On the Use of the Nelder-Mead Simplex Method in Control Design and Systems Theory

The Nelder-Mead simplex method is a well-known algorithm enabling the minimization of functions that are not available in closed-form and that need not be differentiable or convex. Furthermore, it is particularly parsimonious on the number of function evaluations, thus making it preferable to convex optimization paradigms in the case, common when dealing with control design problems, that the objective function of the optimization problem is non-differentiable, non-convex, and its closed-form is not available or difficult to be computed analytically. The main goal of this paper is to show how the joint use of the Nelder-Mead simplex method and the Morrison algorithm can be successfully used to solve relevant and challenging control problems that cannot be easily solved using analytic methods. In particular, it is shown how the problems of strong stabilization, static output feedback stabilization, and design of robust controllers having fixed structure can be framed as optimization problems, which, in turn, can be efficiently solved by coupling the two above mentioned algorithms. The performance of this procedure is compared with state-of-the-art techniques on dozens of static output feedback benchmark case studies, and its effectiveness is demonstrated by several examples.