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

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

SWITCHED LINEAR SYSTEMS [ Hespanha 04 ] Theorem (common weak Lyapunov function): observable for each To handle nonlinear switched systems and non-quadratic weak Lyapunov functions, need a suitable nonlinear observability notion Switched linear system is GAS if infinitely many switching intervals of length s.t..

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

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

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

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

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

SWITCHED NONLINEAR SYSTEMS Theorem (common weak Lyapunov function): s.t. Switched system is GAS if infinitely many switching intervals of length Each system is norm-observable: [Hespanha–L–Sontag–Angeli 05]