The Positive Realness Of A Rational Matrix With Purely Imaginary Zeros

Positive realness is an important concept in system,network and control theory. In this paper, we mainly introduce two structure-preserving methods that can verify the positive realness of a given proper rational matrix H(s) for which H(s) + H~T(—s) has purely imaginary zeros, i.e., the relative Hamiltonian matrix M has purely imaginary eigenvalues.In Algorithm 1, we translate this positive realness problem of rational matrix H(s) to the computation of the stable Lagrangian subspace of Hamiltonian matrix M. By using orthogonal transformations, this Hamiltonian matrix M with purely imaginary eigenvalues can be deflated to a smaller one that has no purely imaginary eigenvalue. Then we compute the stable eigen-space of this reduced Hamiltonian matrix which can be expanded to the stable Lagrangian subspace of M.In Algorithm 2, we first translate the positive realness problem to the symmetric positive definite solution of a continuous-time algebraic Riccati equation. Then we transform the continuous-time algebraic Riccatic equation to a discrete one by using Cayley transformation. Hence, we can use existing iteration algorithm to solve this discrete-time algebraic Riccati equation. In this chapter, we also prove that Algorithm 2 converges to the desired solution globally and linearly.Finally, in chapter 4 of this paper, we get that Algorithm 1 and Algorithm 2 both need O(n~3) flops. Some examples are given to illustrate the performance of these two algorithms.