Stability of Hybrid Automata with Average Dwell Time: An Invariant Approach

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

2. HIOA: A Platform Bridging the Gap Hybrid Systems 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 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 [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 0a11a22…, where: • Each i is a trajectory of the automaton, and • Each (i.lstate, ai ,i+1.fstate) is a discrete step

4. HIOA Model for Switched Systems Switched system: • is a family of systems • is a switching signal • 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

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

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. A A’ Transformation for Stability • Simple stability preserving transformation: • counter Q, for number of extra mode switches • a (reset) timer t • Qmin for the smallest value of Q Theorem:A has average dwell time τa iff Q- Qmin≤ N0in all reachable states of A’. invariant property

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

9. Beyond the CDC paper MILP approach: • Search for counterexample execution by maximizing N(α) - α.length / τaover all executions • Sufficient condition for violating a.d.t. τa: exists a cycle with N(α) - α.length / τa > 0 • This is also necessary condition for some classes of HIOA [Mitra, Liberzon, Lynch, “Verifying average dwell time”, 2004, http://decision.csl.uiuc.edu/~liberzon] Future work: • 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]