### Abstract

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 language | English (US) |
---|---|

Article number | 6482174 |

Pages (from-to) | 1931-1947 |

Number of pages | 17 |

Journal | IEEE Transactions on Automatic Control |

Volume | 58 |

Issue number | 8 |

DOIs | |

State | Published - 2013 |

### Fingerprint

### Keywords

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

### ASJC Scopus subject areas

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

### Cite this

*IEEE Transactions on Automatic Control*,

*58*(8), 1931-1947. [6482174]. https://doi.org/10.1109/TAC.2013.2253253

**Solving large-scale robust stability problems by exploiting the parallel structure of Polya's theorem.** / Kamyar, Reza; Peet, Matthew; Peet, Yulia.

Research output: Contribution to journal › Article

*IEEE Transactions on Automatic Control*, vol. 58, no. 8, 6482174, pp. 1931-1947. https://doi.org/10.1109/TAC.2013.2253253

}

TY - JOUR

T1 - Solving large-scale robust stability problems by exploiting the parallel structure of Polya's theorem

AU - Kamyar, Reza

AU - Peet, Matthew

AU - Peet, Yulia

PY - 2013

Y1 - 2013

N2 - 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.

AB - 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.

KW - Decentralized computing

KW - Large-scale systems

KW - Polynomial optimization

KW - Robust stability

UR - http://www.scopus.com/inward/record.url?scp=84880914601&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84880914601&partnerID=8YFLogxK

U2 - 10.1109/TAC.2013.2253253

DO - 10.1109/TAC.2013.2253253

M3 - Article

AN - SCOPUS:84880914601

VL - 58

SP - 1931

EP - 1947

JO - IEEE Transactions on Automatic Control

JF - IEEE Transactions on Automatic Control

SN - 0018-9286

IS - 8

M1 - 6482174

ER -