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

dss Construction of descriptor state-space models.
dssdata Extraction of matrix-data from a descriptor state-space model.

Building rational transfer functions

RationalTransferFunction Construction of rational transfer function objects.
rtf Building rational transfer functions.

Interconnecting descriptor system models

append Building aggregate models by appending the inputs and outputs.
parallel Connecting models in parallel (also overloaded with +).
series Connecting models in series (also overloaded with *).
horzcat Horizontal concatenation of descriptor system models (also overloaded with [ * * ]).
vertcat Vertical concatenation of descriptor system models (also overloaded with [ *; * ]).

Basic operations on descriptor system models

inv Inversion of a system.
ldiv Left division for two systems (also overloaded with \).
rdiv Right division for two systems (also overloaded with /).
gdual Building the dual of a descriptor system (also overloaded with transpose)
ctranspose Building the conjugate transpose of a system (also overloaded with adjoint and ').
adjoint Building the adjoint of a system.

Basic conversions on descriptor system models

gprescale Balancing of a descriptor system.
c2d Discretization of continuous-time descriptor systems.
dss2rm Rational transfer function matrix of a descriptor system.
dss2pm Polynomial transfer function matrix of a descriptor system.
gbilin Generalized bilinear transformation of a descriptor system.

Some operations on rational transfer functions and matrices

simplify Pole-zero cancellation.
normalize Normalization of a rational transfer function to monic denominator.
confmap Applying a conformal mapping transformation to a rational transfer function or rational transfer function matrix.
zpk Computation of zeros, poles and gain of a rational transfer function.
rtfbilin Generation of common bilinear transformations and their inverses.

Simplification of descriptor system models

gminreal Minimal realization of descriptor systems.
gir Irreducible realization of descriptor systems.
gbalmr Reduced-order approximations of descriptor systems using balancing related methods.
gss2ss Conversion to SVD-like forms without non-dynamic modes.
dss2ss Conversion of descriptor systems to standard form.

Descriptor system analysis

isregular Test whether a descriptor system has a regular pole pencil.
gpole Poles of a descriptor system.
gpoleinfo Poles and pole structure information of a descriptor system.
isproper Test whether a descriptor system is proper.
isstable Test whether a descriptor system is stable.
gzero Zeros of a descriptor system.
gzeroinfo Zeros and zero structure information of a descriptor system.
gnrank Normal rank of the transfer function matrix of a descriptor system.
ghanorm Hankel norm of a proper and stable descriptor system.
gl2norm L2 norm of a descriptor system.
gh2norm H2 norm of a descriptor system.
glinfnorm L∞ norm of a descriptor system.
ghinfnorm H∞ norm of a descriptor system.
gnugap ν-gap distance between two descriptor systems.
freqresp Frequency response of a descriptor system.
timeresp Time response of a descriptor system.
stepresp Step response of a descriptor system.
gbalqual Evaluation of the scaling quality of the matrices of a descriptor system.
pbalqual Evaluation of the scaling quality of the matrices of a linear pencil.

Factorization of descriptor systems

grcf Right coprime factorization with proper and stable factors.
glcf Left coprime factorization with proper and stable factors.
grcfid Right coprime factorization with inner denominator.
glcfid Left coprime factorization with inner denominator.
gnrcf Normalized right coprime factorization.
gnlcf Normalized left coprime factorization.
giofac Inner-outer/QR-like factorization.
goifac Co-outer-co-inner/RQ-like factorization.
grsfg Right spectral factorization of γ^2*I-G'*G.
glsfg Left spectral factorization of γ^2*I-G*G'.

Advanced operations on transfer function matrices

gsdec Additive spectral decompositions.
grnull Right nullspace basis of a transfer function matrix.
glnull Left nullspace basis of a transfer function matrix.
grange Range space basis of a transfer function matrix.
gcrange Coimage space basis of a transfer function matrix.
grsol Solution of the linear rational matrix equation G(λ)*X(λ) = F(λ).
glsol Solution of the linear rational matrix equation X(λ)*G(λ) = F(λ).
grmcover1 Right minimum dynamic cover of Type 1 based order reduction.
glmcover1 Left minimum dynamic cover of Type 1 based order reduction.
grmcover2 Right minimum dynamic cover of Type 2 based order reduction.
glmcover2 Left minimum dynamic cover of Type 2 based order reduction.
ginv Generalized inverses.

Solution of model-matching problems

gnehari Generalized Nehari approximation.
glinfldp Solution of the least distance problem.
grasol Approximate solution of the linear rational matrix equation G(λ)*X(λ) = F(λ).
glasol Approximate solution of the linear rational matrix equation X(λ)*G(λ) = F(λ).

[3] A. Varga, MatrixEquations.jl: Solution of Lyapunov, Sylvester and Riccati matrix equations using Julia. Zenodo: https://doi.org/10.5281/zenodo.3556867.

DescriptorSystems.jl

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