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TOWARDS ROBUST LIE-ALGEBRAIC STABILITY CONDITIONS for SWITCHED LINEAR SYSTEMS

Coordinated Science Laboratory and Dept. of Electrical & Computer Eng., Univ. of Illinois at Urbana-Champaign. TOWARDS ROBUST LIE-ALGEBRAIC STABILITY CONDITIONS for SWITCHED LINEAR SYSTEMS. Daniel Liberzon. European Control Conference, Budapest, Aug 2009. SWITCHED SYSTEMS.

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TOWARDS ROBUST LIE-ALGEBRAIC STABILITY CONDITIONS for SWITCHED LINEAR SYSTEMS

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  1. Coordinated Science Laboratory and Dept. of Electrical & Computer Eng., Univ. of Illinois at Urbana-Champaign TOWARDS ROBUST LIE-ALGEBRAIC STABILITY CONDITIONS for SWITCHED LINEAR SYSTEMS Daniel Liberzon European Control Conference, Budapest, Aug 2009

  2. SWITCHED SYSTEMS Switched system: • is a family of systems • is a switching signal Switching can be: • State-dependent or time-dependent • Autonomous or controlled Discrete dynamics classes of switching signals Properties of the continuous state Details of discrete behavior are “abstracted away” : stability

  3. Asymptotic stability of each subsystem is not sufficient for stability under arbitrary switching STABILITY ISSUE unstable

  4. KNOWN RESULTS: LINEAR SYSTEMS • commuting subsystems: • nilpotent Lie algebras (suff. high-order Lie brackets are 0) • solvable Lie algebras (triangular up to coord. transf.) e.g. • solvable + compact (purely imaginary eigenvalues) [Narendra–Balakrishnan, Agrachev–L] Gurvits, Kutepov, L–Hespanha–Morse, Lie algebra w.r.t. Assuming GES of all modes, GUES is guaranteed for: Quadratic common Lyapunov function exists in all these cases No further extension based on Lie algebra only

  5. KNOWN RESULTS: NONLINEAR SYSTEMS GUAS • Linearization (Lyapunov’s indirect method) • Global results beyond commuting case Recently obtained using optimal control approach [Margaliot–L ’06, Sharon–Margaliot ’07] • Commuting systems Can prove by trajectory analysis[Mancilla-Aguilar ’00] or common Lyapunov function[Shim et al. ’98, Vu–L ’05]

  6. Checkable conditions • In terms of the original data • Independent of representation • Not robust to small perturbations In anyneighborhood of any pair of matrices there exists a pair of matrices generating the entire Lie algebra [Agrachev–L ’01] REMARKS on LIE-ALGEBRAIC CRITERIA How to measure closeness to a “nice” Lie algebra?

  7. EXAMPLE Switching between and (discrete time, or cont. time periodic switching: ) Schur Stable for dwell time : Fact 1 Suppose . If then always stable: Fact 2 When we have where Stable if where is small enough s.t. Fact 3 This generalizes Fact 1 (didn’t need ) and Fact 2 ( )

  8. EXAMPLE Switching between and (discrete time, or cont. time periodic switching: ) Schur Suppose . Stable for dwell time : Fact 1 If then always stable: Fact 2 When we have where

  9. MORE GENERAL FORMULATION Assume switching period :

  10. MORE GENERAL FORMULATION Assume switching period Find smallest s.t.

  11. MORE GENERAL FORMULATION Assume switching period Find smallest s.t. , define by Intuitively, captures: • how far and are from commuting ( ) • how big is compared to ( )

  12. MORE GENERAL FORMULATION Assume switching period Find smallest s.t. , define by already discussed: (“elementary shuffling”) Stability condition: where

  13. SOME OPEN ISSUES • Relation between and Lie brackets of and general formula seems to be lacking

  14. SOME OPEN ISSUES • Relation between and Lie brackets of and Suppose but Elementary shuffling: , not nec. close to but commutes with and Example: • Smallness of higher-order Lie brackets

  15. SOME OPEN ISSUES • Relation between and Lie brackets of and Suppose but Elementary shuffling: , (Gurvits: true for any ) This shows stability for In general, for can define by • Smallness of higher-order Lie brackets not nec. close to but commutes with and Example:

  16. SOME OPEN ISSUES • Relation between and Lie brackets of and can still define but relation with Lie brackets less clear especially when is not a multiple of • Smallness of higher-order Lie brackets • More than 2 subsystems • Optimal way to shuffle • Current work investigates alternative approaches • that go beyond periodic switching

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