A LIE-ALGEBRAIC CONDITION

for STABILITY of

SWITCHED NONLINEAR SYSTEMS

CDC ’04

Michael MargaliotTel Aviv University, Israel

Daniel LiberzonUniv. of Illinois at Urbana-Champaign, USA

SWITCHED vs. HYBRID SYSTEMS

: stability

Switching can be:

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

Hybrid systems give rise to classes of switching signals

Properties of the continuous state

Switched system:

• is a family of systems

• is a switching signal

Further abstraction/relaxation: diff. inclusion, measurable switching

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:

SWITCHED LINEAR SYSTEMS

Extension based only on L.A. is not possible [Agrachev & L ’01]

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)

SWITCHED NONLINEAR SYSTEMS

• Global results beyond commuting case – ???

• Commuting systems

• Linearization (Lyapunov’s indirect method)

=> GUAS [Mancilla-Aguilar,Shim et al., Vu & L]

[Unsolved Problems in Math. Systems and Control Theory]

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)

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:

GENERAL CASE

GAS

Want: polynomial of degree

(proof – by induction on )

bang-bang with switches

THEOREM

Suppose:

• GAS, backward complete, analytic

• s.t.

and

Then differential inclusion is GAS

(and switched system is GUAS)