Download presentation

Presentation is loading. Please wait.

Published byAlfred Pingleton Modified over 3 years ago

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
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)

3
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..

4
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

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

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

7
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

8
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

9
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]

Similar presentations

OK

NONLINEAR CONTROL with LIMITED INFORMATION Daniel Liberzon Coordinated Science Laboratory and Dept. of Electrical & Computer Eng., Univ. of Illinois at.

NONLINEAR CONTROL with LIMITED INFORMATION Daniel Liberzon Coordinated Science Laboratory and Dept. of Electrical & Computer Eng., Univ. of Illinois at.

© 2017 SlidePlayer.com Inc.

All rights reserved.

Ads by Google

Full ppt on electron beam machining videos Ppt on spiritual leadership conference Ppt on electricity in india Ppt on infosys company history Ppt on real numbers for class 9th notes Ppt on artificial intelligence system Ppt on articles of association definition Ppt on asia the continent of contrast Ppt on abo blood grouping reagents Download ppt on coordinate geometry for class 9th 2015