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

Daniel Liberzon Coordinated Science Laboratory

University of Illinois at

Urbana-Champaign

liberzon@uiuc.edu

Sayan Mitra Computer Science and Artificial Intelligence Laboratory

Massachusetts Institute of Technology

mitras@csail.mit.edu

IEEE CDC 2004, Paradise Island, Bahamas

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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]

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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 a1 1 a2 2 …, where:

Each i is a trajectory of the automaton, and

Each (i.lstate, ai , i+1.fstate) is a discrete step

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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

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Stability Under Slow Switchings

t1 12 2

)()( tV t

Assuming Lyapunov functions for the individual modes exist, global asymptotic stability is guaranteed if τa is large enough [Hespanha]

Slow switching:

),( Tt# of switches on average dwell time (τa)

decreasing sequence

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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:

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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

A A’

Theorem: A has average dwell time τa iff Q- Qmin ≤ N0 in all reachable states of A’.

invariant property

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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

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Beyond the CDC paper

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

Search for counterexample execution by maximizing N(α) - α.length / τa over all executions

MILP approach:

Future work:

[Mitra, Liberzon, Lynch, “Verifying average dwell time”, 2004, http://decision.csl.uiuc.edu/~liberzon]

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]