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1

Reproducing Kernels and Riccati Equations

Dym H.

International Journal of Applied Mathematics and Computer Science

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2001

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Vol. 11, no 1

35-53

EN

The purpose of this paper is to exhibit a connection between the Hermitian solutions of matrix Riccati equations and a class of finite dimensional reproducing kernel Krein spaces. This connection is then exploited to obtain minimal factorizations of rational matrix valued functions that are J-unitary on the imaginary axis in a natural way.

2

Reduction of Large Circuit Models Via Low Rank Approximate Gramians

Li J. R., White J.

International Journal of Applied Mathematics and Computer Science

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2001

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Vol. 11, no 5

1151-1171

EN

We describe a model reduction algorithm which is well-suited for the reduction of large linear interconnect models. It is an orthogonal projection method which takes as the projection space the sum of the approximate dominant controllable subspace and the approximate dominant observable subspace. These approximate dominant subspaces are obtained using the Cholesky Factor ADI (CF-ADI) algorithm. We describe an improvement upon the existing implementation of CF-ADI which can result in significant savings in computational cost. We show that the new model reduction method matches moments at the negative of the CF-ADI parameters, and that it can be easily adapted to allow for DC matching, as well as for passivity preservation for multi-port RLC circuit models which come from modified nodal analysis.

3

J-energy preserving well-posed linear systems

Staffans O. J.

International Journal of Applied Mathematics and Computer Science

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2001

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Vol. 11, no 6

1361-1378

EN

The following is a short survey of the notion of a well-posed linear system. We start by describing the most basic concepts, proceed to discuss dissipative and conservative systems, and finally introduce J-energy-preserving systems, i.e., systems that preserve energy with respect to some generalized inner products (possibly semi-definite or indefinite) in the input, state and output spaces. The class of well-posed linear systems contains most linear time-independent distributed parameter systems: internal or boundary control of PDE's, integral equations, delay equations, etc. These systems have existed in an implicit form in the mathematics literature for a long time, and they are closely connected to the scattering theory by Lax and Phillips and to the model theory by Sz.-Nagy and Foias. The theory has been developed independently by many different schools, and it is only recently that these different approaches have begun to converge. One of the most interesting objects of the present study is the Riccati equation theory for this class of infinite-dimensional systems (H2- and Hinfty-theories).

4

Exact Observability of Diagonal Systems With a One-Dimensional Output Operator

Jacob B., Zwart H.

International Journal of Applied Mathematics and Computer Science

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2001

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Vol. 11, no 6

1277-1283

EN

In this paper equivalent conditions for exact observability of diagonal systems with a one-dimensional output operator are given. One of these equivalent conditions is the conjecture of Russell and Weiss (1994). The other conditions are given in terms of the eigenvalues and the Fourier coefficients of the system data.

5

On the bounds on the solutions of the algebraic Lyapunov and Riccati equations

Czornik A., Nawrat A.

Archives of Control Sciences

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2000

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Vol. 10, no. 3/4

197-244

EN

Different types of bounds for solutions of continuous and discrete Lyapunov and Riccati equations obtained up to now are summarized in this paper. Some new bounds are also presented. The efficiency of each bound is illustrated with three numerical examples. A discussion and comparison are given as well. The results may be particularly convenient to get the ready estimate of the solution while solving the equations numerically or to develop theoretical results that rely on these bounds.