On notions and sufficient conditions for forward invariance of sets for hybrid dynamical systems

Forward invariance for hybrid dynamical systems modeled by differential and difference inclusions with state-depending conditions enabling flows and jumps is studied. Several notions of forward invariance are considered and sufficient conditions in terms of the objects defining the system are introduced. In particular, we study forward invariance notions that apply to systems with nonlinear dynamics for which not every solution is unique or may exist for arbitrary long hybrid time. Such behavior is very common in hybrid systems. Lyapunov-based conditions are also proposed for the estimation of invariant sets. Applications and examples are given to illustrate the results. In particular, the results are applied to the estimation of weakly forward invariant sets, which is an invariance property of interest when employing invariance principles to study convergence of solutions.

[1]  Jan H. van Schuppen,et al.  Reachability and control synthesis for piecewise-affine hybrid systems on simplices , 2006, IEEE Transactions on Automatic Control.

[2]  Jean-Pierre Aubin,et al.  Impulse differential inclusions: a viability approach to hybrid systems , 2002, IEEE Trans. Autom. Control..

[3]  Zvi Artstein,et al.  Feedback and invariance under uncertainty via set-iterates , 2008, Autom..

[4]  Edmund M. Clarke,et al.  Computing differential invariants of hybrid systems as fixedpoints , 2008, Formal Methods Syst. Des..

[5]  Martin Buss,et al.  INVARIANCE CONTROL DESIGN FOR CONSTRAINED NONLINEAR SYSTEMS , 2004 .

[6]  Jean-Pierre Aubin,et al.  Viability Theory: New Directions , 2011 .

[7]  Ashish Tiwari,et al.  Generating Polynomial Invariants for Hybrid Systems , 2005, HSCC.

[8]  Martin Buss,et al.  Invariance control for a class of cascade nonlinear systems , 2002, IEEE Trans. Autom. Control..

[9]  Henny B. Sipma,et al.  Constructing invariants for hybrid systems , 2004, Formal Methods Syst. Des..

[10]  Ricardo G. Sanfelice,et al.  Invariance Principles for Hybrid Systems With Connections to Detectability and Asymptotic Stability , 2007, IEEE Transactions on Automatic Control.

[11]  Panos J. Antsaklis,et al.  An invariant‐based approach to the design of hybrid control systems , 2001 .

[12]  Ricardo G. Sanfelice,et al.  Hybrid Dynamical Systems: Modeling, Stability, and Robustness , 2012 .

[13]  Franco Blanchini,et al.  Set invariance in control , 1999, Autom..

[14]  J. Jahn.  Introduction to the Theory of Nonlinear Optimization , 1994 .

[15]  Jessy W. Grizzle,et al.  Hybrid Invariant Manifolds in Systems With Impulse Effects With Application to Periodic Locomotion in Bipedal Robots , 2009, IEEE Transactions on Automatic Control.

[16]  Henk Nijmeijer,et al.  Further results on Lyapunov-like conditions of forward invariance and boundedness for a class of unstable systems , 2009, 53rd IEEE Conference on Decision and Control.

[17]  Ashish Tiwari.  Generating Box Invariants , 2008, HSCC.

[18]  Jean-Pierre Aubin,et al.  Differential Inclusions and Viability Theory , 1982 .

[19]  Maria Domenica Di Benedetto,et al.  Computation of maximal safe sets for switching systems , 2004, IEEE Transactions on Automatic Control.