COMMON WEAK LYAPUNOV FUNCTIONS and OBSERVABILITY

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COMMON WEAK LYAPUNOV FUNCTIONS and
OBSERVABILITY
Daniel Liberzon
Coordinated Science Laboratory and Dept. of
Electrical Computer Eng., Univ. of Illinois at
Urbana-Champaign
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OBSERVABILITY and ASYMPTOTIC STABILITY
Barbashin-Krasovskii-LaSalle theorem
is GAS if s.t.
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Theorem (common weak Lyapunov function)
Switched linear system is GAS if
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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

Joint work with Hespanha, Sontag, and Angeli
No inputs here, but can extend to systems with
inputs
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STATE NORM ESTIMATION
This is a robustified version of
0-distinguishability
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OBSERVABILITY DEFINITION 1 A CLOSER LOOK
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DETECTABILITY vs. OBSERVABILITY
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OBSERVABILITY DEFINITION 2 A CLOSER LOOK
Definition
and
Theorem This is equivalent to definition 1
(small-time obs.)
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SWITCHED NONLINEAR SYSTEMS
Theorem (common weak Lyapunov function)
Switched system is GAS if
HespanhaLSontagAngeli 05