STABILITY OF SWITCHED SYSTEMS

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STABILITY OF SWITCHED SYSTEMS
Daniel Liberzon
Coordinated Science Laboratory and Dept. of
Electrical Computer Eng., Univ. of Illinois at
Urbana-Champaign
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SWITCHED vs. HYBRID SYSTEMS
stability
Properties of the continuous state
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STABILITY ISSUE
unstable
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TWO BASIC PROBLEMS

• Stability for arbitrary switching
• Stability for constrained switching

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TWO BASIC PROBLEMS

• Stability for arbitrary switching
• Stability for constrained switching

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GUAS and COMMON LYAPUNOV FUNCTIONS
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COMMUTING STABLE MATRICES gt GUES
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COMMUTATION RELATIONS and STABILITY
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SWITCHED NONLINEAR SYSTEMS

• Commuting systems

• Global results beyond commuting case ???

Unsolved Problems in Math. Systems and Control
Theory
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SPECIAL CASE
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OPTIMAL CONTROL APPROACH
Associated control system
where
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MAXIMUM PRINCIPLE
(unless )
at most 1 switch
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GENERAL CASE
Theorem suppose

• GAS, backward
  complete, analytic
• s.t.

and
Then switched system is GUAS
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SYSTEMS with SPECIAL STRUCTURE

• Triangular systems
• Feedback systems
• passivity conditions
• small-gain conditions
• 2-D systems

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TRIANGULAR SYSTEMS
Angeli L 00
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FEEDBACK SYSTEMS ABSOLUTE STABILITY
controllable
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FEEDBACK SYSTEMS SMALL-GAIN THEOREM
controllable
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TWO-DIMENSIONAL SYSTEMS
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WEAK LYAPUNOV FUNCTION
Barbashin-Krasovskii-LaSalle theorem
is GAS if s.t.

• (weak Lyapunov
  function)
• is not identically zero along any nonzero
  solution

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COMMON WEAK LYAPUNOV FUNCTION
To extend this to nonlinear switched systems and
nonquadratic common weak Lyapunov functions, we
need a suitable nonlinear observability notion
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NONLINEAR VERSION
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TWO BASIC PROBLEMS

• Stability for arbitrary switching
• Stability for constrained switching

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MULTIPLE LYAPUNOV FUNCTIONS
Useful for analysis of state-dependent switching
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MULTIPLE LYAPUNOV FUNCTIONS
gt GAS
DeCarlo, Branicky
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DWELL TIME
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DWELL TIME
The switching times satisfy
GES
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DWELL TIME
The switching times satisfy
GES
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AVERAGE DWELL TIME
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AVERAGE DWELL TIME
Theorem Hespanha
gt
is GAS
if
GAS is uniform over in this class
Useful for analysis of hysteresis-based switching
logics
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MULTIPLE WEAK LYAPUNOV FUNCTIONS

• .
• observable for each
• s.t. there are infinitely many
• switching intervals of length
• For every pair of switching times
• s.t.
• have

milder than a.d.t.
Extends to nonlinear switched systems as before
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APPLICATION FEEDBACK SYSTEMS
Theorem switched system is GAS if

• s.t. infinitely many switching
  intervals of length
• For every pair of switching times
  at
• which we have

(e.g., switch on levels of equal potential
energy)
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RELATED TOPICS NOT COVERED

• Computational aspects (LMIs, Tempo L)
• Formal methods (work with Mitra Lynch)
• Stochastic stability (Chatterjee L)
• Switched systems with external signals
• Applications to switching control design

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REFERENCES
Lie-algebras and nonlinear switched systems
Margaliot L 04 Nonlinear observability,
LaSalle Hespanha, L, Angeli Sontag 03
(http//decision.csl.uiuc.edu/l
iberzon)