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Robust constrained optimal control for linear systems with linear performance index


M. Morari

AIChE Annual Meeting, Indianapolis, IN, USA, Indianapolis Convention Center

A control system is robust when stability is preserved and the performance specifications are met for a specified range of model variations and a class of noise signals (uncertainty range). Although a rich theory has been developed for the robust control of linear systems, very little is known about the robust control of linear systems with constraints. This type of problem has been addressed in the context of constrained optimal control, and in particular in the context of robust model predictive control (MPC). A typical robust MPC strategy consists of solving a min-max problem to optimize robust performance (the minimum over the control input of the maximum over the disturbance) while enforcing input and state constraints for all possible disturbances. Min-max robust constrained optimal control was originally proposed by Witsenhausen. In the context of robust MPC the problem was tackled by Campo and Morari, and further developed by Allwright and Papavasiliou for SISO FIR plants. Kothare et al. optimize robust performance for polytopic/multi-model and structured feedback uncertainty, Scokaert and Mayne for additive input disturbances, and Lee and Yu for linear time-varying and time-invariant state-space models depending on a vector of parameters belonging to either an ellipsoid or a polyhedron. In all cases, the resulting min-max optimization problems turn out to be computationally demanding, a serious drawback for on-line receding horizon implementation. In this paper we consider discrete-time uncertain linear systems with constraints on inputs and states. We show how state-feedback solutions to open-loop and closed-loop min-max robust constrained optimal control problems based on linear performance index can be computed, off-line, for systems affected by additive norm-bounded input disturbances and/or polyhedral parametric uncertainty. We show that the resulting optimal state-feedback control law is a piecewise affine and continuous function of the state vector and we propose an efficient algorithm for computing it. The algorithm is based on a dynamic programming recursion and a multiparametric linear programming solver. When the optimal control law is implemented in a moving horizon scheme, the on-line computation of the resulting controller is reduced to the evaluation of a piecewise affine function. The technique proposed has both theoretical and practical advantages. From a theoretical point of view, the state-feedback piecewise affine solution gives insight on the action of the controller in different regions of the state space. From a practical point of view, the proposed technique is attractive for a wide range of applications where the simplicity of the on-line computational complexity is a crucial requirement. The approach of this paper relies on multiparametric solvers, and follows the ideas proposed earlier for optimal control of deterministic linear systems and hybrid systems.


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