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COMMUTATION RELATIONS and STABILITY of SWITCHED SYSTEMS

Coordinated Science Laboratory and Dept. of Electrical & Computer Eng., Univ. of Illinois at Urbana-Champaign. COMMUTATION RELATIONS and STABILITY of SWITCHED SYSTEMS. Daniel Liberzon. (commuting Hurwitz matrices). For subsystems – similarly.

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COMMUTATION RELATIONS and STABILITY of SWITCHED SYSTEMS

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  1. Coordinated Science Laboratory and Dept. of Electrical & Computer Eng., Univ. of Illinois at Urbana-Champaign COMMUTATION RELATIONS andSTABILITY of SWITCHED SYSTEMS Daniel Liberzon

  2. (commuting Hurwitz matrices) For subsystems – similarly COMMUTING STABLE MATRICES => GUES

  3. quadratic common Lyapunov function is a common Lyapunov function COMMUTING STABLE MATRICES => GUES Alternative proof: [Narendra–Balakrishnan ’94] . . .

  4. NILPOTENT LIE ALGEBRA => GUES Lie algebra: Lie bracket: For example: (2nd-order nilpotent) Recall: in commuting case In 2nd-order nilpotent case Nilpotent meanssufficiently high-order Lie brackets are 0 Hence still GUES [Gurvits ’95]

  5. SOLVABLE LIE ALGEBRA => GUES Lie’s Theorem: is solvable triangular form Example: quadratic common Lyap fcn exponentially fast exp fast diagonal Larger class containing all nilpotent Lie algebras Suff. high-order brackets with certain structure are 0 [Kutepov ’82, L–Hespanha–Morse ’99]

  6. MORE GENERAL LIE ALGEBRAS Levi decomposition: radical (max solvable ideal) • is compact (purely imaginary eigenvalues) GUES, quadratic common Lyap fcn • is not compact not enough info in Lie algebra: 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]

  7. SUMMARY: LINEAR CASE Assuming GES of all modes, GUES is guaranteed for: • 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) Lie algebra w.r.t. Quadratic common Lyapunov function exists in all these cases Further extension based only on Lie algebra is not possible

  8. Lie bracket of nonlinear vector fields: SWITCHED NONLINEAR SYSTEMS Reduces to earlier notion for linear vector fields (modulo the sign)

  9. GUAS • Linearization (Lyapunov’s indirect method) • Global results beyond commuting case – ? [Unsolved Problems in Math. Systems & Control Theory ’04] SWITCHED NONLINEAR SYSTEMS • Commuting systems Can prove by trajectory analysis[Mancilla-Aguilar ’00] or common Lyapunov function[Shim et al. ’98, Vu–L ’05]

  10. globally asymptotically stable Want to show: is GUAS Will show: differential inclusion is GAS SPECIAL CASE

  11. (original switched system ) Worst-case control law[Pyatnitskiy, Rapoport, Boscain, Margaliot]: fix and small enough OPTIMAL CONTROL APPROACH Associated control system: where

  12. MAXIMUM PRINCIPLE Optimal control: (along optimal trajectory) is linear in GAS (unless ) at most 1 switch

  13. GENERAL CASE Want: polynomial of degree (proof – by induction on ) bang-bang with switches GAS

  14. 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]

  15. 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?

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