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

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

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

4. I TERATIVE A LGORITHMS : PRIOR WORK Algebraic inequalities: Agmon, Motzkin, Schoenberg (1954) Polyak (1964) Yakubovich (1966) Matrix inequalities: Polyak & Tempo (2001) Calafiore & Polyak (2001)

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

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

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

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

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

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