Design of Robust PID Controllers Based on the Scenario Approach

In this paper, the scenario-based optimization approach is used to design a robust controller for linear uncertain systems with bounded parameter uncertainties. The controller structure is assumed to be known. The design methodology is based on the optimization of the integral squared error. Thus, using Laplace transform properties, the assumed performance index is expressed as a function of controller parameters and system uncertainties, and the design problem is written as a robust semi-infinite optimization problem. Then, the scenario approach, which is based on random sampling of constraints, is applied to convert the NP-hard semi-infinite optimization problem into a standard finite nonlinear programming problem. The resulting non-linear optimization problem is then solved by means of a stochastic optimization method. The proposed design approach is illustrated in three application examples by considering both constrained and unconstrained control parameters cases.