The Smoothed Spectral Abscissa for Robust Stability Optimization

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 $\alpha(A)$ gives rise to very difficult optimization problems, since $\alpha(A)$ is not everywhere differentiable and even not everywhere Lipschitz. We therefore propose a new stability measure, namely, the smoothed spectral abscissa $\tilde\alpha_{\epsilon}(A)$, which is based on the inversion of a relaxed $H_2$-type cost function. The regularization parameter $\epsilon$ allows tuning the degree of smoothness. For $\epsilon$ approaching zero, the smoothed spectral abscissa converges towards the nonsmooth spectral abscissa from above so that $\tilde\alpha_{\epsilon}(A)\leq0$ guarantees asymptotic stability. Evaluation of the smoothed spectral abscissa and its derivatives w.r.t. matrix parameters $x$ 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: On the one hand, the minimization of the smoothed spectral abscissa $\tilde\alpha_{\epsilon}(A(x))$ as a function of the matrix parameters for a fixed value of $\epsilon$, and, on the other hand, the maximization of $\epsilon$ such that the stability requirement $\tilde\alpha_{\epsilon}(A(x))\leq0$ is still satisfied. The latter problem can be interpreted as an $H_2$-norm minimization problem, and its solution additionally implies an upper bound on the corresponding $H_\infty$-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.