GRADIENT ALGORITHMS for

COMMON LYAPUNOV FUNCTIONS

Daniel LiberzonUniv. of Illinois at Urbana-Champaign, U.S.A.

Roberto Tempo IEIIT-CNR, Politecnico di Torino, Italy

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 :

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)

ITERATIVE ALGORITHMS: PRIOR WORK

Algebraic inequalities:

Agmon, Motzkin, Schoenberg (1954)

Polyak (1964)

Yakubovich (1966)

Matrix inequalities:

Polyak & Tempo (2001)

Calafiore & Polyak (2001)

GRADIENT ALGORITHMS: 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.

GRADIENT ALGORITHMS: PRELIMINARIES

– convex differentiable real-valued functional on the

space of symmetric matrices,

Gradient:

( is unit eigenvector of with eigenvalue )

1.

2.

– convex in

given

GRADIENT ALGORITHMS: 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)

GRADIENT ALGORITHMS: 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):

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