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

James V. Burke, Adrian S. Lewis, Michael L. Overton
2003 IFAC Proceedings Volumes  
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
more » ... 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.
doi:10.1016/s1474-6670(17)35659-8 fatcat:yjzwi3wfl5eefda62x2fkxfzqm