Robust Mini-Max regulator for uncertain nonlinear polynomial systems

In this paper, we present a solution to the problem of the quadratic Mini-Max regulator for polynomial uncertain systems. The main characteristic of this type of problems is that the parameters describing the dynamic of the nonlinear plant depend on a vector of unknown parameters, which belongs to a finite parametric set, and the solution is given in terms of the worst case scenario. That is to say, the result of the application of a certain control input (in terms of the cost function value) is associated with the worst or least favorable value of the unknown parameter. Based on the general necessary conditions for mini-max optimality, a closed form for the control is provided, which makes use of a p-linear form tensor representation of the polynomial system. In its turn, this allows one to present the solution in a way similar to the so called Riccati technique. The final control is shown to be a convex combination (with some weights) of the optimal controls for each fixed parameter of the polynomial system. A numerical example for the robust regulation of the well-known Duffing equation which represents a typical (and challenging to control) nonlinear polynomial system is presented, to show the effectiveness of our approach.