Explicit H∞ controllers for 4th order single-input single-output systems with parameters and their application to the two mass-spring system with damping

Abstract

The purpose of this paper is to explicitly characterize H∞ controllers for 4th order single-input single-output (SISO) systems in terms of their coefficients considered as unknown parameters. In the SISO case, computing H∞ controllers requires to find the real positive definite solution of an algebraic Riccati equation (ARE). Due to the system parameters, no purely numerical method can be used to find such a solution, and thus parametric H∞ controllers. Using elimination techniques for zero-dimensional polynomial systems, we first give a rational parametrization of all the solutions of the ARE. Then, as the problem reduces to solving polynomials of degree 4, closed-form solutions are obtained for all the solutions of this ARE by using expressions by radicals. Using the concept of discriminant variety, we then show that the maximal real root of one of these polynomials is encoded by two different closed-form expressions depending on the values of the system parameters, which yields to different positive definite solution of the ARE. The above results are then used to explicitly compute the H∞ criterion γopt and H∞ controllers in terms of the system parameters. Finally, we study in detail a particular system: the two-mass-spring system with damping. Due to the low number of parameters, we can plot the variations of γopt in function of the parameters, compute approximations of γopt at a working point, and derive the expression of a weight function of the parameters to set γopt to a desired value.