Dk).
Since u(k#1Dk)"F(k)x(k#1Dk) and P(k) is a feasible solution to the optimization problem at time k#1, while u(k#1Dk#1) and P(k#1) is the optimal solution to the same problem, one has x(k#1Dk#1)TQx(k#1Dk#1)#u(k#1Dk#1)T
Minimization of x(kDk)TQx(kDk)#u(kDk)TRu(kDk)# x(k#1Dk)TP(k)x(k#1Dk) with P(k)'0 is equivalent to min c u(k@k),P(k),c subject to
(E.1)
x(kDk)TQx(kDk)#u(kDk)TRu(kDk) # [A(p(k))x(kDk)#B(p(k))u(kDk)]T
]Ru(k#1Dk#1)# x(k#2Dk#1)T
]P(k)[A(p(k))x(kDk)#B(p(k))u(kDk)]4c.
]P(k#1)x(k#2Dk#1)
(E.2)
By using Schur complements and de"ning QK ~1(k)"P(k), this is equivalent to
4x(k#1Dk)TQx(k#1Dk)#u(k#1Dk)Tu(k#1Dk) (D.4)
#x(k#2Dk)TP(k)x(k#2Dk).
min c u(k@k),QK ,c
(E.3)
subject to
A
c
B
[A(p(k))x(kDk)#B(p(k))u(kDk)]T x(kDk)TQ0.5 u(kDk)TR0.5
A(p(k))x(kDk)#B(p(k))u(kDk)
QK (k)
0
0
Q0.5x(kDk)
0
I
0
R0.5u(kDk)
0
0
I
50
(E.4)
Y. Lu, Y. Arkun / Automatica 36 (2000) 527}540
which proves (24). Similarly, the inequality '(k)('(k!1) or x(kDk)TQx(kDk)#u(kDk)TRu(kDk) # [A(p(k))x(kDk)#B(p(k))u(kDk)]TP(k)[A(p(k))x(kDk) #B(p(k))u(kDk)]('(k!1) can be expressed as
A
'(k!1)
539
Arkun, Y., Banerjee, A., & Lakshmanan, N. M. (1998). Self scheduling MPC using LPV models. In R. Berber, Nonlinear model based control, NATO ASI Series. Dordrecht: Kluwer Academic Publishers. Banerjee, A., Arkun, Y., Ogunnaike, B., & Pearson, R. (1997). Estimation of nonlinear systems using linear multiple models. AICHE Journal, 43(5), 1204}1226. Bitmead, R. R., Gevers, M., & Wertz, V. (1990). Adaptive optimal control: The thinking man's GPC. Englewood Cli!s, NJ: Prentice-Hall.
B
[A(p(k))x(kDk)#B(p(k))u(kDk)]T x(kDk)TQ0.5 u(kDk)TR0.5
A(p(k))x(kDk)#B(p(k))u(kDk)
QK (k)
0
0
Q0.5x(kDk)
0
I
0
R0.5u(kDk)
0
0
I
which proves (25). (26) can be proven as done in Appendix A to prove (15). When the scheduling feedback control law (9) is used, the proof holds with the appropriate changes discussed at the end of Appendix A.
Appendix F. Proof of Theorem 5 De"ne '(x(kDk)) as a candidate Lyapunov function and check Lyapunov stability criteria on it. First establish that '(x(k))"x(kDk)TQx(kDk)#u(kDk)TRu(kDk) #x(k#1Dk)TP(k)x(k#1Dk)50 and '(x(k))"0 if only if x(k)"x(kDk)"0. Since Q, R and P(k) are positive de"nite, it is true that '(x(k))'0 for all x(k)O0. If x(k)"0, since the minimum value the objective function '(k) can take is zero, the optimal solution of input is u(kDk)"0. This results in x(k#1Dk)"0 and '(k)"0. On the other hand, when '(x(k))"0, the terms x(kDk)TQx(kDk), u(kDk)TRu(kDk), and x(k#1Dk)TP(k)x(k#1Dk) must be all zero. Therefore, we know x(k)"0 since Q'0. Secondly, it is known that '(k)('(k!1) for k51 holds whenever the optimization is feasible. Based on the above two conditions, '(k) is monotonically decreasing and bounded below by zero. Thus it converges to zero, which means both x(k) and u(k) must converge to zero and the controller is asymptotically stabilizing.
References Apkarian, P., Gahinet, P., & Becker, G. (1995). Self-scheduled H = control of linear parameter-varying systems. Automatica, 31, 1251}1261.
'0
(E.5)
Boyd, S., Ghaoui, L. E., Feron, E., & Balakrishnan, V. (1994). Linear matrix inequalities in system and control theory. Philadelphia: SIAM. Chen, H., & AllgoK wer, R. (1998). Quasi-in"nite horizon nonlinear model predictive control scheme with guaranteed stability. Automatica, 34(10), 1205}1217. Chmielewski, D., & Manousiouthakis, V. (1996). On constrained in"nite-time linear quadratic optimal control. Systmes & Control Letters, 29, 121}129. Cutler, C. R., & Ramaker, B. L. (1980). Dynamic matrix control * a computer control algorithm. 1980 Joint automatic control conference (pp. WP5-B/6). Kothare, M. V., Balakrishnan, V., & Morari, M. (1996). Robust constrained model predictive control using linear matrix inequalities. Automatica, 32(10), 1361}1379. Liu, R. W. (1968). Convergent systems. IEEE Transactions on Automatic Control, AC-13, 384}391. Michalska, H., & Mayne, D. Q. (1993). Robust receding horizon control of constrained nonlinear systems. IEEE Transaction on Automatic Control, 38(11), 1623}1633. Rawlings, J. B., & Muske, K. R. (1993). The stability of constrained receding horizon control. IEEE Transactions on Automatic Control, 38(10), 1512}1516. Richalet, J. (1980). General principles of scenario predictive control. 1980 Joint automatic control conference (pp. FA9-A/4). Scokaert, P. O. M., & Rawlings, J. B. (1998). Constrained linear quadratic regulation. IEEE Transactions on Automatic Control, 43(8), 1163}1169. Shamma, J. S., & Athans, M. (1991). Gain scheduling: Potential hazards and possible remedies. Proceedings of the American control conference (pp. 516}521). Wu, F., & Packard, A. (1995). LQG control design for LPV systems. Proceedings of the American control conference (pp. 4440}4444).
Yaman Arkun is the Dean of the College of Engineering at KOC University in Istanbul, Turkey. He received his B.S. from University of Bosphorous, Istanbul, Turkey, and his M.S. and Ph.D. from University of Minnesota. He was on the faculty at RPI from 1979 to 1986. Before he moved to Turkey he was at Georgia Institute of Technology from 1986 to 1999 as Professor of Chemical Engineering. Dr. Arkun has held industrial visiting positions at Tennessee Eastman, Du Pont and Weyerhaeuser companies. He is the North American Editor of the Journal of
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Y. Lu, Y. Arkun / Automatica 36 (2000) 527}540
Process Control. He has served as Editor of Automatica, editorial board member of International Journal of Control, Trustee and Secretary of CACHE, and CAST Director. He is the 1986 recipient of the Donald P. Eckman Award given by the American Automatic Control Council. He has chaired the Systems and Control Area 10b of CAST (1988}1990) and served as the AIChE Director to American Automatic Control Council (1989}1991). His research interest is in process dynamics, modeling and control. In particular he is interested in robust, nonlinear and predictive control, and synthesis of control systems for large-scale plants.
Yaohui Lu was born in Tangshan, China, in 1970. He received the B.Sc. and M.Sc. degree in Chemical Engineering from Tianjin University, China in 1992 and 1995 respectively. He is now a Ph.D. student in School of Chemical Engineering of Georgia Institute of Technology. His current research interests include nonlinear process and model predictive control, system dynamics, and robust stability theory.
