1. 1 of 12 COMMUTATORS, ROBUSTNESS, and STABILITY of SWITCHED LINEAR SYSTEMS SIAM Conference on Control & its Applications, Paris, July 2015 Daniel Liberzon Univ. of Illinois at Urbana-Champaign, U.S.A. Joint work with Yuliy Baryshnikov TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: A A A A A

2. 2 of 12 SWITCHED SYSTEMS Switched 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” : stabilityProperties of the continuous state Discrete dynamics classes of switching signals

3. STABILITY ISSUE unstable Asymptotic stability of individual systems is not sufficient for stability under arbitrary switching 3 of 12

4. COMMUTING STABLE MATRICES => GUES For matrices – similarly (commuting Hurwitz matrices) Switched linear system: 4 of 12

5. ...... quadratic common Lyapunov function [ Narendra–Balakrishnan ’94 ] COMMUTING STABLE MATRICES => GUES Alternative proof: is a common Lyapunov function 5 of 12

6. SUMMARY of KNOWN RESULTS Lie algebra w.r.t. No further extension based on Lie algebra only Quadratic common Lyapunov function exists in all these cases Stability is preserved under arbitrary switching for: commuting matrices: [ Narendra–Balakrishnan, nilpotent Lie algebras (suff. high-order Lie brackets are 0) e.g. Gurvits, solvable Lie algebras (triangular up to coord. transf.) Kutepov,L–Hespanha–Morse, solvable + compact (purely imaginary eigenvalues) Agrachev–L ] 6 of 12

7. 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 How to capture closeness to a “nice” Lie algebra? 7 of 12

8. ALMOST COMMUTING vs. NEAR COMMUTING Halmos’s problem (1976): is a pair of (self-adjoint) matrices which almost commute always close to a pair of commuting matrices? Łojasiewicz (1978): general theorem relating the value of a function to the distance to its zero set Ji–Kollar–Shiffman (1992): more specific Łojasiewicz bounds for polynomials or maxima of finitely many polynomials Hastings–Loring (2010): Łojasiewicz-type bounds for self-adjoint and unitary matrices with bounded norm Pearcy–Shields (1978): positive result, one matrix must be self-adjoint Exel–Loring (1978): counterexample, uses unitary matrices of growing dimension Lin (1997): positive result for self-adjoint matrices with bounded norm Kachkovskiy–Safarov (2014): quantitative bound for Lin’s theorem 8 of 12

9. 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 9 of 12

10. 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 10 of 12

11. 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 11 of 12

12. CONCLUSIONS Asymptotic: admissible commutator size goes to 0 at rate of stability margin raised to power Quantitative: explicit estimates are possible in principle, but checking LMI feasibility is probably more numerically efficient Open questions: Relation to approach in [ Agrachev–Baryshnikov–L, SCL, 2012 ] Switched nonlinear systems (e.g. polynomial) Compact but infinite matrix families (trade-off between and ) Qualitative: switched linear system remains stable if commutation relations are only approximately satisfied Results: 12 of 12