### Abstract

Stabilization by static output feedback {SOF) is a long-standing open problem in control: given an n by n matrix A and rectangular matrices B and C, find a p by q matrix K such that A + BKC is stable. Low-order controller design is a practically important problem that can be cast in the same framework, with {p+k)(q + k) design parameters instead of pq, where k is the order of the controller, and k << n. Robust stabilization further demands stability in the presence of perturbation and satisfactory transient as well as asymptotic system response. We formulate two related nonsmooth, nonconvex optimization problems over K, respectively with the following objectives: minimization of the ϵ-pseudospeetral abscissa of A + BKC, for a fixed ϵ ≥ 0, and maximization of the complex stability radius of A + BKC. Finding global optimizers of these functions is hard, so we use a recently developed gradient sampling method that approximates local optimizers. For modest-sized systems, local optimization can be carried out from a large number of starting points with no difficulty. The best local optimizers may then be investigated as candidate solutions to the static output feedback or low-order controller design problem. We show results for two problems published in the control literature. The first is a turbo-generator example that allows us to show how different choices of the optimization objective lead to stabilization with qualitatively different properties, conveniently visualized by pseudospectral plots. The second is a well known model of a Boeing 767 aircraft at a flutter condition, For this problem, we are not aware of any SOF stabilizing K published in the literature. Our method was not only able to find an SOF stabilizing K, but also to locally optimize the complex stability radius of A + BKC. We also found locally optimizing order-1 and order-2 controllers for this problem. All optimizers are visualized using pseudospectral piots.

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

Pages (from-to) | 175-181 |

Number of pages | 7 |

Journal | IFAC Proceedings Volumes (IFAC-PapersOnline) |

Volume | 36 |

Issue number | 11 |

DOIs | |

State | Published - Jan 1 2003 |

Event | 4th IFAC Symposium on Robust Control Design, ROCOND 2003 - Milan, Italy Duration: Jun 25 2003 → Jun 27 2003 |

### Fingerprint

### Keywords

- H-infinity norm
- Low-order controller
- Nonsmooth optimization
- Pseudospectra
- Spectral abscissa
- Stability radius
- Static output feedback

### ASJC Scopus subject areas

- Control and Systems Engineering

### Cite this

*IFAC Proceedings Volumes (IFAC-PapersOnline)*,

*36*(11), 175-181. https://doi.org/10.1016/S1474-6670(17)35659-8

**A nonsmooth, nonconvex optimization approach to Robust Stabilization by static output feedback and Low-order controllers.** / Burke, James V.; Lewis, Adrian S.; Overton, Michael.

Research output: Contribution to journal › Conference article

*IFAC Proceedings Volumes (IFAC-PapersOnline)*, vol. 36, no. 11, pp. 175-181. https://doi.org/10.1016/S1474-6670(17)35659-8

}

TY - JOUR

T1 - A nonsmooth, nonconvex optimization approach to Robust Stabilization by static output feedback and Low-order controllers

AU - Burke, James V.

AU - Lewis, Adrian S.

AU - Overton, Michael

PY - 2003/1/1

Y1 - 2003/1/1

N2 - Stabilization by static output feedback {SOF) is a long-standing open problem in control: given an n by n matrix A and rectangular matrices B and C, find a p by q matrix K such that A + BKC is stable. Low-order controller design is a practically important problem that can be cast in the same framework, with {p+k)(q + k) design parameters instead of pq, where k is the order of the controller, and k << n. Robust stabilization further demands stability in the presence of perturbation and satisfactory transient as well as asymptotic system response. We formulate two related nonsmooth, nonconvex optimization problems over K, respectively with the following objectives: minimization of the ϵ-pseudospeetral abscissa of A + BKC, for a fixed ϵ ≥ 0, and maximization of the complex stability radius of A + BKC. Finding global optimizers of these functions is hard, so we use a recently developed gradient sampling method that approximates local optimizers. For modest-sized systems, local optimization can be carried out from a large number of starting points with no difficulty. The best local optimizers may then be investigated as candidate solutions to the static output feedback or low-order controller design problem. We show results for two problems published in the control literature. The first is a turbo-generator example that allows us to show how different choices of the optimization objective lead to stabilization with qualitatively different properties, conveniently visualized by pseudospectral plots. The second is a well known model of a Boeing 767 aircraft at a flutter condition, For this problem, we are not aware of any SOF stabilizing K published in the literature. Our method was not only able to find an SOF stabilizing K, but also to locally optimize the complex stability radius of A + BKC. We also found locally optimizing order-1 and order-2 controllers for this problem. All optimizers are visualized using pseudospectral piots.

AB - Stabilization by static output feedback {SOF) is a long-standing open problem in control: given an n by n matrix A and rectangular matrices B and C, find a p by q matrix K such that A + BKC is stable. Low-order controller design is a practically important problem that can be cast in the same framework, with {p+k)(q + k) design parameters instead of pq, where k is the order of the controller, and k << n. Robust stabilization further demands stability in the presence of perturbation and satisfactory transient as well as asymptotic system response. We formulate two related nonsmooth, nonconvex optimization problems over K, respectively with the following objectives: minimization of the ϵ-pseudospeetral abscissa of A + BKC, for a fixed ϵ ≥ 0, and maximization of the complex stability radius of A + BKC. Finding global optimizers of these functions is hard, so we use a recently developed gradient sampling method that approximates local optimizers. For modest-sized systems, local optimization can be carried out from a large number of starting points with no difficulty. The best local optimizers may then be investigated as candidate solutions to the static output feedback or low-order controller design problem. We show results for two problems published in the control literature. The first is a turbo-generator example that allows us to show how different choices of the optimization objective lead to stabilization with qualitatively different properties, conveniently visualized by pseudospectral plots. The second is a well known model of a Boeing 767 aircraft at a flutter condition, For this problem, we are not aware of any SOF stabilizing K published in the literature. Our method was not only able to find an SOF stabilizing K, but also to locally optimize the complex stability radius of A + BKC. We also found locally optimizing order-1 and order-2 controllers for this problem. All optimizers are visualized using pseudospectral piots.

KW - H-infinity norm

KW - Low-order controller

KW - Nonsmooth optimization

KW - Pseudospectra

KW - Spectral abscissa

KW - Stability radius

KW - Static output feedback

UR - http://www.scopus.com/inward/record.url?scp=85058215332&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85058215332&partnerID=8YFLogxK

U2 - 10.1016/S1474-6670(17)35659-8

DO - 10.1016/S1474-6670(17)35659-8

M3 - Conference article

AN - SCOPUS:85058215332

VL - 36

SP - 175

EP - 181

JO - IFAC-PapersOnLine

JF - IFAC-PapersOnLine

SN - 2405-8963

IS - 11

ER -