Browsing by Author "Zhou, Y."

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    Bounds on Eigenvalue Decay Rates and Sensitivity of Solutions to Lyapunov Equations
    (2002-06) Sorensen, Danny C.; Zhou, Y.
    Balanced model reduction is a technique for producing a low dimensional approximation to a linear time invariant system. An important feature of balanced reduction is the existence of an error bound that is closely related to the decay rate of the eigenvalues of certain system Gramians. Rapidly decaying eigenvalues imply low dimensional reduced systems. New bounds are developed for the eigen-decay rate of the solution of Lyapunov equation AP + PAT =- BBT. These bounds take into account the low rank right hand side structure of the Lyapunov equation. They are valid for any diagonalizable matrix A. Numerical results are presented to illustrate the effectiveness of these bounds when the eigensystem of A is moderately conditioned. We also present a bound on the norm of the solution P when A is diagonalizable and derive bounds on the conditioning of the Lyapunov operator for generalA.
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    On the Decay Rate of Hankel Singular Values and Related Issues
    (2001-05) Antoulas, A.C.; Sorensen, D.C.; Zhou, Y.
    This paper investigates the decay rate of the Hankel singular values of linear dynamical systems. This issue is of considerable interest in model reduction by means of balanced truncation, for instance, since the sum of the neglected singular values provides an upper bound for an appropriate norm of the approximation error. The decay rate involves a new set of invariants associated with a linear system, which are obtained by evaluating a modified transfer function at the poles of the system. These considerations are equivalent to studying the decay rate of the eigenvalues of the product of the solutions of two Lyapunov equations. The related problem of determining the decay rate of the eigenvalues of the solution to one Lyapunov equation will also be addressed. Very often these eigenvalues like the Hankel singular values, are decaying rapidly. This fact has motivated the development of several algorithms for computing low rank approximate solutions to Lyapunov equations. However, until now, conditions assuring rapid decay have not been well understood. Such conditions are derived here by relating the solution to a numerically low rank Cauchy matrix determined by the poles of the system. Bounds explaining rapid decay rates are obtained under some mild conditions.