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- W2160261378 abstract "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 $tildealpha_{epsilon}(A)$, 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 $tildealpha_{epsilon}(A)leq0$ 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 $tildealpha_{epsilon}(A(x))$ 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 $tildealpha_{epsilon}(A(x))leq0$ 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." @default.
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- W2160261378 date "2009-01-01" @default.
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- W2160261378 title "The Smoothed Spectral Abscissa for Robust Stability Optimization" @default.
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- W2160261378 doi "https://doi.org/10.1137/070704034" @default.
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