NONLINEAR OBSERVABILITY NOTIONS and STABILITY of SWITCHED SYSTEMS CDC ’02 João Hespanha Univ. of California at Santa Barbara Daniel Liberzon Univ. of Illinois at Urbana-Champaign Eduardo Sontag Rutgers University

MOTIVATING REMARKS  Several ways to define observability (equivalent for linear systems)  Related issues: observer design or state-norm estimation detectability vs. observability LaSalle’s invariance principle (says that largest unobservable set wrt )  Goal: investigate these with nonlinear tools

STATE NORM ESTIMATION (observability Gramian) where for some In particular, this implies 0-distinguishability

SMALL-TIME vs. LARGE-TIME OBSERVABILITY The properties and are NOT equivalent Counterexample:

INITIAL-STATE vs. FINAL-STATE OBSERVABILITY The properties and are equivalent Reason: for FC systems, and for UO systems Contrast with

DETECTABILITY vs. OBSERVABILITY Detectability is Hurwitz small Observability can have arbitrary eigenvalues Detectability (OSS): where Observability: can be chosen to decay arbitrarily fast

DETECTABILITY vs. OBSERVABILITY (continued) and This is equivalent to small-time observability defined before OSS admits equivalent Lyapunov characterization: For observability, must have arbitrarily rapid growth Observability:

LASALLE THEOREM for SWITCHED SYSTEMS finite index set Assume that for each : 1. pos. def. rad. unbdd function s.t. 2.The system is small-time observable: Collection of systems:

LASALLE THEOREM (continued) Then the switched system is GAS – piecewise const switching signal For the switched system assume: 3. s.t. there are infinitely many switching intervals of length 4.For every pair of switching times s.t. have

SUMMARY  Proposed observability definitions for nonlinear systems in terms of comparison functions  Investigated implications and equivalences among them  Used them to obtain a LaSalle-like stability theorem for switched systems  General versions of results apply to systems with inputs