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 U.S.A.
DISC HS, June 2003
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SWITCHED vs. HYBRID SYSTEMS
stability
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STABILITY ISSUE
Asymptotic stability of each subsystem is
necessary for stability
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STABILITY ISSUE
Asymptotic stability of each subsystem is
necessary but not sufficient for stability
(This only happens in dimensions 2 or higher)
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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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GLOBAL UNIFORM ASYMPTOTIC STABILITY
GUAS is Lyapunov stability
plus asymptotic convergence
Reduces to standard GAS notion for non-switched
systems
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COMPARISON FUNCTIONS
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COMMON LYAPUNOV FUNCTION
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COMMON LYAPUNOV FUNCTION (continued)
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CONVEX COMBINATIONS
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SWITCHED LINEAR SYSTEMS
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COMMUTING STABLE MATRICES gt GUES

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LIE ALGEBRAS and STABILITY
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SOLVABLE LIE ALGEBRA gt GUES
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MORE GENERAL LIE ALGEBRAS
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NONLINEAR SYSTEMS

• Nothing is known beyond this

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REMARKS on LIE-ALGEBRAIC CRITERIA
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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
Recall for linear systems, triangular gt GUAS
For nonlinear systems, not true in general
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INPUT-TO-STATE STABILITY (ISS)
Nonlinear systems
For switched systems, triangular ISS gt GUAS
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FEEDBACK SYSTEMS ABSOLUTE STABILITY
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FEEDBACK SYSTEMS SMALL-GAIN THEOREM
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TWO-DIMENSIONAL SYSTEMS
Necessary and sufficient conditions for
GUES known since 1970s
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WEAK LYAPUNOV FUNCTION
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COMMON WEAK LYAPUNOV FUNCTION
Extends to nonlinear switched systems and
nonquadratic common weak Lyapunov functions
using a suitable nonlinear observability notion
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TWO BASIC PROBLEMS

• Stability for arbitrary switching
• Stability for constrained switching

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MULTIPLE LYAPUNOV FUNCTIONS
Very useful for analysis of state-dependent
switching
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MULTIPLE LYAPUNOV FUNCTIONS
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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
average dwell time
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SWITCHED LINEAR SYSTEMS

• GUES over all with large enough
• Finite induced norms for
• The case when some subsystems are unstable

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STATE-DEPENDENT SWITCHING
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STATE-DEPENDENT SWITCHING
Switched system unstable for some no
common
But switched system is stable for (many) other
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MULTIPLE WEAK LYAPUNOV FUNCTIONS
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STABILIZATION by SWITCHING
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STABILIZATION by SWITCHING
both unstable
Assume stable
for some
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UNSTABLE CONVEX COMBINATIONS
Can also use multiple Lyapunov functions
LMIs
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REFERENCES
Branicky, DeCarlo, Hespanha