A perturbation method for optimizing matrix stability

We present the first practical perturbation method for optimizing matrix stability using spectral abscissa minimization. Using perturbation theory for a matrix with simple eigenvalues and coupling this with linear programming, we successively reduce the spectral abscissa of a matrix until it reaches a local minimum. Optimality conditions for a local minimizer of the spectral abscissa are provided and proved for both the affine matrix problem and the output feedback control problem. Experiments show that this novel perturbation method is efficient, especially for a matrix with the majority of whose eigenvalues are already located in the left half of the complex plane. Moreover, unlike most available methods, the method does not require the introduction of Lyapunov variables. The method is illustrated for a small size matrix from an affine matrix problem and is then applied to large matrices actually arising from more sophisticated control problems used in the design of the Boeing 767 jet and a nuclear powered turbo-generator.