# DescriptorSystems.jl

A descriptor system is a generalized state-space representation of the form

Eλx(t) = Ax(t) + Bu(t),
y(t)   = Cx(t) + Du(t),

where x(t) is the state vector, u(t) is the input vector, and y(t) is the output vector, and where λ is either the differential operator λx(t) = dx(t)/dt for a continuous-time system or the advance operator λx(t) = x(t + ΔT) for a discrete-time system with the sampling time ΔT. In all what follows, we assume E is square and possibly singular, and the pencil A − λE is regular (i.e., det(A − λE) ̸≡ 0). If E = I, we call the above representation a standard state-space system.

The corresponding input-output representation is

Y(λ) = G(λ)U(λ),

where, depending on the system type, λ = s, the complex variable in the Laplace transform for a continuous-time system, or λ = z, the complex variable in the Z-transform for a discrete-time system, Y(λ) and U(λ) are the Laplace- or Z-transformed output and input vectors, respectively, and G(λ) is the rational transfer function matrix (TFM) of the system, defined as

                -1
G(λ) = C(λE − A)  B + D.

It is well known that the descriptor system representation is the most general description for a linear time-invariant system. Continuous-time descriptor systems arise frequently from modelling interconnected systems containing algebraic loops or constrained mechanical systems which describe contact phenomena. Discrete-time descriptor representations are frequently used to model economic processes. A main apeal of descriptor system models is that the manipulation of rational and polynomial matrices can be easily performed via their descriptor system representations, since each rational or polynomial matrix can be interpreted as the TFM of a descriptor system. For an introductory presentation of the main concepts, see [1].

The theoretical background for the analysis of descriptor systems closely relies on investigating the properties of certain linear matrix pencils, as the regular pole pencil P(λ) = A-λE, or the generally singular system matrix pencil S(λ) = [A-λE B; C D]. Therefore, the main analysis tools of descriptor systems are pencil manipulation techniques (e.g., reductions to various Kronecker-like forms), as available in the MatrixPencils package [2]. Among the main applications of pencil manipulation algorithms, we mention the computation of minimal nullspace bases, the computation of poles and zeros, the determination of the normal rank of polynomial and rational matrices, computation of various factorizations of rational matrices, as well as the solution of linear equations with polynomial or rational matrices. Important additional computational ingredients in these applications are tools for solving matrix equations, as various Lyapunov, Sylvester and Riccati equations. These tools are provided by the MatrixEquations package [3].

The available functions in the DescriptorSystems.jl package cover both standard and descriptor systems with real or complex coefficient matrices. The current version of the package includes the following functions:

Building descriptor system state-space models

Building rational transfer functions

Interconnecting descriptor system models

Basic operations on descriptor system models

Basic conversions on descriptor system models

Some operations on rational transfer functions and matrices

Simplification of descriptor system models

Descriptor system analysis

Factorization of descriptor systems

Advanced operations on transfer function matrices

Solution of model-matching problems

Andreas Varga