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COMMON WEAK LYAPUNOV FUNCTIONS and OBSERVABILITY

COMMON WEAK LYAPUNOV FUNCTIONS and OBSERVABILITY. Daniel Liberzon. Coordinated Science Laboratory and Dept. of Electrical & Computer Eng., Univ. of Illinois at Urbana-Champaign. ( weak Lyapunov function).

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COMMON WEAK LYAPUNOV FUNCTIONS and OBSERVABILITY

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  1. COMMON WEAK LYAPUNOV FUNCTIONS and OBSERVABILITY Daniel Liberzon Coordinated Science Laboratory and Dept. of Electrical & Computer Eng., Univ. of Illinois at Urbana-Champaign

  2. (weak Lyapunov function) • is not identically zero along any nonzero solution (observability with respect to ) Example: => GAS observable OBSERVABILITY and ASYMPTOTIC STABILITY Barbashin-Krasovskii-LaSalle theorem: is GAS if s.t.

  3. SWITCHED LINEAR SYSTEMS [Hespanha ’04] • s.t. . • observable for each • infinitely many switching intervals of length To handle nonlinear switched systems and non-quadratic weak Lyapunov functions, need a suitable nonlinear observability notion Theorem (common weak Lyapunov function): Switched linear system is GAS if

  4. OBSERVABILITY: MOTIVATING REMARKS Several ways to define observability (equivalent for linear systems) Benchmarks: • observer design or state norm estimation • detectability vs. observability • LaSalle’s stability theorem for switched systems Joint work with Hespanha, Sontag, and Angeli No inputs here, but can extend to systems with inputs

  5. STATE NORM ESTIMATION where (observability Gramian) Observability definition #1: where This is a robustified version of 0-distinguishability

  6. OBSERVABILITY DEFINITION #1: A CLOSER LOOK Small-time observability: Large-time observability: Initial-state observability: Final-state observability:

  7. DETECTABILITY vs. OBSERVABILITY Detectability is Hurwitz Observability can have arbitrary eigenvalues A natural detectability notion is output-to-state stability (OSS): [Sontag-Wang] where Observability def’n #2: OSS, and can decay arbitrarily fast

  8. OSS admits equivalent Lyapunov characterization: For observability, should have arbitrarily rapid growth OBSERVABILITY DEFINITION #2: A CLOSER LOOK Definition: and Theorem:This is equivalent to definition #1 (small-time obs.)

  9. SWITCHED NONLINEAR SYSTEMS • s.t. • Each system • infinitely many switching intervals of length is norm-observable: Theorem (common weak Lyapunov function): Switched system is GAS if [Hespanha–L–Sontag–Angeli ’05]

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