##### Department of Mathematics,

University of California San Diego

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### Engineering (controls group)

## Michael Overton

#### Courant Institute of Mathematical Sciences,NYU

## Optimizing Stability and Controllability via Pseudospectra

##### Abstract:

A matrix is stable if its eigenvalues are in the left half of the complex plane. More practical stability measures include the pseudospectral abscissa (maximum real part of the pseudospectrum) and the distance to instability (minimum norm perturbation required to make a stable matrix unstable). Likewise, the classical definition of controllability is not as useful as a measure of the distance to uncontrollability. Matrices often arise in applications as parameter dependent. Optimization of stability or controllability measures over parameters is challenging because the objective functions are nonsmooth and nonconvex. We solve such optimization problems, locally at least, via a novel method based on gradient sampling. One of our stability optimization examples is a difficult problem from the control literature: finding stable low-order controllers for a model of a Boeing 767 at a flutter condition. We also give a controllability optimization example and explain its connection with an interesting open question: how many connected components are possible for pseudospectra of rectangular matrices? joint work with James V. Burke, University of Washington, Seattle, WA Adrian S. Lewis, Simon Fraser University, Burnaby, BC, Canada

Host: Bill Helton

### November 12, 2003

### 10:30 AM

### EBU-II 584E

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