An example is presented in order to illustrate that nominally exponentially stable discrete-time PieceWise Affine (PWA) systems can have zero robustness  to arbitrarily small additive disturbances. It is shown that this is mainly due to the absence of a continuous Lyapunov function. The importance of this issue cannot be overstated since nominally stabilizing controllers are affected by perturbations when applied in practice. Moreover, many of the methods for synthesizing stabilizing state-feedback controllers for discrete-time PWA systems employ discontinuous (piecewise quadratic) Lyapunov functions, for example see [2, 3]. The discrete-time input-to-state stability framework  is used in order to develop an a posteriori robustness test, which boils down to solving a finite number of linear programming problems. The test can be used to check whether a specific nominally stable PWA system, possibly with a discontinuous Lyapunov function, has some (inherent) robustness to additive disturbances or not.
|Title of host publication||Proceedings of the 25th Benelux meeting on Systems and Control, 13-15 March 2006, Heeze, The Netherlands|
|Place of Publication||Heeze, Netherlands|
|Publication status||Published - 2006|
|Event||25th Benelux Meeting on Systems and Control, March 13-15, 2006, Heeze, The Netherlands - Heeze, Netherlands|
Duration: 13 Mar 2006 → 15 Mar 2006
|Conference||25th Benelux Meeting on Systems and Control, March 13-15, 2006, Heeze, The Netherlands|
|Period||13/03/06 → 15/03/06|