MPC for stable linear systems with model uncertainty☆
Introduction
In the model predictive control strategy, at each sample step, an optimal sequence of control inputs, which minimizes an open loop cost function, is computed. The optimization problem also includes hard constraints on the inputs and soft constraints on the outputs or states. In the output-tracking problem, the cost function is defined as the sum of the squared differences between the predicted output at sampling instants and the output reference value. The weighted sum of the inputs is also included.
A modeling approach frequently adopted in model predictive controller (MPC) considers a discrete-time state-space model in the incremental form (Lee, Morari, & Garcia, 1994)where is the vector of states, is the vector of inputs, Δuk=uk−uk−1 is the input increment, k is the present time step and is the vector of outputs. and C are matrices with appropriate dimensions. In the MPC literature, robustness is sought considering certain classes of model uncertainty. From the practical point of view, the following classes can be considered relevant:
(i) Multi-model system (Badgwell, 1997), where the true matrices (A,B,C) of , are unknown but lie in a set . Each j corresponds to a particular operating point of the system.
(ii) Polytopic system, where matrices (A,B,C) are assumed to lie in a polytopic set (Kothare, Balakrishnan, & Morari, 1996)A particular sub-class of this kind of model uncertainty assumes that uncertainty concentrates on the input distribution matrix. In this case, B is such thatand A and C are assumed to be known and invariant (Lee & Cooley, 2000).
In the design of MPC, robust stability is an important issue (Mayne, Rawlings, Rao, & Scokaert, 2000; Morari & Lee, 1999). The target is to design a controller that is stable independent of the operating conditions, which usually alters the process model. For the nominal case, where only the most likely model is considered by the controller, there are several methods to obtain a stable MPC. A popular approach considers that the output and input horizons are finite, and stability is obtained through the inclusion of a terminal state constraint (Keerthi & Gilbert, 1988; Meadows, Henson, Eaton, & Rawlings, 1995; Polak & Yang, 1993). This method cannot be usually extended to the robust output-tracking problem because the terminal constraint cannot be simultaneously satisfied by all the possible process models. Another usual method to obtain a stable MPC is based on an infinite output horizon (Rawlings & Muske, 1993). For stable systems, the infinite horizon open loop cost can be expressed as a finite horizon cost with the inclusion of a terminal state penalty, which has to be computed through the solution of a Lyapunov equation. The extension of this approach to the robust multi-model MPC was proposed by Badgwell (1997) with the inclusion of contracting constraints for the costs associated with the possible plants. In that method, it was assumed that, for the computed input sequence, the state goes to zero at infinite time for all possible plant models. This assumption is not usually true for the output-tracking problem where, in most of the cases, the system steady state is not known. To overcome this problem, Ralhan (1999) proposed a two-stage approach. In the upper stage, targets for the inputs are searched such that the offset in the system output is minimized. These targets are passed to the second stage for the robust dynamic optimization. Although robust stability is not guaranteed, the author showed that if the system reaches a steady state, it will be the correct steady state. Kothare et al. (1996) proposed a min–max predictive control algorithm where the worst-case optimal linear state feedback is treated as an LMI problem. The approach was extended by Lu and Arkun (2000) to polytopic linear parameter-varying systems and for the scheduling MPC problem. Rodrigues and Odloak (2000) proposed a solution to the output feedback robust MPC through the explicit inclusion of a Lyapunov-type inequality constraint in the control optimization problem. Lee and Cooley (2000) extended the approach of Rawlings and Muske (1993) to the robust regulator problem of a system with time-varying input matrix.
The method proposed in this work can be considered as a further extension of the infinite horizon MPC (IHMPC) of Rawlings and Muske (1993) to systems with model uncertainty. When compared to the method of Badgwell (1997) we can identify the following improvements:
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The state-space model on which the controller is based improves the system representation usually adopted in the MPC literature. With this model, for a finite control horizon, the infinite output horizon MPC cost function can be integrated explicitly. The approach becomes more efficient as the computation of the control sequence does not include the solution of the Lyapunov equation to compute the terminal state penalty. The on-line solution of this equation may be time consuming for large systems.
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The usual MPC cost function is modified to guarantee that the cost remains bounded for all possible models of the process. This is done through the inclusion of slack variables in the output error. With this approach, we prove that the robust controller drives the true plant to the desired reference values.
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This work overcomes one of the major barriers to the practical implementation of robust MPC: the need to know the system steady state. Hence, it can be applied to the output-tracking problem and to the regulator problem with unknown disturbances.
Section snippets
Revisited IHMPC
MPC is usually based on a discrete state-space model as shown in , . In the output-tracking problem, the IHMPC cost can be defined as follows:where is the output prediction at time instant k+j made at time is the desired output reference, m is the control horizon, and are positive definite weighting matrices. The controller that is based on the minimization of the above cost function corresponds to the IHMPC (
IHMPC for systems with model uncertainty
Here we assume that the parameters of the model represented by , are uncertain, which means that model matrices A, B and C are not exactly known. Recalling the definitions of these matrices, we observe that the state matrix A includes matrix F, which depends on the system poles only. The input matrix B depends on F and on the step response coefficients D0 and Dd. Hence, if uncertainty concentrates on the gains of the system then only the input matrix B results uncertain. However, if the system
Examples
In this section, we present simulation results that illustrate the performance of the robust MPC developed in this work. Although we could not find in the control literature an example showing the application of an existing robust MPC to the output-tracking case, we tried to compare the proposed robust MPC to other existing controllers. In the simulations performed here, two chemical processes were studied. The first example is characteristic of the oil refining industry (Rodrigues & Odloak,
Conclusion
In this paper, we presented a robust stable controller based on the infinite horizon MPC. The main strength of the paper is related to the solution of the output-tracking problem for systems with model uncertainties. This work extends existing approaches that are usually focused on the regulator problem. The proposed strategy is based on a state-space model, which was tailored to predict the system output as a continuous function of time. This model representation allows the integration of the
Acknowledgements
Support for this work was provided by Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP) under Grant 96/08087-0 and by Conselho Nacional de Desenvolvimento Cientı́fico e Tecnológico (CNPq) under Grant 300860/97-9.
Marco Antônio Rodrigues was born in Arcos, MG, Brazil, in 1970. He received the B.Sc. degree in chemical engineering from the Federal University of Uberlândia in 1993, and the M.Sc. and Ph.D. degrees in chemical engineering from the University of São Paulo, USP, in 1996 and 2001, respectively. Currently, he is a postdoctoral fellow at the Laboratory of Process Control and Simulation (LSCP) of the Department of Chemical Engineering of the Polytechnic School of USP. His current research interests
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Marco Antônio Rodrigues was born in Arcos, MG, Brazil, in 1970. He received the B.Sc. degree in chemical engineering from the Federal University of Uberlândia in 1993, and the M.Sc. and Ph.D. degrees in chemical engineering from the University of São Paulo, USP, in 1996 and 2001, respectively. Currently, he is a postdoctoral fellow at the Laboratory of Process Control and Simulation (LSCP) of the Department of Chemical Engineering of the Polytechnic School of USP. His current research interests include model predictive control, robust stability, fault-tolerant control and their applications to chemical plants.
Darci Odloak is a Professor at the Department of Chemical Engineering of the Polytechnic School of the University of São Paulo. He received a M.Sc. from the University of Rio de Janeiro (COPPE) in 1977 and a Ph.D. from the University of Leeds-UK in 1980. He worked for Petrobras from 1973 to 1990 as a process engineer and from 1991 to 1996 as the head of the Advanced Control Group that developed and implemented an in-house advanced control package in the main oil refineries of Brazil. His present research interest is robust model predictive control, fault-tolerant control and integration of control and real time optimization.
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This paper was not presented at any IFAC meeting. This paper was recommended for publication in revised form by Associate Editor Frank Allgöwer under the direction of Editor Sigurd Skogestad.