A Descriptor Systems Toolbox for MATLAB

We describe a recently developed Descriptor Systems Toolbox implemented under Matlab. This Toolbox relies on the object oriented approach for control systems analysis and design provided within the standard Control Toolbox of Matlab. The basic approach to develop the Descriptor Systems Toolbox was to exploit the powerful matrix and system object manipulation features of Matlab via flexible and functionally rich high level m-functions, while simultaneously enforcing highly efficient and numerically sound structure exploiting computations via the mex -function technology of Matlab to solve critical numerical problems. The mex -functions are based on Fortran codes from Lapack and Slicot. 1 Why a Descriptor Systems Toolbox? It is well-known that a descriptor system of the form Eẋ(t) = Ax(t) +Bu(t) y(t) = Cx(t) +Du(t) with E square and possibly singular and with A− λE a regular matrix pencil, is the most general description for a linear time-invariant continuous-time system. Such systems arise frequently from modelling interconnected systems even with standard tools like Simulink (recall the ”algebraic loop” warning). Descriptor models are also common in modelling constrained mechanical systems (e.g., contact problems). Moreover, the descriptor representation is necessary to perform some operations even with standard systems like conjugation or inversion. Discrete-time descriptor representations are frequently used to model economic processes. The Descriptor Systems Toolbox is primarily intended to provide an extended functionality for the Control Toolbox of Matlab by allowing the manipulation of descriptor systems, the most general class of linear system models. Although these models are formally supported in the Control Toolbox, systems with singular E are not allowed. This is why some functions in the Descriptor Systems Toolbox represent simply extensions of functions already present in the Control Toolbox. The other functions are new and allow, for the first time, a convenient user-friendly operation to solve the most complicated dynamic analysis problems, as for example, the determination of complete Kronecker-structure of a linear pencil. The Descriptor Systems Toolbox is also useful for manipulating rational and polynomial matrices. Recall that each rational matrix R(λ) can be seen as the transfer-function matrix (TFM) of a continuousor discrete-time descriptor system. Thus, each R(λ) can be equivalently realized by a descriptor system quadruple (A− λE,B,C,D) satisfying R(λ) = C(λE −A)−1B +D, where λ = s for a continuous-time descriptor realization or λ = z for discrete-time descriptor realization. It is widely accepted that most of numerical operations on rational matrices, and in particular on polynomial matrices, are best done by manipulating instead the matrices of the corresponding descriptor systems descriptions. Many operations on standard matrices have nice generalizations for rational matrices. Straightforward generalizations are the rank, determinant, inverse and several generalized inverses. The conjugate transposition of a complex matrix M∗ generalizes to the conjugation of a rational matrix R∗(λ), where R∗(s) = R (−s) or R∗(z) = R (1/z), while the full-rank, inner-outer and spectral factorizations can be seen as generalizations of the familiar LU, QR and Cholesky factorizations, respectively. Many aspects for scalar polynomials and rational functions, as for example, poles and zeros, minimum degree coprime factorizations, normalized coprime factorizations or spectral factorization have nontrivial generalizations for polynomial and rational matrices.