Stability and invariance analysis of uncertain PWA systems based on linear programming

This paper analyzes stability of discrete-time uncertain piecewise-affine systems whose dynamics are defined on a bounded set χ that is not necessarily invariant. The objective is to prove the uniform asymptotic stability of the origin and to find an invariant domain of attraction. This goal is attained by defining a suitable extended dynamics (which is partially fictitious), and by using a numerical procedure based on linear programming. The theoretical results are based on the definition of a piecewise-affine, possibly discontinuous, Lyapunov function.

[1]  A. Bemporad,et al.  Stability and Invariance Analysis of Approximate Explicit MPC based on PWA Lyapunov Functions , 2011 .

[2]  Mato Baotic,et al.  Stabilizing low complexity feedback control of constrained piecewise affine systems , 2005, Autom..

[3]  Stephen P. Boyd,et al.  Linear Matrix Inequalities in Systems and Control Theory , 1994 .

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

[5]  Alberto Bemporad,et al.  A survey on explicit model predictive control , 2009 .

[6]  M. Morari,et al.  Stability analysis of hybrid systems with a linear performance index , 2004, 2004 43rd IEEE Conference on Decision and Control (CDC) (IEEE Cat. No.04CH37601).

[7]  Panos J. Antsaklis,et al.  Synthesis of uniformly ultimate boundedness switching laws for discrete-time uncertain switched linear systems , 2003, 42nd IEEE International Conference on Decision and Control (IEEE Cat. No.03CH37475).

[8]  G. Zhai,et al.  Quadratic stabilizability of switched linear systems with polytopic uncertainties , 2003 .

[9]  Frank Allgöwer,et al.  Nonlinear model predictive control : towards new challenging applications , 2009 .

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

[11]  Franco Blanchini,et al.  Set-theoretic methods in control , 2007 .

[12]  James E. Falk,et al.  Delaunay partitions in Rn applied to non-convex programs and vertex/facet enumeration problems , 2005, Comput. Oper. Res..

[13]  B. Schutter,et al.  On hybrid systems and closed-loop MPC systems , 2001, Proceedings of the 40th IEEE Conference on Decision and Control (Cat. No.01CH37228).

[14]  E. Yaz.  Linear Matrix Inequalities In System And Control Theory , 1998, Proceedings of the IEEE.

[15]  W. P. M. H. Heemels,et al.  Lyapunov Functions, Stability and Input-to-State Stability Subtleties for Discrete-Time Discontinuous Systems , 2009, IEEE Transactions on Automatic Control.

[16]  Mircea Lazar,et al.  Model predictive control of hybrid systems : stability and robustness , 2006 .

[17]  Manfred Morari,et al.  Analysis of discrete-time piecewise affine and hybrid systems , 2002, Autom..

[18]  M. Morari,et al.  A survey on stability analysis of discrete-time piecewise affine systems , 2005 .

[19]  A. Papachristodoulou,et al.  Analysis of switched and hybrid systems - beyond piecewise quadratic methods , 2003, Proceedings of the 2003 American Control Conference, 2003..

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

[21]  David Q. Mayne,et al.  Invariant approximations of the minimal robust positively Invariant set , 2005, IEEE Transactions on Automatic Control.