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서울대학교 제어계측신기술연구소

Model Predictive Control:

Yesterday, Today and Tomorrow

Jinhoon Choi

Engineering Research Center for Advanced Control and Instrumentation

Seoul National University

Introduction

Model predictive control, (MPC) also called as receding horizon control or moving horizon control, is a technique to implement the finite horizon open loop optimal control sequences in the infinite horizon feedback setting. The main idea of MPC existed since the early 60's when the open loop optimal control problem was vigorously studied [7]. However, MPC was not an interesting control strategy in those days because the infinite horizon linear quadratic optimal control problem admits a nice linear feedback type optimal solution. Nevertheless, in the late 70's, the chemical engineers found that MPC has a significant advantage over the other control strategies [6]. Indeed they showed that MPC can handle effectively various constraints that are always present in any practical control problems. The control input cannot assume its value outside a bounded region due to the physical limitations of manipulated variables. Moreover, the states of the plant are usually desired to lie in a designated area of the state space because of safety, environmental regulation and so on. Even if there existed a number of powerful linear control strategies for unconstrained linear plants, all these linear control strategies are hardly effective in the presence of the constraints.  Hence, the flexible constraints handling capability of MPC is an important advantage in addressing real world control problems. Moreover, constrained MPC has proven its usefulness in practice. Hence, since the late 70s, constrained MPC has been the core research topic within the chemical engineering control community. During the last decade, the stability properties of constrained model predictive regulation (MPR) started to be established. However, some of the claimed stability properties are not completely proven. Moreover, it is recently shown that constrained MPC possesses the exponential stability                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   properties that are stronger than those claimed before [3]. Hence, in this paper, we first review the existing stability theory for constrained MPR. We then summarize a complete stability theory for constrained MPR. Finally, the future directions in MPC research is identified.

Yesterday

The main idea of MPC existed since the early 60's as mentioned in the introduction section. However, the stability of MPC was dubious at that time. Indeed as mentioned in [1], the closed loop stability with MPC is not guaranteed in general. The first comprehensive stability result for unconstrained MPR is established by Kwon and Pearson [10] in the late 70s. They proposed to constrain the terminal state to be zero in the open loop optimal control problem. With this constraint, they were able to establish the asymptotic stability of the closed loop system composed of a controllable linear plant and unconstrained MPR. This terminal state constraint idea is later extended to the hard constrained nonlinear system case by Keerthi and Gilbert [9]. In the early 90's, an alternate stabilizing MPR formulation is proposed by Rawlings and Muske [13]. Instead of the terminal state constraints, they adopted an infinite horizon cost function and claimed that the resulting constrained MPR is asymptotically stable. A system is  asymptotically stable if and only if it is attractive and stable. However, Rawlings and Muske proved only the attractivity of the closed loop system. For hard state constrained MPR, the stability can be guaranteed only inside the region defined by the hard state constraints since the control is not defined outside. To circumbent this drawback, Rawlings and Muske proposed to drop some of the hard state constraints so that the quadratic program associated with MPR is feasible. On the other hand, Zheng and Morari [16] adopted the state constraint softening idea by Zafiriou and Chiou [15] and claimed the asymptotic stability of the closed loop system. But, similar to the result by Rawlings and Muske, they showed the attractivity only. Besides these stability results for constrained state feedback MPR, the stability of the constrained output feedback MPR is also addressed. Based on the stability theory of perturbed systems, Muske and Rawlings [12] adopted a nonconstructive approach to the stability of the output feedback case. When a state feedback law is combined with an asymptotic observer, the key requirement for the perturbed system stability theory to be applicable is the Lipschitz continuity of the state feedback control law. The result in [8] is identified as a vehicle to establish the Lipschitz continuity of hard constrained state feedback MPR with the state constraint dropping. The result in [8] requires that the gradient of the binding constraints be linearly independent. However, this condition is not verified in [12]. Indeed, as shown in [3], there exists an example where this requirement is not satisfied. Hence, the nonconstructive approach is not in general applicable. On the otherhand, Zheng and Morari [16] adopted a constructive approach and claimed the asymptotic stability of the closed loop system composed of a stable linear plant and mixed constrained output feedback MPR. However, again only the                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           attractivity is established.

Today

A complete stability result for mixed constrained MPR has been recently established by Choi and Kwon [3]. They showed that mixed constrained MPR is exponentially stable for both state and output feedback cases, under the assumption that the linear plant in the absence of the input saturation is stabilizable and detectable. Firstly, it is shown that a stable (marginally stable) system is globally (semi-globally) exponentially stabilizable by mixed constrained state feedback MPR because the optimal cost function is an appropriate Lyapunov function for the closed loop system. Moreover, any unstable system is shown to be exponentially stabilizable by mixed constrained state feedback MPR in any compact subset of the constrained asymptotically stabilizable set. On the other hand, the sum of the optimal objective associated with state feedback MPR and a positive definite quadratic term of the observer error forms a Lyapunov function from which one can conclude the the global (semi-global) [local] exponential stability of the closed loop system composed of a stable (marginally stable) [unstable] linear plant and mixed constrained output feedback MPR. For the marginally stable and unstable plant cases, the domain of attraction is shown to be the set of all initial conditions for which mixed constrained MPR is defined throughout the trajectory. Notice that this is the largest possible domain of attraction since, otherwise, there exists a sampling time where the control is not defined.

Tomorrow

As shown in the previous section, a complete stability theory for MPR is now established. In this section, we will identify the topics for which the MPC research heads after the completion of stability theory for MPR.

Constrained linear quadratic optimal regulation (CLQOR) can be viewed as an MPR since the solution of the CLQOR problem is not given as a feedback type. Recently, two CLQOR algorithms are proposed [4], [14]. However, we found that there exists a more effcient algorithm. Moreover, the stability of CLQOR is not completely established yet. Hence, CLQOR will be one of the most important research topics in MPC research

Another interesting control problem other than the regulation is the disturbance attenuation. The unconstrained model predictive disturbance attenuation (MPDA) problem is already established [11]. However, similar to the MPR case, unconstrained MPDA is not very interesting because there exists a number of powerful infinite horizon disturbance attenuation algorithms. Hence, the incorporation of the constraints into MPDA problem is important and will become one of the core research topics in MPC research.

The model uncertainty was identified as a key reason why the modern control algorithms failed in practice in the late 70s. Since then, the robustness of various control strategies were one of the hottest issues in control                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                 research. In this context, the robustness of the constrained MPC is also an inportant issue in constrained MPC. Robust MPC in the presence of the structured uncertainties is already addressed [2]. However, there still exist a number of robustness issues in MPC such as robust MPC in face of the unstructured uncertainties.

An alternate strategy to cope with the structured uncertainty is the adaptation. The adaptation idea in MPC is not new since one of the algorithm is proposed for adaptive control [5]. However, the theoretical background for adaptive MPC is not well established. Hence, adaptive MPC will be one of the focal research topics in MPC research.

MPC possesses a great potential as a practical control algorithm. However, it is usually computationally demanding. Hence, the most important issue in MPC practice is to reduce the computation time in the existing MPC algorithms and to come up with new model predictive control algorithms that require less computation time.

References

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3.	Choi, J. and Kwon, W. H.: IEEE Tran. Auto. Cont., submitted.

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11.	Lee J.-W., Kwon, W. H. and Lee, J. H.: Int. J. Cont., accepted.

12.	Muske, K. R. and Rawlings, J.: J. Process Cont., 3, 85(1993).

13.	Rawlings, J. and Muske, K. R.: IEEE Tran. Auto. Cont., 38, 1512(1993).

14.	Scokaert, P. O. M. and Rawlings, J. B.: IEEE Tran. Auto. Cont.,	submitted.

15.	Zafiriou, E. and Chiou, H. W.: Proc. 1993 Amer. Cont. Conf., 372(1993).

16.	Zheng, A. and Morari, M.: IEEE Tran. Auto. Cont., 40, 1818(1995).