A Nonsmooth, Nonconvex Optimization Approach to Robust Stabilization by Static Output Feedback and Low-order Controllers

Stabilization by static output feedback (SOF) is a long-standing open problem in control: given an n by n matrix A and rectangular matrices B and C, find a p by q matrix K such that A + BKC is stable. Low-order controller design is a practically important problem that can be cast in the same framework, with (p+k)(q+k) design parameters instead of pq, where k is the order of the controller, and k << n. Robust stabilization further demands stability in the presence of perturbation and satisfactory transient as well as asymptotic system response. We formulate two related nonsmooth, nonconvex optimization problems over K, respectively with the following objectives: minimization of the -pseudospectral abscissa of A+BKC, for a fixed ≥ 0, and maximization of the complex stability radius of A + BKC. Finding global optimizers of these functions is hard, so we use a recently developed gradient sampling method that approximates local optimizers. For modest-sized systems, local optimization can be carried out from a large number of starting points with no difficulty. The best local optimizers may then be investigated as candidate solutions to the static output feedback or low-order controller design problem. We show results for two problems published in the control literature. The first is a turbo-generator example that allows us to show how different choices of the optimization objective lead to stabilization with qualitatively different properties, conveniently visualized by pseudospectral plots. The second is a well known model of a Boeing 767 aircraft at a flutter condition. For this problem, we are not aware of any SOF stabilizing K published in the literature. Our method was not only able to find an SOF stabilizing K, but also to locally optimize the complex stability radius of A + BKC. We also found locally optimizing order–1 and order–2 controllers for this problem. All optimizers are visualized using pseudospectral plots.