NONLINEAR OBSERVABILITY NOTIONS and STABILITY of SWITCHED SYSTEMS

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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
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MOTIVATING REMARKS

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

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STATE NORM ESTIMATION
where
(observability Gramian)
for some
In particular, this implies 0-distinguishability
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SMALL-TIME vs. LARGE-TIME OBSERVABILITY
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INITIAL-STATE vs. FINAL-STATE OBSERVABILITY
The properties
and
are equivalent
Reason
for FC systems, and
for UO systems
Contrast with
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DETECTABILITY vs. OBSERVABILITY
Detectability is
Hurwitz
small small
Observability can have
arbitrary eigenvalues
Detectability (OSS)
where
Observability can be chosen to decay
arbitrarily fast
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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
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LASALLE THEOREM for SWITCHED SYSTEMS
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LASALLE THEOREM (continued)
piecewise const switching signal
Then the switched system is GAS
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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