The Optimal H Norm of a Parametric System Achievable Using a Static Feedback Controller

In recent years, algorithms based on Computer Algebra ([1]-[3]) have been introduced into a range of control design problems because of the capacity to handle unknown parameters as indeterminates. This feature of algorithms in Computer Algebra reduces the costs of computer simulation and the trial and error process involved, enabling us to design and analyze systems more theoretically with the behavior of given parameters. In this paper, we apply Computer Algebra algorithms to H∞ control theory, representing one of the most successful achievements in post-modern control theory. More specifically, we consider the H∞ norm minimization problem using a state feedback controller. This problem can be formulated as follows: Suppose that we are given a plant described by the linear differential equation dx/dt = Ax + B1w + B2u, z = Cx + Du, where A, B1, B2, C, D are matrices whose entries are polynomial in an unknown parameter k. We apply a state feedback controller u = -Fx to the plant, where F is a design parameter, and obtain the system dx/dt = (A-B2F)x + B1w, z = (C-DF)x. Our task is to compute the minimum H∞ norm of the transfer function G(s)(=(C-DF)(sI-A + B2F)-1B1) from w to z achieved using a static feedback controller u = -Fx, where F is a constant matrix. In the H∞ control theory, it is only possible to check if there is a controller such that ||G(s)||∞ < γ is satisfied for a given number γ, where ||G(s)||∞ denotes the H∞ norm of the transfer function G(s). Thus, a typical procedure to solve the H∞ optimal problem would involve a bisection method, which cannot be applied to plants with parameters. In this paper, we present a new method of solving the H∞ norm minimization problem that can be applied to plants with parameters. This method utilizes QE (Quantifier Elimination) and a variable elimination technique in Computer Algebra, and expresses the minimum of the H∞ norm as a root of a bivariate polynomial. We also present a numerical example to illustrate each step of the algorithm.