Application of Bernstein Branch-and-Bound Method to PID Controls with Maximum Stability Degree

Abstract

The stability degree of a stable control system is defined to be the negative of the largest real part of the zeros of its characteristic equation. In this paper, we consider the problem of tuning parameters of PID controller for delay-free linear time-invariant systems to maximize the degree of closed-loop stability. By applying the Bernstein branch-and-bound (BBB) method to test the existence of the stable region in the parameter space, a procedure is proposed to design maximum-stability PID controllers for linear time-invariant delay-free systems. The applicability of the BBB method here is based on fact that the stable region is characterized by a set of multi-variate polynomials in parameters, which is derived from the Lienard-Chipart criterion for a Hurwitz polynomial. An example of designing a PD controller with maximum stability-degree for a fourth-order system is given to verify the proposed approach.