RATRIX : A RATional matRIXcalculator for computer aided analysis and

synthesis of linear multivariable control systems

P. Tzekis, N.P. Karampetakis and A.I. Vardulakis

Department of MathematicsAristotle University of Thessaloniki

Thessaloniki 54006, Greece

•Why we develop this program ?

•Program description.

•Examples of use.

Overview

Symbolic computation programs

Why we develop this Why we develop this program ?program ?

Programs that handle both numbers and symbols such as Mathematica, Maple, Matlab, MACSYMA, Reduce,..

Advantages of symbolic computation programs

•Symbolic storage.(Variables can be stored in exact form I.e. 1/3 instead of 0.333)

•Inbuilt procedures(Existing procedures for special areas of mathematics)

•Programming Language(High Level programming languages allowing procedures to be written)

Why we develop this Why we develop this program ?program ?

Disadvantages of symbolic computation programs

•Large size of memory they use.

•Slow speed they have.

•No existing procedures for the study of rational matrices and its applications in analysis and design of control systems.(Except the polynomial toolbox of Matlab, created by PolyX)

•Require knowledge of the procedures from the user.

Why we develop this Why we develop this program ?program ?

How to overcome these disadvantages ?

Why we develop this Why we develop this program ?program ?

User friendly environmentProcedures for rational matrices and control

RATRIX

Description of the main window ?

Program Program DescriptionDescription

Shortcuticons

Menu

Matrices

Procedures

Mapleenvironment

Results

Program Program DescriptionDescription

Description of the main procedures

Program Program DescriptionDescription

Description of the main procedures

Program Program DescriptionDescription

You can save your session !

Program Program DescriptionDescription

You can use the kernel of Maple !

Program Program DescriptionDescription

Benefits of RATRIX

• The user friendly Windows based interface.• Internal use of the powerful kernel of Maple.• Is working over the four well known rings.(polynomials, proper rational functions, proper and Shur stable rational functions and proper and Hurwitz stable rational functions)• The user can work both on the windows application environment (beginner) and the standard Maple environment (advanced).

Creation of a Creation of a matrixmatrix

EXAMPLESEXAMPLESCreate a matrix

Complete the entries of the Complete the entries of the matrixmatrix

Use the icons

Find the Smith McMillan form Find the Smith McMillan form in in ΩΩSS

Smith McMillan Form

The Smith Form

The name of theprocedure

The left transforming matrix The left transforming matrix U(s) is proper and Hurwitz U(s) is proper and Hurwitz StableStable

We can check that the We can check that the condition T=USV is satisfied condition T=USV is satisfied

A right MFD of T(s) in A right MFD of T(s) in ΩΩSS

MFD

The McMillan Degree of T(s) The McMillan Degree of T(s)

McMillanDegree

Find a polynomial matrix Find a polynomial matrix solution of D1*X+N1*Y=T solution of D1*X+N1*Y=T

Dioph. Equ.

Define the matrices A,B,C Define the matrices A,B,C

Find the polynomial solution Find the polynomial solution of A*X+B*Y=C of A*X+B*Y=C

Check if the solutions X,Y Check if the solutions X,Y satisfy the condition satisfy the condition A*X+B*Y=CA*X+B*Y=C

Find a Hurwitz stable Find a Hurwitz stable stabilizing compensator R for stabilizing compensator R for the matrix A.the matrix A.

StabilizingCompensators

The compensator is too The compensator is too arbitrary. arbitrary.

We select specific values for We select specific values for the arbitrary parameters. the arbitrary parameters.

and the solution is …. and the solution is ….

Find the finite decoupling Find the finite decoupling zeros of the PMD defined by zeros of the PMD defined by A,B,C. A,B,C.

DecouplingZeros

You can save your session. You can save your session.

with the extension .con with the extension .con

Conclusions

•The paper has presented a user-friendly

Windows based application program for the

manipulation of rational matrices and the

solution of basic Analysis and Synthesis

problem of linear systems.

•This program can be used for educational,

research and industrial uses.