Explicit H∞ controllers for 1st to 3rd order single-input single-output systems with parameters

The purpose of this paper is to explicitly characterize H∞ controllers for single-input single-output (SISO) systems of order 1, 2 and 3 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. 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 of zero-dimensional polynomial systems, we first give a parametrization of all the solutions of the algebraic Riccati equation associated with the H∞ control problem. Since the problem reduces to solving an univariate polynomial of degree less than or equal to 4, closed-form solutions are then obtained for the solutions of the algebraic Riccati equation by means of radicals. Using the concept of discriminant variety, we show that the maximal real root of this polynomial is always defined by the same closed-form expression, which yields the positive definite solution of the algebraic Riccati equation. Finally, we use the above results to explicitly compute the H∞ criteria γpt and H∞ controllers in terms of the system parameters and study them with respect to parameter variations.