Invariant sets computation for convex difference inclusions systems

In this paper we introduce the Convex Difference Inclusion (CDI) systems as a modeling framework useful to analyze set-theory and invariance-related issues for nonlinear and uncertain systems. The dynamics of a CDI system is given by a set-valued map whose values are convex, compact subsets of the space and are determined by convex bounding functions. Necessary and sufficient boundary-type conditions for invariance and contractiveness, characteristic of the linear systems, are given for the CDI systems. Lyapunov functions are proved to be induced by contractive sets for CDI systems, as in the linear context. A computational procedure for obtaining polytopic invariant and contractive sets for nonlinear systems, based on the properties of the CDI systems, is presented.

[1]  T. Alamo,et al.  Convex invariant sets for discrete-time Lur'e systems , 2009, Autom..

[2]  F. Blanchini.  Ultimate boundedness control for uncertain discrete-time systems via set-induced Lyapunov functions , 1994, IEEE Trans. Autom. Control..

[3]  E. F. Camacho,et al.  On the computation of convex robust control invariant sets for nonlinear systems , 2010, Autom..

[4]  Eduardo F. Camacho,et al.  On the computation of invariant sets for constrained nonlinear systems: An interval arithmetic approach , 2003, 2003 European Control Conference (ECC).

[5]  Isabelle Queinnec,et al.  Ultimate bounded stability and stabilization of linear systems interconnected with generalized saturated functions , 2011, Autom..

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

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

[8]  Sophie Tarbouriech,et al.  Polyhedral regions of local stability for linear discrete-time systems with saturating controls , 1999, IEEE Trans. Autom. Control..

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

[10]  E. Gilbert,et al.  Theory and computation of disturbance invariant sets for discrete-time linear systems , 1998 .

[11]  Eduardo F. Camacho,et al.  A new concept of invariance for saturated systems , 2006, Autom..

[12]  Per-olof Gutman,et al.  Admissible sets and feedback control for discrete-time linear dynamical systems with bounded controls and states , 1984, The 23rd IEEE Conference on Decision and Control.

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

[14]  Riccardo Scattolini,et al.  A stabilizing model-based predictive control algorithm for nonlinear systems , 2001, Autom..

[15]  J. Aubin,et al.  Differential inclusions set-valued maps and viability theory , 1984 .

[16]  R. Schneider.  Convex Bodies: The Brunn–Minkowski Theory: Minkowski addition , 1993 .

[17]  D. Bertsekas.  Infinite time reachability of state-space regions by using feedback control , 1972 .

[18]  David Q. Mayne,et al.  Constrained model predictive control: Stability and optimality , 2000, Autom..

[19]  Eduardo F. Camacho,et al.  On the computation of local invariant sets for nonlinear systems , 2007, 2007 46th IEEE Conference on Decision and Control.

[20]  L. Gurvits.  Stability of discrete linear inclusion , 1995 .

[21]  Manfred Morari,et al.  Robust constrained model predictive control using linear matrix inequalities , 1994, Proceedings of 1994 American Control Conference - ACC '94.

[22]  Basil Kouvaritakis,et al.  Nonlinear model predictive control with polytopic invariant sets , 2003, Autom..

[23]  María M. Seron,et al.  Systematic ultimate bound computation for sampled-data systems with quantization , 2007, Autom..

[24]  Jean-Pierre Aubin,et al.  Viability theory , 1991 .

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

[26]  Tingshu Hu,et al.  Analysis and design for discrete-time linear systems subject to actuator saturation , 2002, Syst. Control. Lett..

[27]  M. BravoJ.,et al.  On the computation of invariant sets for constrained nonlinear systems , 2005 .

[28]  Stephen P. Boyd,et al.  Convex Optimization , 2004, Algorithms and Theory of Computation Handbook.