SIAM Journal on Optimization vol:20 issue:1 pages:156-171
This paper 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 α(A) gives rise to very difficult optimization problems, since α(A) is not everywhere differentiable, and even not everywhere Lipschitz. 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 ε allows to tune the degree of smoothness. For ε approaching zero, the smoothed spectral abscissa α ε(A) converges towards the nonsmooth spectral abscissa from above, so that α ε(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 α ε(A) for a fixed value of ε , and the maximization of ε such that α ε(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 H∞-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.