1. A DESCRIPTOR SYSTEMS PACKAGE FOR MATHEMATICA
A.I. Vardulakis, N. P. Karampetakis, E. Antoniou, P. Tzekis and S. Vologiannidis
Department of Mathematics
Aristotle University of Thessaloniki
Thessaloniki 54006, Greece

2. Outline of the presentation
Control System Professional
Polynomial Control Systems
Descriptor Control Systems

3. Mathematica and Control Control System Professional
Control System Professional handles linear systems described by state-space equations and proper transfer functions.
Time-Domain Response Analysis
System Interconnections
Controllability and Observabillity
Realizations Construction and Conversion
Feedback Control Systems Design
Optimal Control Systems Design
Linearization tools
Control System Professional is a collection of Mathematica programs that extend Mathematica to solve a wide range of control system problems. Its main features are:
Time-Domain Response Analyses
Provides symbolic analysis capabilities for solving state equations
Simulates system behavior numerically
Examples include step, impulse, and ramp response functions
System interconnections
Serial (cascade), parallel, and feedback interconnections
Controllability and Observability
Determines and computes controllability and observability
Computes the controllability and observability matrices and Gramians
Solves the continuous and discrete matrix Lyapunov equations
Computes the dual to the input system
Realizations Construction and Conversion
Selects controllable and observable subsystems; finds the minimal realizations
Computes the Kalman and Jordan canonical forms
Constructs internally balanced realizations
Cancels common zero-pole pairs in transfer functions
Converts between equivalent realizations with similarity transforms
Feedback Control Systems Design
Using Ackermann's formula or the robust algorithm, computes the feedback for the eigenvalue assignment
Finds the estimator for state reconstruction
Optimal Control Systems Design
Designs the optimal feedback for linear system and quadratic cost functions (i.e., solves continuous and discrete linear-quadratic optimal control problems)
Finds the optimal solution to the output regulator problem
Solves the continuous and discrete algebraic Riccati equations
Finds the discrete equivalent to the continuous regulator
Finds the Kalman estimator and filter for stochastic systems
Collection of Classical Methods
Frequency response plotting routines handle arbitrary transfer functions and rational polynomial transfer functions
Generates Bode, Nyquist, and Nichols plots
Singular value plot reduces complexity for MIMO systems
Plots and animates the root loci, providing information on their direction and evolution
Nonlinear Control Systems
Provides a linearization tool for constructing linear models of nonlinear systems
Makes rational polynomial approximations of nonlinear systems

4. Mathematica and Control Polynomial Control Systems
Polynomial Control Systems developed by Prof. Munro handles the general class of polynomial matrix descriptions (PMDs).
Model transformations
System analysis
System design
Another package that deals with control problems is “Polynomial Control Systems” developed by Professor Munro.
It deals with the general class of polynomial matrix descriptions.
Model transformations
State space, transfer-function (matrix), left or
right matrix-fraction form, and Rosenbrock's system
matrix in state-space or polynomial form.

5. Objectives of the descriptor systems package
Extend the functionality of the Control Systems Professional package in order to handle descriptor state space representations and improper transfer functions.
Manipulation of polynomial and rational matrices
Introduction of descriptor state space systems as data objects
Extension of the functions of CSP concerning
System analysis
Time-Domain Response Analysis
Synthesis and design techniques
Maintain compatibility with the existing infrastructure of Control Systems Professional and Polynomial Control Systems.
The descriptor systems package is an addon to control systems professional developed by the authors in cooperation with Wolfram Research.
The objectives of the our package are
To extend the functionality of Control systems Professional in order to handle descriptor state space representations.
Specifically
we created a subpackage concerning the manipulations of polynomial and rational matrices over several rings
We introduced descriptor state space data objects in Mathematica
We extended many of the functions of control system professional to handle descriptor state space objects. Therefore, many of the functions of Control systems Professional concerning SA
TDR
SADT
are also available for descriptor state space systems.
While developing this project, we tried to maintain compatibility with the existing infrastructure of CSS and that of PCS in order to provide the end-user with a unified experience.
CSP
PCS
DCS

6. Manipulation of polynomial and rational matrices
New functions for the study of rings of rational functions with poles in a prescribed region of the complex plane as well as for rational matrices with entries coming from these rings
the ring of rational functions with no poles in the complex plane (polynomials) (ForbiddenPolesArea->FiniteComplex)
the ring of rational functions with no poles at infinity (proper functions) (ForbiddenPolesArea->InfinityPoint)
the ring of rational functions with no poles in the extended right half complex plane (proper and Hurwitz stable rational functions) (ForbiddenPolesArea->HurwitzStable)
the ring of rational functions with no poles outside the unit circle (proper and Schur stable rational functions)
(ForbiddenPolesArea->SchurStable)
One of the most important parts of the project was the implementation of functions for the study of rings of rational functions with poles in a prescribed region of the complex plane as well as for rational matrices with entries coming from these rings
The rings that have been implemented up to now are .
All rings are described in out function using the option forbidddd in the parentheses
We will see some of those functions in the following examples.
Please note that it is possible for the interested user to easily define other rings.
a) determine whether a rational function (or a matrix) belongs to a particular ring b) calculate the quotient and remainder of a division between two rational functions over a particular ring, c) compute the greatest common divisor and least common multiple of functions (rational matrices) over a particular ring, d) check whether two rational functions (matrices) are coprime over a particular ring, e) compute structural invariants of a rational matrix such as the Smith form of a rational matrix in a particular ring, the Smith - McMillan form of a rational matrix using unimodular matrices over a particular ring, f) determine particular and general solutions of rational matrix Diophantine equations over specific rings

7. Manipulation of polynomial and rational matrices Problems studied over different rings
Division between two rational functions
Greatest common divisor and least common multiple
Coprimeness
Smith - McMillan form
Solutions of rational matrix Diophantine equations

8. Descriptor State Space Models
A brief reminder of the definition of descriptor system representations.
A Descriptor state space system data object named “DescriptorStateSpace” has been added in Mathematica.
Transformations between Transfer functions, Descriptor State Space and PMDs are available

9. Descriptor State Space Models The descriptor state-space model of a simple RLC circuit.
Consider the following simple RLC circuit (Dai 1989) + - -V s ( t ) V C L R I(
In the remaining part of this paper we will present some simple examples of the functionality of the descriptor systems package.
Consider the following RLC circuit.
Where R, L, C stand for resistor, inductor and capacity quantities respectively. The corresponding voltages are named Vr, Vl, Vc
R, L, C stand for the resistor, inductor and capacity quantities respectively. VS is the voltage source (control input), and VR, VL, VC are the corresponding voltages.

10. Descriptor State Space Models The descriptor state-space model of a simple RLC circuit. Definition
a={{0,1,0,0},{1/C,0,0,0},{-R,0,0,1},{0,1,1,1}};
b={{0},{0},{0},{-1}};
c={{0,0,1,0}};
dss=DescriptorStateSpace[e, a, b, c] B E A
We begin by defining the appropriate matrices names me,ma,mb, and mc that correspond to the matrices E, A, B, C and feed them as input to the descriptorstatespace function.
The ouput of the function is a typesetted object. The upper right matrix corresponds to E, the one to its left to A etc.
In order to transform the descriptor state space system to a transferfunction object the Transferfunction function is used. D C
TransferFunction[dss]

11. Descriptor State Space Models The descriptor state-space model of a simple RLC circuit.
Using the function EquationForm that takes as input the descriptor state space system, and the names of the pseudostate variables we want to use, Mathematica displays the descriptor StateSpace objects in the familiar descriptor state-space equations.
The function Weirstrass Canonical form, returns the Weirstrass Canonical form of the system where we can see that the system has no poles at infinity.
Note that as input we provided the RLC system with R=L=C=1

12. System Analysis Properties
Determination of the structural invariants and properties of descriptor systems
controllability, reachability and observability matrices
finite and infinite decoupling zeros
finite and infinite system poles and zeros
finite and infinite invariant zeros
finite and infinite transmission poles and zeros
Controllability, reachability, observability, detectability, stabilizability, stability tests.
The next section presents through some examples the functions available for descriptor systems concerning system analysis properties.
Briefly we denote some of the key functions.
We have implemented functions concerning the

13. Descriptor State Space Models Analysis of the descriptor state-space model of a simple RLC circuit. Zeros-Poles
. The Smith McMillan form of the pencil
McMillanDecomposition[s*e - a, s][[1]]//Factor
. The Smith McMillan form of the pencil at infinity
McMillanDecomposition[s*e - a, s, ForbiddenPolesArea -> InfinityPoint][[1]]
The Smith McMillan decomposition functions retuns the Smith McMillan form of the matrix over several rings of rational matrices, as well as the transforming matrices.
Using the function without the option ForbiddenPolesArea, returns the normal Smith McMillan form over the the ring of rational functions.
The Smiith McMillan form at infinity can be found using the option ForbiddenPolesArea -> InfinityPoint.
Note that the pencil sE-A has no zeros at infinity and thus the system has no poles at infinity as we have already mentioned before (Weierstrass form).
No zeros at infinity

14. Descriptor State Space Models Analysis of the descriptor state-space model of a simple RLC circuit. Zeros-Poles
Infinite transmission poles-zeros (Infinite poles-zeros of )
tf=TransferFunction[dss];
McMillanDecomposition[tf[s], s, ForbiddenPolesArea -> InfinityPoint][[1]]
One transmission zero at infinity of order 2
Infinite input-decoupling zeros (Infinite zeros of )
sc = AppendRows[s*e-a, b] ;
McMillanDecomposition[sc, s, ForbiddenPolesArea -> InfinityPoint][[1]]
The invariants described above, are implemented in the sequel by certain functions.....
No decoupling zeros at infinity

15. Descriptor State Space Models Analysis of the descriptor state-space model of a simple RLC circuit. Controllability-Observability
Consider the RLC circuit with R=L=C=1.
dssrlc=dss/.{R->1, L->1, C->1};
Cm=ControllabilityMatrix[dssrlc] Controllable[dssrlc]

16. Time domain responses
Symbolic approach (StateResponse and OutputResponse)
When supplied the input and the initial conditions, attempts to calculate the state and output response respectively.
Simulation based approach (SimulationPlot)
Approximate numerical solutions.
Provides symbolic analysis capabilities for solving state equations
Simulates system behavior numerically
Examples include step, impulse, and ramp response functions

17. Time domain responses Response of the descriptor state-space model of a simple RLC circuit. State Response
dssrlc=dss/.{R->0.5, C->0.4, L->1};
x0 = {0,0,0,0};ut = {DiracDelta[t]};
xd=StateResponse[dssrlc,ut,t,InitialConditions->x0]//N
is the unit step function

18. Time domain responses Response of the descriptor state-space model of a simple RLC circuit. State Response
Plot[Evaluate[xd/.DiracDelta->Gaussian],{t,-0.01,12},
PlotStyle->{RGBColor[0,1,0],RGBColor[1,0,0],RGBColor[0,0,1],
RGBColor[1,0,1]},PlotRange->All] I VL VC VR

19. Design Synthesis Techniques
Stabilizing compensator design, asymptotic tracking, model matching and disturbance rejection.
Descriptor system interconnections such as series, parallel, feedback and generic interconnection.
Pole assignment techniques

20. Design Synthesis Techniques Pole assignment of a simple RLC circuit.
Assign the poles of the system to {p1, p2} by constant state feedback
f=StateFeedbackGains[dss, {p1, p2}, Method->FiniteDescriptorPoleAssignment]
McMillanDecomposition[s*e-(a+b.f), s][[1]]//Factor
The pencil corresponding to the closed loop system.

21. Outline of the presentation
Control System Professional
Polynomial Control Systems
Descriptor Control Systems
Manipulation of polynomial and rational matrices
Extension of the functions of CSP concerning
System analysis
Time-Domain Response Analysis
Synthesis and design techniques

22. A DESCRIPTOR SYSTEMS PACKAGE FOR MATHEMATICA
Acknowledgements
Thanks to Wolfram Research and especially to Dr. Igor Bakshee for their interest and valuable help.
Further development
Advanced Numerical methods for descriptor control systems.