# Descriptor Systems

A descriptor system representation of the form

with E square and possibly singular and with A-sE a regular matrix pencil, is the most general description for a linear time-invariant continu­ous-time system. Such systems arise when modelling interconnected systems, constrained mechanical systems (e.g., contact problems) or performing system operations like conjugation or inversion.

Descriptor system descriptions frequently appear when solving computational problems in the analysis and design of standard linear systems. The numerically reliable solution of many standard control problems like the solu­tion of Riccati equations, computation of system zeros, design of fault detection and isolation filters (FDI), etc. relies on using descriptor system techniques.

Many algorithm for standard systems as for example stabilization techniques, factorization methods, minimal realization, model reduction, etc. have been extended to the more general descriptor system descriptions. An important application of these algorithms is the numeri­cally reliable computation with rational and polynomial matrices via equivalent descriptor representations. Recall that each rational matrix R(s) can be seen as the transfer-function matrix of a continuous- or discrete-time descriptor system. Thus, each R(s) can be equivalently realized by a descriptor system quadruple (A-sE, B, C, D) satisfying

Many operations on standard matrices (e.g., finding the rank, determinant, inverse or generalized inverses), or the solution of linear matrix equa­tions can be performed for rational matrices as well using descriptor system techniques. Other important applications of descriptor techniques are the computation of inner-outer and spectral factorisations, or minimum degree and normal­ized coprime factorisations of polynomial and rational matrices.

The development of reliable numerical algorithms for analysis and synthesis of descriptor systems has been an active area of research in the last decade. For an overview of computational methods and software for descriptor systems see [1].

## Analysis

For the analysis of structural properties of descriptor systems like controllability, observability or minimality, algorithms based on orthogonal reduction of particular system pencils to appropriate staircase forms can be employed. Numerically stable algorithms for the structural analysis of descriptor systems have been proposed in [2]. The highly efficient, numerically stable structure exploiting algorithms for computing zeros of descriptor systems [3] as well as of various Kronecker-like forms [4] represent universal structural analysis tools of both standard and descriptor systems. The algorithms for minimal realization and pole-zero computations are the basis for numerically reliable approaches to perform conversions between various representations of generalized systems (polynomial, descriptor, rational) [5].

## Synthesis

The synthesis methods for descriptor systems rely on algorithms for state-feedback stabilization using pole assignment and linear-quadratic techniques. Pole assignment and stabilization methods based on a generalized real Schur form technique, have been proposed in [6]. These techniques can also be employed to compute coprime factorizations of rational matrices. To fully exploit the essential non-uniqueness of the multi-input pole assignment problem, a parametric optimization based approach has been developed in [7] to solve this problem. This new approach uses a generalized Sylvester system based parametrization to solve a minimum norm robust pole assignment problem for a descriptor system.

## Descriptor system techniques for fault detection

By using descriptor systems techniques the fault detection and isolation synthesis problems can be solved in the most general setting. Several computational algorithms have been developed for the special need of fault detection. For example, the nullspace methods for the synthesis of fault detectors are based on numerically reliable algorithms for computing minimal proper rational nullspaces of rational matrices. Suitable algorithms based on reduction to a Kronecker-like form are described in [10,11]. Furthermore, the computation of least order detectors requires special computational algorithms for determining minimum dynamic covers. Such algorithms have been developed for descriptor systems [12] using extensions of the orthogonal reduction algorithms to compute controllability or observabilitystaircaseforms [2].

# DESCRIPTOR SYSTEMS toolbox (current version 1.06)

The DESCRIPTOR SYSTEMS toolbox has been developed:

• to enhance the MATLAB Control Toolbox by handling the most general linear system representations
• to solve in a numerically reliable way many standard control problems by using descriptor system techniques
• to manipulate in a numerically reliable way rational and polynomial matrices
• to extend the capabilities of basic MATLAB with matrix pencil methods

The toolbox is based partly on the RASP-DESCRIPT collection of routines [1] and partly on the free control software library SLICOT [9], both implemented in Fortran 77. The underlying algorithms encompass computations like the determination of complete Kronecker structure of linear pencils, generalized pole assignment and stabilization, model conversions (descriptor state-space to rational/polynomial representations), general rational factorisations (inner-outer, normalized coprime), generalized inverses (left/right, week, Moore-Penrose), solution of systems of equations with rational matrices etc. This package illustrates the new trend in CACSD to employ high quality, robust control software written in high level languages (e.g., Fortran) in user-friendly environments like MATLAB via appropriate gateways (MEX-functions).

The toolbox main strengths are the guaranteed numerical reliability achieved by careful selection of employed algorithms and software, and the high computational efficiency achieved by implementing critical structure exploiting computations as MEX-functions based on high quality robust numerical software available in the Fortran libraries LAPACK, SLICOT and RASP-DESCRIPT. The toolbox fully supports both standard and descriptor systems by employing specific algorithms. Also all functions are available for both continuous- and discrete-time systems. User friendly operation is achieved by object oriented manipulations based on the MATLAB Control Toolbox system objects.

Version 1.0 of the Descriptor Systems Toolbox is described in [8], where the underlying algorithms are also indicated and several examples illustrating the basic operations are given. The implementations of all functions exploit the best of MATLAB and Fortran programming, by trying to balance the matrix manipulation power of MATLAB with the intrinsic high efficiency of carefully implemented structure-exploiting Fortran codes available in LAPACK and SLICOT. This approach illustrates the possibility of turning high complexity structure-exploiting algorithms into numerically robust and user-friendly CACSD software. This paradigm is applicable to the development of high performance computer-aided control engineering environments.

Related publications:

[1] Varga, A.:

Numerical algorithms and software tools for analysis and modelling of descriptor systems. Prepr. of 2nd IFAC Workshop on System Structure and Control, Prague, Czechoslovakia, pp. 392-395, 1992.

[2] Varga, A.:

Computation of irreducible generalized state-space realizations.Kybernetika, 26:89-106, 1989.

[3] Misra P., Van Dooren P., Varga A.:

Computation of structural invariants of generalized state space systems. Automatica, vol. 30, pp. 1921-1936, 1994.

[4] Varga, A.:

Computation of Kronecker-like forms of a system pencil: applications, algorithms and software. Proc. of IEEE International Symposium on Computer Aided Control System Design, CACSD'96, Dearborn, MI, CACSD '96, pp. 77-82, 1996.

[5] Varga, A.:

Computation of transfer function matrices of generalized state-space models. Int. J. Control, 50:2543-2561, 1989.

[6] Varga, A.:

On stabilization methods of descriptor systems. System & Control Letters, vol. 24, pp. 133-138, 1995.

[7] Varga, A.:

A numerically reliable approach to robust pole assignment for descriptor systems. Future Generation Computer Systems, vol. 19, pp. 1221-1230, 2003.

[8] Varga, A.:

A DESCRIPTOR SYSTEMS toolbox for MATLAB. Proc. of IEEE International Symposium on Computer Aided Control System Design, CACSD'2000, Anchorage, Alaska, 2000.

[9] Van Huffel, S., Sima, V., Varga, A., Hammarling, S., Delebecque, F.:

High-performance numerical software for control. IEEE Control Systems Magazine, vol. 24, pp. 60-76, 2004.

[10] Varga, A.:

On computing least order fault detectors using rational nullspace bases. Proc. of IFAC Symp. SAFEPROCESS'03, Washington DC, USA, 2003.

[11] Varga, A.:

On computing nullspace bases - a fault detection perspective.Proc. of the 17th IFAC World Congress, Seoul, Korea, 2008.

[12] Varga, A.:

Reliable algorithms for computing minimal dynamic covers for descriptor systems. Proc. of MTNSâ€™04, Leuven , Belgium, 2004.

Contact
 Dr.-Ing. Andreas VargaGerman Aerospace CenterInstitute of System Dynamics and Control, Aircraft Systems Dynamics Tel: +49 8153 28-2407 Fax: +49 8153 28-1441E-Mail: Andreas.Varga@dlr.de