TY - JOUR

T1 - The smoothed spectral abscissa for robust stability optimization

AU - Vanbiervliet, J.

AU - Vandereycken, B.

AU - Michiels, W.

AU - Vandewalle, S.

AU - Diehl, M.

PY - 2008

Y1 - 2008

N2 - This note concerns the stability optimization of (parameterized) matrices A(x), a problem typically arising in the design of fixed-order or fixed-structured feedback controllers. It is well known that the minimization of the spectral abscissa function a (A) gives rise to very difficult optimization problems, since a (A) is not everywhere differentiable, and even not everywhere Lipschitz. In this note we therefore propose a new stability measure, namely the smoothed spectral abscissa, which is based on the inversion of a relaxed H2-type cost function. A regularization parameter e allows to tune the degree of smoothness. For e approaching zero, the smoothed spectral abscissa a e (A) converges towards the nonsmooth spectral abscissa from above, so that a e (A) = 0 guarantees asymptotic stability. Evaluation of the smoothed spectral abscissa and its derivatives w.r.t. the matrix parameters can be performed at the cost of solving a primal-dual Lyapunov equation pair, allowing for an efficient integration into a derivative based optimization framework. Two optimization problems are considered: the minimization in function of the parameters x of the smoothed spectral abscissa a e (A) for a fixed value of e , and the maximization of e such that a e (A) = 0 is still satisfied. The latter problem can be interpreted as a H2-norm minimization problem, and its solution additionally implies an upper bound on the corresponding H8 -norm, or a lower bound on the distance to instability. In both cases additional equality and inequality constraints on the variables can be naturally taken into account in the optimization problem.

AB - This note concerns the stability optimization of (parameterized) matrices A(x), a problem typically arising in the design of fixed-order or fixed-structured feedback controllers. It is well known that the minimization of the spectral abscissa function a (A) gives rise to very difficult optimization problems, since a (A) is not everywhere differentiable, and even not everywhere Lipschitz. In this note we therefore propose a new stability measure, namely the smoothed spectral abscissa, which is based on the inversion of a relaxed H2-type cost function. A regularization parameter e allows to tune the degree of smoothness. For e approaching zero, the smoothed spectral abscissa a e (A) converges towards the nonsmooth spectral abscissa from above, so that a e (A) = 0 guarantees asymptotic stability. Evaluation of the smoothed spectral abscissa and its derivatives w.r.t. the matrix parameters can be performed at the cost of solving a primal-dual Lyapunov equation pair, allowing for an efficient integration into a derivative based optimization framework. Two optimization problems are considered: the minimization in function of the parameters x of the smoothed spectral abscissa a e (A) for a fixed value of e , and the maximization of e such that a e (A) = 0 is still satisfied. The latter problem can be interpreted as a H2-norm minimization problem, and its solution additionally implies an upper bound on the corresponding H8 -norm, or a lower bound on the distance to instability. In both cases additional equality and inequality constraints on the variables can be naturally taken into account in the optimization problem.

U2 - 10.1137/070704034

DO - 10.1137/070704034

M3 - Article

VL - 20

SP - 156

EP - 171

JO - SIAM Journal on Optimization

JF - SIAM Journal on Optimization

SN - 1052-6234

IS - 1

ER -