Decentralized computation for robust stability of large-scale systems with parameters on the hypercube

In this paper, we propose a parallel algorithm to solve the problem of robust stability of systems with large state-space and with large number of uncertain parameters. The dependence of the system on the parameters is polynomial and the parameters are assumed to lie in a hypercube. Although the parameters are assumed to be static, the method can also be applied to systems with time-varying parameters. The algorithm relies on a variant of Polya's theorem which is applicable to polynomials with variables inside a multi-simplex. The algorithm is divided into formulation and solution subroutines. In the formulation phase, we construct a large-scale semidefinite programming problem with structured elements. In the solution phase, we use a structured primal-dual approach to solve the structured semidefinite programming problem. In both subroutines, computation, memory and communication are efficiently distributed over hundreds and potentially thousands of processors. Numerical tests confirm the accuracy and scalability of the proposed algorithm.