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

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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 liberz
on_at_uiuc.edu
Sayan Mitra Computer Science and Artificial
Intelligence Laboratory


Massachusetts Institute of Technology mitras_at_csa
il.mit.edu
IEEE CDC 2004, Paradise Island, Bahamas
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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
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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 , ?i1.fstate) is a discrete
  step

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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

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Stability Under Slow Switchings
t
Slow switching
Assuming Lyapunov functions for the individual
modes exist, global asymptotic stability is
guaranteed if ta is large enough Hespanha
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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

Theorem A has average dwell time ta iff Q- Qmin
N0 in all reachable states of A.
invariant property
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Case Study Hysteresis Switch
Inputs
Initialize
Find
?
no
yes

• 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

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Beyond the CDC paper
MILP approach

• Search for counterexample execution by maximizing
  N(a) - a.length / ta over all executions

• Sufficient condition for violating a.d.t. ta
  exists a cycle with N(a) - a.length / ta gt 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