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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
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•Why we develop this program ?
•Program description.
•Examples of use.
Overview
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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,..
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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 ?
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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 ?
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How to overcome these disadvantages ?
Why we develop this Why we develop this program ?program ?
User friendly environmentProcedures for rational matrices and control
RATRIX
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Description of the main window ?
Program Program DescriptionDescription
Shortcuticons
Menu
Matrices
Procedures
Mapleenvironment
Results
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Program Program DescriptionDescription
Description of the main procedures
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Program Program DescriptionDescription
Description of the main procedures
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Program Program DescriptionDescription
You can save your session !
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Program Program DescriptionDescription
You can use the kernel of Maple !
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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).
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Creation of a Creation of a matrixmatrix
EXAMPLESEXAMPLESCreate a matrix
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Complete the entries of the Complete the entries of the matrixmatrix
Use the icons
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Find the Smith McMillan form Find the Smith McMillan form in in ΩΩSS
Smith McMillan Form
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The Smith Form
The name of theprocedure
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The left transforming matrix The left transforming matrix U(s) is proper and Hurwitz U(s) is proper and Hurwitz StableStable
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We can check that the We can check that the condition T=USV is satisfied condition T=USV is satisfied
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A right MFD of T(s) in A right MFD of T(s) in ΩΩSS
MFD
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The McMillan Degree of T(s) The McMillan Degree of T(s)
McMillanDegree
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Find a polynomial matrix Find a polynomial matrix solution of D1*X+N1*Y=T solution of D1*X+N1*Y=T
Dioph. Equ.
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Define the matrices A,B,C Define the matrices A,B,C
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Find the polynomial solution Find the polynomial solution of A*X+B*Y=C of A*X+B*Y=C
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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
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Find a Hurwitz stable Find a Hurwitz stable stabilizing compensator R for stabilizing compensator R for the matrix A.the matrix A.
StabilizingCompensators
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The compensator is too The compensator is too arbitrary. arbitrary.
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We select specific values for We select specific values for the arbitrary parameters. the arbitrary parameters.
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and the solution is …. and the solution is ….
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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
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You can save your session. You can save your session.
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with the extension .con with the extension .con
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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.