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

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

MODELS of HYBRID SYSTEMS

…

[Van der Schaft–Schumacher ’00] [Proceedings of HSCC]

continuous

discrete[Nešić–L ‘05]

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

STABILITY ISSUE

unstable

Asymptotic stability of each subsystem is

not sufficient for stability

TWO BASIC PROBLEMS

• Stability for arbitrary switching

• Stability for constrained switching

TWO BASIC PROBLEMS

• Stability for arbitrary switching

• Stability for constrained switching

GLOBAL UNIFORM ASYMPTOTIC STABILITY

GUAS is: Lyapunov stability

plus asymptotic convergence

GUES:

quadratic is GUES

where is positive definite

is GUAS if (and only if) s.t.

COMMON LYAPUNOV FUNCTION

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

COMMUTING STABLE MATRICES => GUES

For subsystems – similarly

(commuting Hurwitz matrices)

...

quadratic common Lyapunov function[Narendra–Balakrishnan ’94]

COMMUTING STABLE MATRICES => GUES

Alternative proof:

is a common Lyapunov function

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

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

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

SWITCHED NONLINEAR SYSTEMS

Lie bracket of nonlinear vector fields:

Reduces to earlier notion for linear vector fields(modulo the sign)

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

SPECIAL CASE

globally asymptotically stable

Want to show: is GUAS

Will show: differential inclusion

is GAS

OPTIMAL CONTROL APPROACH

Associated control system:

where

(original switched system )

Worst-case control law [Pyatnitskiy, Rapoport, Boscain, Margaliot]:

fix and small enough

MAXIMUM PRINCIPLE

is linear in

at most 1 switch

(unless )

GAS

Optimal control:(along optimal trajectory)

GENERAL CASE

GAS

Want: polynomial of degree

(proof – by induction on )

bang-bang with switches

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

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?

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

FEEDBACK SYSTEMS: SMALL-GAIN THEOREM

controllable

Small-gain theorem:

quadratic common Lyapunov function

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)

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. .

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]