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Ensuring Stability in Hybrid Automata with Average Dwell Time: An Invariant Approach

This paper presents a method to analyze the stability of hybrid automata through an invariant approach, focusing on the average dwell time property. By transforming the automaton, we derive conditions ensuring that average dwell time becomes an invariant property, enabling robust stability verification using theorem proving and model checking techniques. The implications of this research extend to various applications, including switched systems and supervisory control in uncertain environments, where compliance with average dwell time is critical for system stability.

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Ensuring Stability in Hybrid Automata with Average Dwell Time: An Invariant Approach

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  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]

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