## Selection of basic software tools for structured matrix decompositions and perturbations

### Paul Van Dooren

SLICOT Working Note 1999-9: June 1999.

In this note a survey is given of areas of systems and control where structured matrix problems are important. In identification we mention four different types of data collection : impulse response, input-output pairs, frequency response and covariance data. In each of those, the identification problem can be rewritten in terms of structured matrix problems for which there exist fast decompositions. The use of structured matrix decompositions should yield an improvement in speed of computations. In analysis and design one encounters eigenvalue problems with specific structure such as cyclic, Hamiltonian and symplectic matrices. For those problems it is important to use structure preserving decompositions, mainly to improve the numerical accuracy of the computations, although these algorithms typically yield improved computational complexities as well. We also list the key numerical routines that should be provided in the SLICOT library in order to tackle most of the problems mentioned in this note.

## Model reduction routines for SLICOT

### Andras Varga

NICONET Report 1999-8: June 1999.

We report on the newest developments of model reduction software for SLICOT. Three enhanced accuracy model reduction algorithms belonging to the class of methods based on or related to balancing techniques form the basis of model reduction software in SLICOT. These methods are primarily intended for the reduction of linear, stable, continuous- or discrete-time systems. However, in combination with additive spectral decomposition or coprime factorization techniques the basic methods can be employed to reduce unstable systems too. The implemented computational methods for reduction of stable and unstable systems, and the associated software available in SLICOT are presented. Performance comparisons performed using appropriate interface software to user-friendly environments like MATLAB and Scilab show the superiority of SLICOT model reduction tools over existing model reduction software.

## DTLEX - A collection of benchmark examples for discrete-time Lyapunuv equations

### Daniel Kressner, Volker Mehrmann and Thilo Penzl

SLICOT Working Note 1999-7: June 1999.

This paper describes the benchmark collection DTLEX, that contains test examples of discrete-time algebraic Lyapunov equations. These matrix equations are also known as Stein equations. The main focus of DTLEX is on scalable benchmark examples depending on parameters, which affect the conditioning of the equation. Such examples are particularly useful for the assessment of the complexity and the accuracy of numerical solution methods.

## CTLEX - A collection of benchmark examples for continuous-time Lyapunuv equations

### Daniel Kressner, Volker Mehrmann and Thilo Penzl

SLICOT Working Note 1999-6: June 1999.

This paper describes the benchmark collection CTLEX, that contains test examples of continuous-time algebraic Lyapunov equations. The main focus of this collection is on scalable benchmark examples depending on parameters, which affect the conditioning of the equation. Such examples are particularly useful for the assessment of the complexity and the accuracy of numerical solution methods.

## Fortran 77 routines for H-infinity and H2 design of discrete-time linear control systems

### Petko Hr. Petkov, Da-Wei Gu and Mihail M. Konstantinov

NICONET Report 1999-5: May 1999.

We present Fortran 77 subroutines intended for state-space design of H-infinity (sub)optimal controllers and H2 optimal controllers for linear discrete-time control systems.

The subroutines make use of LAPACK and BLAS libraries and produce estimates of the condition numbers of the matrices which are to be inverted and estimates of the condition numbers of the matrix Ricatti equations which are to be solved in the computation of the controllers. The subroutines will be included in the SLICOT library.