Structured controller design for closed‐loop D ‐stability in convex/non‐convex regions: Mixed integer‐linear programming approach

Abstract

Here it is assumed that the characteristic function of a linear control system, which is in the form of a polynomial and its coefficients are desired affine functions of unknown parameters of controller, is given. It is also assumed that the transfer function of controller has a desired order and structure. Hence, some of the coefficients of the characteristic polynomial may not depend on some or any of the controller parameters. The main problem under consideration is to calculate the parameters of the controller such that all roots of the characteristic equation lie (if possible) inside the desired D‐stability region, which can be any convex/non‐convex connected/disconnected subset of complex plane. This problem has important applications in control theory. For example, non‐convex D‐stability regions appear when designing a controller for a fractional‐order system is aimed. It is shown that this problem, which is still open even in dealing with general convex regions, is exactly equivalent to a set of linear algebraic equalities/inequalities in mixed real‐integer variables, which can be solved efficiently by using the available software. Application of the proposed method for optimal controller design is also studied and three numerical examples are presented.