the role of lie brackets in stability of linear and nonlinear switched systems

THE ROLE OF LIE BRACKETS IN

STABILITY OF LINEAR AND NONLINEAR

SWITCHED SYSTEMS

Daniel Liberzon

Coordinated Science Laboratory andDept. of Electrical & Computer Eng.,Univ. of Illinois at Urbana-Champaign

Semi-plenary lecture, MTNS, Blacksburg, VA, 7/31/08 1 of 24

THE ROLE OF LIE BRACKETS IN

STABILITY OF LINEAR AND NONLINEAR

SWITCHED SYSTEMS

Daniel Liberzon

Coordinated Science Laboratory andDept. of Electrical & Computer Eng.,Univ. of Illinois at Urbana-Champaign

SWITCHED vs. HYBRID SYSTEMS

Switching can be:

• State-dependent or time-dependent• Autonomous or controlled

Further abstraction/relaxation: differential inclusion, measurable switching

Details of discrete behavior are “abstracted away”

: stabilityProperties of the continuous state

Switched system:

• is a family of systems• is a switching signal

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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

GLOBAL UNIFORM ASYMPTOTIC STABILITY

GUAS is Lyapunov stability

plus asymptotic convergence

GUES:

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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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LIE ALGEBRAS and STABILITY

Nilpotent means suff. high-order Lie brackets are 0e.g.

is nilpotent if s.t.

is solvable if s.t.

Lie algebra:

Lie bracket:

Nilpotent GUES [Gurvits ’95]

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SOLVABLE LIE ALGEBRA => GUES

Example:

quadratic common Lyap fcn diagonal

exponentially fast

0

exp fast

[L–Hespanha–Morse ’99]

Lie’s Theorem: is solvable triangular form

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MORE GENERAL LIE ALGEBRAS

Levi decomposition:

radical (max solvable ideal)

There exists one set of stable generators for which

gives rise to a GUES switched system, and another

which gives an unstable one

[Agrachev–L ’01]

• is compact (purely imaginary eigenvalues) GUES,

quadratic common Lyap fcn

• is not compact not enough info in Lie algebra:

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SUMMARY: LINEAR CASE

Extension based only on the Lie algebra is not possible

Lie algebra w.r.t.

Quadratic common Lyapunov function exists in all these cases

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)

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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 and Control Theory]

• 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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SINGULARITY

Need: nonzero on ideal generated by

(strong extremality)

At most 2 switches GAS

Know: nonzero on

strongly extremal(time-optimal control for auxiliary system in )

constant control(e.g., )

Sussmann ’79:

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GENERAL CASE

GAS

Want: polynomial of degree

(proof – by induction on )

bang-bang with switches

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THEOREM

Suppose:

• GAS, backward complete, analytic

• s.t.

and

Then differential inclusion is GAS,

and switched system is GUAS [Margaliot–L ’06]

Further work in [Sharon–Margaliot ’07]19 of 24

REMARKS on LIE-ALGEBRAIC CRITERIA

• Checkable conditions

• In terms of the original data

• Independent of representation

• Not robust to small perturbations

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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?

EXAMPLE

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(discrete time, or cont. time periodic switching: )

Suppose .

Fact 1 Stable for dwell time :

Fact 2 If then always stable:

This generalizes Fact 1 (didn’t need ) and Fact 2 ( )

Switching between and

Schur

Fact 3 Stable if where is small enough s.t.

When we have where

EXAMPLE

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(discrete time, or cont. time periodic switching: )

Suppose .

Fact 1 Stable for dwell time :

Fact 2 If then always stable:

Switching between and

Schur

When we have where

MORE GENERAL FORMULATION

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Assume switching period :


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Find smallest s.t.

Assume switching period


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Intuitively, captures:

• how far and are from commuting ( )

• how big is compared to ( )

, define byFind smallest s.t.


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Stability condition: where

already discussed: (“elementary shuffling”)


, define byFind smallest s.t.

SOME OPEN ISSUES

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• Relation between and Lie brackets of and

general formula seems to be lacking

SOME OPEN ISSUES

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Suppose but

Elementary shuffling: ,

not nec. close to but commutes with and

• Smallness of higher-order Lie brackets

Example:

SOME OPEN ISSUES


Suppose but

Elementary shuffling: ,

not nec. close to but commutes with and

This shows stability for

Example:

In general, for can define by

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(Gurvits: true for any )

SOME OPEN ISSUES


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• More than 2 subsystems

can still define but relation with Lie brackets less clear

• Optimal way to shuffle

especially when is not a multiple of

Thank you for your attention!

the role of lie brackets in stability of linear and nonlinear switched systems

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