TOWARDS ROBUST LIE-ALGEBRAIC STABILITY

CONDITIONS for SWITCHED LINEAR SYSTEMS

49th CDC, Atlanta, GA, Dec 2010

Daniel LiberzonUniv. of Illinois, Urbana-Champaign, USA

Yuliy BaryshnikovBell Labs, Murray Hill, NJ, USA

Andrei A. AgrachevS.I.S.S.A., Trieste, Italy

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SWITCHED 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”

: stabilityProperties 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 under arbitrary switching

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KNOWN RESULTS: LINEAR SYSTEMS

Lie algebra w.r.t.

No further extension based on Lie algebra only

Quadratic common Lyapunov function exists in all these cases

Stability is preserved under arbitrary switching for:

• commuting subsystems:

[Narendra–Balakrishnan,

• nilpotent Lie algebras (suff. high-order Lie brackets are 0)e.g.

Gurvits,

• solvable Lie algebras (triangular up to coord. transf.)

Kutepov, L–Hespanha–Morse,

• solvable + compact (purely imaginary eigenvalues)

Agrachev–L]4 of 11

KNOWN RESULTS: 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

Recently obtained using optimal control approach

[Margaliot–L ’06, Sharon–Margaliot ’07]

• Commuting systems

GUAS

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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 capture closeness to a “nice” Lie algebra?

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DISCRETE TIME: BOUNDS on COMMUTATORS

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– Schur stable

Let

GUES

Idea of proof: take , , consider

GUES:

DISCRETE TIME: BOUNDS on COMMUTATORS

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– Schur stable

GUES:

Let

GUES

Idea of proof: take , , consider

DISCRETE TIME: BOUNDS on COMMUTATORS

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inductionbasis

contraction small

– Schur stable

GUES:

Let

GUES

Idea of proof: take , , consider

DISCRETE TIME: BOUNDS on COMMUTATORS

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Accounting for linear dependencies among – easy

– Schur stable

GUES:

Let

GUES

Extending to higher-order commutators – hard

DISCRETE TIME: BOUNDS on COMMUTATORS

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For they commute GUES, common Lyap fcn

– Schur stable

GUES:

Let

GUES

Example:

Our approach still GUES for

Perturbation analysis based on GUES for

CONTINUOUS TIME: LIE ALGEBRA STRUCTURE

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compact set of Hurwitz matrices

GUES: Lie algebra:

Levi decomposition: ( solvable, semisimple)

Switched transition matrix splits as where

and

Let and

GUESrobust condition butnot constructive

CONTINUOUS TIME: LIE ALGEBRA STRUCTURE

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more conservative buteasier to verifyGUES

There are also intermediate conditions

GUES:

Levi decomposition: ( solvable, semisimple)

Switched transition matrix splits as where

and

Let and

compact set of Hurwitz matrices

Lie algebra:

CONTINUOUS TIME: LIE ALGEBRA STRUCTURE

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Levi decomposition:

Switched transition matrix splits as

Previous slide: small GUES

But we also know: compact Lie algebra (not nec. small) GUES

Cartan decomposition: ( compact subalgebra)

Transition matrix further splits: where

and

Let

GUES

CONTINUOUS TIME: LIE ALGEBRA STRUCTURE

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Example:

GUES

Levi decomposition:

Cartan decomposition:

SUMMARY

Stability is preserved under arbitrary switching for:

• commuting subsystems:

• solvable Lie algebras (triangular up to coord. transf.)

• solvable + compact (purely imaginary eigenvalues)

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approximately

approximately

approximately