Solving large-scale robust stability problems by exploiting the parallel structure of Polya's theorem

Reza Kamyar, Matthew Peet, Yulia Peet

Research output: Contribution to journalArticlepeer-review

11 Scopus citations


In this paper, we propose a distributed computing approach to solving large-scale robust stability problems on the simplex. Our approach is to formulate the robust stability problem as an optimization problem with polynomial variables and polynomial inequality constraints. We use Polya's theorem to convert the polynomial optimization problem to a set of highly structured linear matrix inequalities (LMIs). We then use a slight modification of a common interior-point primal-dual algorithm to solve the structured LMI constraints. This yields a set of extremely large yet structured computations. We then map the structure of the computations to a decentralized computing environment consisting of independent processing nodes with a structured adjacency matrix. The result is an algorithm which can solve the robust stability problem with the same per-core complexity as the deterministic stability problem with a conservatism which is only a function of the number of processors available. Numerical tests on cluster computers and supercomputers demonstrate the ability of the algorithm to efficiently utilize hundreds and potentially thousands of processors and analyze systems with 100+ dimensional state-space. The proposed algorithms can be extended to perform stability analysis of nonlinear systems and robust controller synthesis.

Original languageEnglish (US)
Article number6482174
Pages (from-to)1931-1947
Number of pages17
JournalIEEE Transactions on Automatic Control
Issue number8
StatePublished - 2013


  • Decentralized computing
  • Large-scale systems
  • Polynomial optimization
  • Robust stability

ASJC Scopus subject areas

  • Control and Systems Engineering
  • Computer Science Applications
  • Electrical and Electronic Engineering


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