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NONLINEAR OBSERVABILITY NOTIONS and STABILITY of SWITCHED SYSTEMS

NONLINEAR OBSERVABILITY NOTIONS and STABILITY of SWITCHED SYSTEMS. Jo ã o Hespanha Univ. of California at Santa Barbara. Daniel Liberzon Univ. of Illinois at Urbana-Champaign. Eduardo Sontag Rutgers University. CDC ’02. MOTIVATING REMARKS. Several ways to define observability

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NONLINEAR OBSERVABILITY NOTIONS and STABILITY of SWITCHED SYSTEMS

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

  2. 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

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

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

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

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

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

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

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

  10. 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

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