A symbolic-numeric method for the parametric H ∞ loop-shaping design problem

In this paper, we present a symbolic-numeric method for solving the H∞ loop-shaping design problem for a low order single-input single-output system with parameters. Due to the system parameters, no purely numerical algorithm can indeed solve the problem. Using Gröbner basis techniques and the rational univariate representation of zero-dimensional algebraic varieties, we first give a parametrization of all the solutions of the two algebraic Riccati equations associated with the H∞ control problem. Then, following the works of [1], [9] on the spectral factorization problem, a certified symbolic-numeric algorithm is obtained for the computations of the positive definite solutions of these two algebraic Riccati equations. Finally, we present a certified symbolic-numeric algorithm which solves the H∞ loop-shaping design problem for the above class of systems. This algorithm is illustrated with a standard example.