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INTRODUCTION to SWITCHED SYSTEMS ;
STABILITY under ARBITRARY SWITCHING
Daniel Liberzon
Coordinated Science Laboratory andDept. of Electrical & Computer Eng.,Univ. of Illinois at Urbana-Champaign
IAAC Workshop, Herzliya, Israel, June 1, 2009
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electric circuit
SWITCHED and HYBRID SYSTEMS
Hybrid systems combine continuous and discrete dynamics
Which practical systems are hybrid?
Which practical systems are not hybrid?
More tractable models of continuous phenomena
thermostat
stick shift walking
cell division
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MODELS of HYBRID SYSTEMS
…
[Van der Schaft–Schumacher ’00] [Proceedings of HSCC]
continuous
discrete[Nešić–L ‘05]
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SWITCHED vs. HYBRID SYSTEMSSwitched system:
• is a family of systems
• is a switching signal
Switching can be:
• State-dependent or time-dependent• Autonomous or controlled
Details of discrete behavior are “abstracted away”
: stability and beyondProperties of the continuous state
Discrete dynamics classes of switching signals
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STABILITY ISSUE
unstable
Asymptotic stability of each subsystem is
not sufficient for stability
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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
GUES:
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quadratic is GUES
where is positive definite
is GUAS if (and only if) s.t.
COMMON LYAPUNOV FUNCTION
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OUTLINE
Stability criteria to be discussed:
• Commutation relations (Lie algebras)
• Feedback systems (absolute stability)
• Observability and LaSalle-like theorems
Common Lyapunov functions will play a central role
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COMMUTING STABLE MATRICES => GUES
For subsystems – similarly
(commuting Hurwitz matrices)
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...
quadratic common Lyapunov function[Narendra–Balakrishnan ’94]
COMMUTING STABLE MATRICES => GUES
Alternative proof:
is a common Lyapunov function
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Nilpotent means sufficiently high-order Lie brackets are 0
NILPOTENT LIE ALGEBRA => GUES
Lie algebra:
Lie bracket:
Hence still GUES [Gurvits ’95]
For example:
(2nd-order nilpotent)
In 2nd-order nilpotent case
Recall: in commuting case
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SOLVABLE LIE ALGEBRA => GUES
Example:
quadratic common Lyap fcn diagonal
exponentially fast
[Kutepov ’82, L–Hespanha–Morse ’99]
Larger class containing all nilpotent Lie algebras
Suff. high-order brackets with certain structure are 0
exp fast
Lie’s Theorem: is solvable triangular form
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SUMMARY: LINEAR CASE
Lie algebra w.r.t.
Assuming GES of all modes, GUES is guaranteed for:
• commuting subsystems:
• nilpotent Lie algebras (suff. high-order Lie brackets are 0)e.g.
• solvable Lie algebras (triangular up to coord. transf.)
• solvable + compact (purely imaginary eigenvalues)
No further extension based on Lie algebra only [Agrachev–L ’01]
Quadratic common Lyapunov function exists in all these cases
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SWITCHED NONLINEAR SYSTEMS
Lie bracket of nonlinear vector fields:
Reduces to earlier notion for linear vector fields(modulo the sign)
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SWITCHED NONLINEAR SYSTEMS
• Linearization (Lyapunov’s indirect method)
Can prove by trajectory analysis [Mancilla-Aguilar ’00]
or common Lyapunov function [Shim et al. ’98, Vu–L ’05]
• Global results beyond commuting case – ?
[Unsolved Problems in Math. Systems & Control Theory ’04]
• Commuting systems
GUAS
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SPECIAL CASE
globally asymptotically stable
Want to show: is GUAS
Will show: differential inclusion
is GAS
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OPTIMAL CONTROL APPROACH
Associated control system:
where
(original switched system )
Worst-case control law [Pyatnitskiy, Rapoport, Boscain, Margaliot]:
fix and small enough
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MAXIMUM PRINCIPLE
is linear in
at most 1 switch
(unless )
GAS
Optimal control:(along optimal trajectory)
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GENERAL CASE
GAS
Want: polynomial of degree
(proof – by induction on )
bang-bang with switches
[Margaliot–L ’06, Sharon–Margaliot ’07]
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REMARKS on LIE-ALGEBRAIC CRITERIA
• Checkable conditions
• In terms of the original data
• Independent of representation
• Not robust to small perturbations
In any neighborhood of any pair of matricesthere exists a pair of matrices generating the entire Lie algebra [Agrachev–L ’01]
How to measure closeness to a “nice” Lie algebra?
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FEEDBACK SYSTEMS: ABSOLUTE STABILITY
Circle criterion: quadratic common Lyapunov function
is strictly positive real (SPR):
For this reduces to SPR (passivity)
Popov criterion not suitable: depends on
controllable
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FEEDBACK SYSTEMS: SMALL-GAIN THEOREM
controllable
Small-gain theorem:
quadratic common Lyapunov function
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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)
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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. .
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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]