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

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

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:

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) Further extension based only on Lie algebra is not possible 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)

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 [ Margaliot–L ’06 ] Further work in [ 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 matrices there exists a pair of matrices generating the entire Lie algebra [ Agrachev–L ’01 ] How to measure closeness to a “nice” Lie algebra?

ŁOJASIEWICZ INEQUALITY real analytic function – zero set of Then for every compact, s.t. Meaning: if then When is a maximum of finitely many polynomials, explicit bounds on the exponent can be obtained

ŁOJASIEWICZ APPLIED TO OUR SET – UP – finite set of Hurwitz matrices Suppose where is Frobenius norm Let This is max of polynomials of degree 4 in coefficients of (or can use sum to get a single polynomial) Let be distance from to nearest -tuple of pairwise commuting matrices ( Frobenius norm of difference between stacked matrices ) Łojasiewicz inequality gives i.e., with Higher-order commutators (near nilpotent / solvable) – similar

PERTURBATION ANALYSIS commute (or generate nilpotent or solvable Lie algebra) common Lyapunov function which is still negative definite if For original matrices: For commuting and solvable cases, specific known constructions of can be used to estimate the right-hand side [ Baryshnikov–L, CDC ’13 ] depend on (# of matrices) and on compact set where they live can be explicitly estimated estimating is a bit more difficult [ Ji–Kollar–Shiffman ] stability of is ensured if

DISCRETE TIME: BOUNDS on COMMUTATORS induction basis contraction small – Schur stable GUES: Let GUES Idea of proof: take,, consider