Let M(y) be a matrix whose entries are polynomial in y, λ(y) and v(y) be a set of eigenvalue and eigenvector of M(y). Then, λ(y) and v(y) are algebraic functions of y, and λ(y) and v(y) have their power series expansionsλ(y) = β0 + β1 y + + βk yk + (βj C),(1) v(y) = γ0 + γ1 y + + γk yk + (γj Cn), (2)provided that y=0 is not a singular point of λ(y) or v(y). Several algorithms are already proposed to compute the above power series expansions using Newton's method (the algorithm in [4]) or the Hensel construction (the algorithm in[5],[12]). The algorithms proposed so far compute high degree coefficients βk and γk, using lower degree coefficients βj and γj (j=0,1,,k-1). Thus with floating point arithmetic, the numerical errors in the coefficients can accumulate as index k increases. This can cause serious deterioration of the numerical accuracy of high degree coefficients βk and γk, and we need to check the accuracy. In this paper, we assume that given matrix M(y) does not have multiple eigenvalues at y=0 (this implies that y=0 is not singular point of λ(y) or v(y)), and presents an algorithm to estimate the accuracy of the computed power series βi,γj in (1) and (2). The estimation process employs the idea in [9] which computes a coefficient of a power series with Cauchy's integral formula and numerical integrations. We present an efficient implementation of the algorithm that utilizes Newton's method. We also present a modification of Newton's method to speed up the procedure, introducing tuning parameter p. Numerical experiments of the paper indicates that we can enhance the performance of the algorithm by 1216%, choosing the optimal tuning parameter p.

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 = 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 = -F x to the plant, where F is a design parameter, and obtain the system = (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.

H∞ optimal control is one of the most successful achievements in the post modern control theory. In the H∞ optimal control, we design a controller that minimizes the H∞ norm of a given system. Although the algorithms to solve the problem have already been reported, they focus on numerical systems (systems without any unknown parameters) and, can not be applied for parametric systems (systems with unknown parameters). Given a parametric system, this paper presents an algorithm to compute the optimal H∞ norm of the system achieved by an output feedback controller. The optimal H∞ norm is expressed as , where φ(k) denotes a root of a bivariate polynomial. A numerical example is given to show the effectiveness of the algorithm.