On bounds for the existence of a bistable controller

Abstract

By S(r) we shall denote the set of polynomials nonzero on the closed disk {z : |z|⩽r} . The problem to be studied is given as follows. Avoidance Problem. Given polynomials A and B, nonnegative integers n, m, and positive numbers r 1, r 2, r 3, determine if there exist P∈ S(r 1) , Q∈ S(r 2) with degrees n and m, respectively, such that AP+ BQ is in S(r 3) . It is shown that the Avoidance Problem for r 1= r 2= r 3=1 is equivalent to the well-known Bistable Stabilization Problem for a large class of control systems. Also, a general class of optimization problems for solving the Avoidance Problem is introduced. The class of optimization problems includes problems with sets more general than disks S(r) . In addition, an algorithm is proposed for solving these optimization problems, and numerical experience is reported. As an application, bistable controllers for the plant P β(z)=z/(z 2+β) are shown to exist for values of β as low as 0.0246, thus improving on previously reported results by Blondel et al. for this difficult case. Furthermore, it is shown how in some cases it is possible to find optimal controllers by using a combination of numerical and algebraic techniques, and a variant of the algorithm already introduced. As an example, a low order controller is found for P β for the case when β=7−4 3 ≈0.0717 . This method could be applied to other problems where the plant depends on a parameter.