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Polynomial Methods for Systems, Signals and Control
Polynomial Toolbox 2.0 for MATLAB 5
The Polynomial Toolbox 2.0 is a package for systems, signals and control analysis and design based on advanced polynomial methods. It consists of as many as 222 M-files in MATLAB code and is easy to use.
Polynomials and polynomial matrices play an important role in linear system theory. They do not only arise because polynomials cannot be avoided, but also because from first principles multivariable linear systems are modeled by sets of differential equations in the input u and the output y of the form
where A and B are polynomial matrices in the differential operator d/dt (or the delay operator for discrete-time systems). Polynomial matrix models do not replace state space and frequency domain descriptions but provide a powerful additional tool.
To define polynomial matrices is as simple as typing
|» A=[1 1+s; 1-s 2*s];|
|» B=[s 0; 0 s];|
It is even easier to compute with them
|1 + s 1 + s|
|1 - s 3s|
|s s + s^2|
|s - s^2 2s^2|
|-1 + 2s + s^2|
For larger matrices the Polynomial Matrix Editor is available.
Polynomial Matrix Editor
The Polynomial Toolbox implements new original algorithms that are fast and reliable. This includes linear matrix polynomial equation solvers based on Sylvester matrices, the application of FFT for rank, determinant and other functions, a variety of new algorithms for spectral factorization and much more.
The modeling of single-input single-output LTI systems often leads to differential equations
with A and B polynomials or, equivalently, to the transfer function
of the system. For multi-input multi-output systems A and B become polynomial matrices and the transfer matrix is expressed in polynomial matrix fraction (PMF) form
The Polynomial Toolbox provides many macros for PMFs such as conversion between left and right fractions (lmf2rmf, rmf2lmf), properties testing (isprime, ispropper, isstable), and zero-pole plots (zpplot).
Feedback control system design by polynomial methods naturally introduces linear polynomial matrix equations such as
is the controller transfer matrix. The Polynomial Toolbox offers numerous solvers for the equations suitably named axbyc, xaybc, axybc, and alike.
Next to the linear equations, optimum design problems naturally call for quadratic equations with polynomial matrices such as spectral, J-spectral or even nonsymmetrical factorizations (spf, spcof, fact).
The Polynomial Toolbox offers simple programs for classical analysis. Its built-in convertors make all the tools of Control System Toolbox available for systems described by PMFs. In addition a wide range of macros is provided to test robustness of various kinds for systems with parametric uncertainties, including single parameter stability margins (stabint), interval polynomials (kharit, khplot), and polytopic uncertainties (ptopplot, ptopex, etc.)
Robust stability analysis for interval polynomials
Frequency domain solutions of many famous and proven design methods are directly provided, including
|dsshinf||sub- and optimal H-infinity design|
|mixeds||mixed sensitivity problem|
|stab||all stabilizing controllers|
Many other design routines can easily be developed based upon the basic polynomial matrix macros.
Numerous convertors enable direct cooperation with the Control System Toolbox
|lti2lmf, lti2rmf||LTI objects to PMF|
|ss, tf, zpk||PMF to LTI objects|
|lmf2dss, rmf2dss||PMF to descriptor systems|
|dss2lmf, dss2rmf||descriptor systems to PMF|
and the Symbolic Math Toolbox
|sym||polynomial matrix to symbolic format|
|sym2pol||symbolic format to polynomial matrix|
A SIMULINK 2 block set for LTI systems described by polynomial matrix fractions is also provided.
Users of the Polynomial Toolbox include control engineers involved in control systems analysis and design, communication engineers with an interest in filter design, and university teachers engaged in a variety of courses in linear systems, signals, and control.
© Copyright 1998 by PolyX, Ltd. All rights reserved. Polynomial Toolbox is a trademark of PolyX, Ltd. MATLAB and SIMULINK are registered trademarks of The MathWorks, Inc. Other product or brand names are trademarks or registered trademarks of their respective holders.
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