### Abstract

This paper concentrates on a risk-adjusted version of the well known quadratic stabilization problem for uncertain linear systems. For a wide class of probability density functions and state equation structures for the uncertain parameters, the main result of this paper is as follows: With nominally determined quadratic Lyapunov function V(x) = x^{T}Px, the set of controller gains K_{ε} guaranteeing quadratic Lyapunov instability risk level 0≤ε≤1 or less is convex. Hence, this so-called Probabilistic Design Problem reduces to a convex program. One of the ramifications of this result involves the issue of high-gain control. It is demonstrated that for small values of the risk probability ε, the controller gains which are required can be much smaller than their counterpart obtained via classical robustness theory.

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

Pages (from-to) | 430-434 |

Number of pages | 5 |

Journal | Proceedings of the American Control Conference |

Volume | 1 |

State | Published - 1999 |

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### All Science Journal Classification (ASJC) codes

- Control and Systems Engineering

### Cite this

*Proceedings of the American Control Conference*,

*1*, 430-434.

}

*Proceedings of the American Control Conference*, vol. 1, pp. 430-434.

**On convexity of the Probabilistic Design Problem for quadratic stabilizability.** / Barmish, B. R.; Lagoa, Constantino Manuel.

Research output: Contribution to journal › Article

TY - JOUR

T1 - On convexity of the Probabilistic Design Problem for quadratic stabilizability

AU - Barmish, B. R.

AU - Lagoa, Constantino Manuel

PY - 1999

Y1 - 1999

N2 - This paper concentrates on a risk-adjusted version of the well known quadratic stabilization problem for uncertain linear systems. For a wide class of probability density functions and state equation structures for the uncertain parameters, the main result of this paper is as follows: With nominally determined quadratic Lyapunov function V(x) = xTPx, the set of controller gains Kε guaranteeing quadratic Lyapunov instability risk level 0≤ε≤1 or less is convex. Hence, this so-called Probabilistic Design Problem reduces to a convex program. One of the ramifications of this result involves the issue of high-gain control. It is demonstrated that for small values of the risk probability ε, the controller gains which are required can be much smaller than their counterpart obtained via classical robustness theory.

AB - This paper concentrates on a risk-adjusted version of the well known quadratic stabilization problem for uncertain linear systems. For a wide class of probability density functions and state equation structures for the uncertain parameters, the main result of this paper is as follows: With nominally determined quadratic Lyapunov function V(x) = xTPx, the set of controller gains Kε guaranteeing quadratic Lyapunov instability risk level 0≤ε≤1 or less is convex. Hence, this so-called Probabilistic Design Problem reduces to a convex program. One of the ramifications of this result involves the issue of high-gain control. It is demonstrated that for small values of the risk probability ε, the controller gains which are required can be much smaller than their counterpart obtained via classical robustness theory.

UR - http://www.scopus.com/inward/record.url?scp=0033285781&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0033285781&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:0033285781

VL - 1

SP - 430

EP - 434

JO - Proceedings of the American Control Conference

JF - Proceedings of the American Control Conference

SN - 0743-1619

ER -