Performance analysis of piecewise linear systems and model predictive control systems

Bemporad and Morari (1999) provided a tool for obtaining the explicit solution of constrained model predictive control (MPG) problems by showing that the control law is a continuous piecewise affine (PWA) function of the state vector. Therefore, the feedback interconnection between the MPC controller and a linear system, or a PWA system (e.g., a PWA approximation of a nonlinear system), is a PWA system. For discrete-time PWA and hybrid systems, the present authors (2000) presented an algorithm for verification/reachability analysis. In this paper, we formulate the performance analysis problem of closed-loop PWA systems (including MPC feedback loops where the prediction model and the plant model could be different) as a reachability analysis problem, and use our algorithm to obtain a tool for characterizing (i) the set of states for which the evolution is feasible, (ii) the domain of stability, (iii) the performance of the closed-loop.

[1]  E. Gilbert,et al.  Optimal infinite-horizon feedback laws for a general class of constrained discrete-time systems: Stability and moving-horizon approximations , 1988 .

[2]  K. T. Tan,et al.  Linear systems with state and control constraints: the theory and application of maximal output admissible sets , 1991 .

[3]  Eduardo Sontag.  Nonlinear regulation: The piecewise linear approach , 1981 .

[4]  A. Morse,et al.  Basic problems in stability and design of switched systems , 1999 .

[5]  Thomas A. Henzinger,et al.  HYTECH: a model checker for hybrid systems , 1997, International Journal on Software Tools for Technology Transfer.

[6]  Ilya Kolmanovsky,et al.  Maximal output admissible sets for discrete-time systems with disturbance inputs , 1995, Proceedings of 1995 American Control Conference - ACC'95.

[7]  M. Kantner.  Robust stability of piecewise linear discrete time systems , 1997, Proceedings of the 1997 American Control Conference (Cat. No.97CH36041).

[8]  M. Branicky.  Multiple Lyapunov functions and other analysis tools for switched and hybrid systems , 1998, IEEE Trans. Autom. Control..

[9]  John Lygeros,et al.  Controllers for reachability specifications for hybrid systems , 1999, Autom..

[10]  Alberto Bemporad,et al.  Control of systems integrating logic, dynamics, and constraints , 1999, Autom..

[11]  Alberto Bemporad,et al.  The explicit linear quadratic regulator for constrained systems , 2003, Autom..

[12]  Alberto Bemporad,et al.  Optimization-Based Verification and Stability Characterization of Piecewise Affine and Hybrid Systems , 2000, HSCC.

[13]  Alberto Bemporad,et al.  Observability and controllability of piecewise affine and hybrid systems , 2000, IEEE Trans. Autom. Control..

[14]  Eduardo D. Sontag,et al.  Interconnected Automata and Linear Systems: A Theoretical Framework in Discrete-Time , 1996, Hybrid Systems.

[15]  John N. Tsitsiklis,et al.  Complexity of stability and controllability of elementary hybrid systems , 1999, Autom..

[16]  Olaf Stursberg,et al.  A Case Study in Tool-Aided Analysis of Discretely Controlled Continuous Systems: The Two Tanks Problem , 1997, Hybrid Systems.

[17]  Thomas A. Henzinger,et al.  HYTECH: A Model Checker for Hybrid Systems , 1997, CAV.