GRADIENT ALGORITHMS for COMMON LYAPUNOV FUNCTIONS

1. GRADIENT ALGORITHMS for COMMON LYAPUNOV FUNCTIONS Daniel Liberzon Univ. of Illinois at Urbana-Champaign, U.S.A. Roberto Tempo IEIIT-CNR, Politecnico di Torino, Italy CDC ’03

2. PROBLEM Given Hurwitz matrices and matrix , find matrix : Analytical results: • hard to come by (beyond ) • require special structure • can handle large finite families • provide limited insight LMI methods: • gradient descent iterations • handle inequalities sequentially Our approach: Goal: algorithmic approach with theoretical insight Motivation: stability of uncertain and switched systems

3. MOTIVATING EXAMPLE In the special case when matrices commute: (Narendra & Balakrishnan, 1994) quadratic common Lyapunov function . . . Nonlinear extensions: Shim et al. (1998), Vu & L (2003)

4. ITERATIVE ALGORITHMS: PRIOR WORK Matrix inequalities: Polyak & Tempo (2001) Calafiore & Polyak (2001) Algebraic inequalities: Agmon, Motzkin, Schoenberg (1954) Polyak (1964) Yakubovich (1966)

5. GRADIENT ALGORITHMS: PRELIMINARIES – convex differentiable real-valued functional on the space of symmetric matrices, Examples: 1. (need this to be a simple eigenvalue) 2. ( is Frobenius norm, is projection onto matrices)

6. GRADIENT ALGORITHMS: PRELIMINARIES – convex differentiable real-valued functional on the space of symmetric matrices, – convex in given 1. Gradient: ( is unit eigenvector of with eigenvalue ) 2.

7. GRADIENT ALGORITHMS: DETERMINISTIC CASE – finite family of Hurwitz matrices – arbitrary symmetric matrix ( – suitably chosen stepsize) Theorem: Solution , if it exists, is found in a finite number of steps distance from to solution set decreases at each correction step Idea of proof: – visits each index times Gradient iteration:

8. GRADIENT ALGORITHMS: PROBABILISTIC CASE – compact (possibly infinite) family – picked using probability distribution on s.t. every relatively open subset has positive measure Gradient iteration (randomized version): Theorem: Solution , if it exists, is found in a finite number of steps with probability 1 Idea of proof: still get closer with each correction step correction step is executed with prob. 1

9. SIMULATION EXAMPLE vertices Deterministic gradient: ( ineqs): 10,000 iterations (a few seconds) ( ineqs): 10,000,000 iterations (a few hours) Randomized gradient gives faster convergence Both are quite easy to program Compare: quadstab (MATLAB) stacks when Interval family of triangular Hurwitz matrices:

10. CONCLUSIONS Open issues: • performance comparison of different methods • addressing existence of solutions • optimal schedule of iterations • additional requirements on solution Contribution: • Gradient iteration algorithms for solving • simultaneous Lyapunov matrix inequalities • Deterministic convergence for finite families, • probabilistic convergence for infinite families