## The smoothed spectral abscissa for robust stability optimization

by J. Vanbiervliet, B. Vandereycken, W. Michiels, S. Vandewalle, M. Diehl

#### 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 $\tilde\alpha_{\epsilon}(A)$, 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.

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#### Reference:

J. Vanbiervliet, B. Vandereycken, W. Michiels, S. Vandewalle, M. Diehl, "The smoothed spectral abscissa for robust stability optimization", In SIAM J. Optim., vol. 20, no. 1, pp. 156-171, 2009.

#### Bibtex Entry:

@article{Vanbiervliet_VMVD_2009, 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 $\tilde\alpha_{\epsilon}(A)$, 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.}, Author = {Vanbiervliet, J. and Vandereycken, B. and Michiels, W. and Vandewalle, S. and Diehl, M.}, Doi = {10.1137/070704034}, Journal = {SIAM J. Optim.}, Number = {1}, Pages = {156--171}, Pdf = {http://www.unige.ch/math/vandereycken/papers/published_Vanbiervliet_VMVD_2009.pdf}, Title = {The smoothed spectral abscissa for robust stability optimization}, Volume = {20}, Year = {2009}}