1 Stability of Hybrid Automata with Average Dwell Time: An Invariant Approach Daniel Liberzon Coordinated Science Laboratory University of Illinois at Urbana-Champaign Sayan Mitra Computer Science and Artificial Intelligence Laboratory Massachusetts Institute of Technology IEEE CDC 2004, Paradise Island, Bahamas

2 HIOA: math model specification  Expressive: few constraints on continuous and discrete behavior  Compositional: analyze complex systems by looking at parts  Structured: inductive verification  Compatible: application of CT results e.g. stability, synthesis HIOA: A Platform Bridging the Gap Control Theory: Dynamical system with boolean variables  Stability  Controllability  Controller design Computer Science: State transition systems with continuous dynamics  Safety verification  model checking  theorem proving Hybrid Systems [Lynch, Segala, Vaandrager]

3 Hybrid I/O Automata  V= U  Y  X: input, output, internal variables  Q: states, a set of valuations of V  : start states  A = I  O  H: input, output, internal actions  D  Q  A  Q: discrete transitions  T: trajectories for V, functions describing continuous evolution  Execution (fragment): sequence  0 a 1  1 a 2  2 …, where:  Each  i is a trajectory of the automaton, and  Each (  i.lstate, a i,  i+1.fstate) is a discrete step

4  Switched system modeled as HIOA:  Each mode is modeled by a trajectory definition  Mode switches are brought about by actions  Usual notions of stability apply  Stability theorems involving Common and Multiple Lyapunov functions carry over Switched system:  is a family of systems  is a switching signal HIOA Model for Switched Systems

5 Stability Under Slow Switchings t Assuming Lyapunov functions for the individual modes exist, global asymptotic stability is guaranteed if τ a is large enough [Hespanha] Slow switching: # of switches on average dwell time ( τ a ) decreasing sequence

6 Verifying Average Dwell Time  Average dwell time is a property of the executions of the automaton Invariant approach:  Transform the automaton A  A’ so that the a.d.t property of A becomes an invariant property of A’.  Then use theorem proving or model checking tools to prove the invariant(s) Invariant I(s) proved by base case : induction discrete: continuous:

7 Transformation for Stability  Simple stability preserving transformation:  counter Q, for number of extra mode switches  a (reset) timer t  Q min for the smallest value of Q AA’ Theorem: A has average dwell time τ a iff Q- Q min ≤ N 0 in all reachable states of A’. invariant property

8 Case Study: Hysteresis Switch Initialize Find no yes ? Inputs:  Under suitable conditions on (compatible with bounded noise and no unmodeled dynamics), can prove a.d.t. See CDC paper for details  Used in switching (supervisory) control of uncertain systems

9 Beyond the CDC paper  Sufficient condition for violating a.d.t. τ a : exists a cycle with N(α) - α.length / τ a > 0  This is also necessary condition for some classes of HIOA  Search for counterexample execution by maximizing N(α) - α.length / τ a over all executions MILP approach: Future work: [Mitra, Liberzon, Lynch, “Verifying average dwell time”, 2004,  Input-output properties (external stability)  Supporting software tools [Kaynar, Lynch, Mitra]  Probabilistic HIOA [Cheung, Lynch, Segala, Vaandrager] and stability of stochastic switched systems [Chatterjee, Liberzon, FrA01.1]