Takuya Kitamoto

Parametric Computation of H α Norm of a System

While the computation of H_infin norm of a system is one of the most important theme in H_infin control theory, most of the algorithms proposed so far are numerical ones and cannot be applied to the system with a parameter. In this paper, we propose an algorithm to compute H_infin norm of a system with a parameter k. The H_infin norm is expressed in the form of 1/radic(Psi_q(f(q, k), l_i)) (v_i<k<v_i+1), 1/radic(Psi_q(f(q, k), m_i)) (k=v_i) (v_iisinR, l_i, m_iisinN), where Psi _q(f(q, k), l) denotes l-th real root of bivariate polynomial f(q, k) with respect to q

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.

On the Computation of the Determinant of a Generalized Vandermonde Matrix

“Vandermonde” matrix is a matrix whose (i,j)th entry is in the form of \(x_i^j\). The matrix has a lot of applications in many fields such as signal processing and polynomial interpolations. This paper generalizes the matrix, and let its (i,j) entry be f j (x i ) where f j (x) is a polynomial of x. We present an efficient algorithm to compute the determinant of the generalized Vandermonde matrix. The algorithm is composed of two sub-algorithms: the one that depends on given polynomials f j (x) and the one that does not. The latter algorithm (the one does not depend on f j (x)) can be performed beforehand, and the former (the one that depends on f j (x)) is mainly composed of the computation of determinants of numerical matrices. Determinants of the generalized Vandermonde matrices can be used, for example, to compute the optimal H ∞ and H 2 norm of a system achievable by a static feedback controller (for details, see [18],[19]).

On the Hankel singular values for a parametric system

Given a system with a parameter k, its Hankel singular values, denoted by sigma_i(k) (i = 1,ldrldrldr, n), are naturally functions of k. In this paper, we show that sigma_i(k) can be expressed as a root of a bivariate polynomial f (x, k) with respect to x, and present an algorithm to compute the polynomial f (x, k). We then apply the expression of sigma_i(k) to examine the extrema and the asymptotic behaviors of sigma_i(k). We also show that the ratio sigma_i(k)/sigma_j(k) of two distinct Hankel singular values can also be expressed as a root of bivariate polynomial. This gives us a systematic method to examine various properties such as the extrema or the asymptotic behaviors of the ratio sigma_i(k)/sigma_j(k). Considering that the ratio sigma_i(k)/sigma_j(k) is quite important information for dasiabalanced model reductionpsila, we can utilize the properties for a balanced model reduction of a parametric system.

On Computation of a Power Series Root with Arbitrary Degree of Convergence

AbstractGiven a bivariate polynomial f(x, y), let ☎i(y) be a power series root of f(x, y) = 0 with respect tox, i.e., ☎i(y) is a function ofy such thatf(☎i(y),,y) = 0. If ☎i(y) is analytic aty = 0, then we have its power series expansion(1)

The Optimal H∞ Norm of a Parametric System Achievable by an Output Feedback Controller

\phi (y) = \alpha _0 + \alpha _1 y + \alpha _2 y^2 + \cdots + \alpha _r y^r + \cdots .

Efficient Computation of the Characteristic Polynomial of a Polynomial Matrix

Let ☎i(k)(y) denote ☎i(y) truncated atyk, i.e.,(2)