PUBLICATIONS - National Conferences
| Title: |
Development of a Robust Eigenvalue Assignment Toolbox Using a Kharitonov Based Approach |
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Author: |
M T Söylemez and N Munro |
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Year: |
1997 (10) |
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Conference: |
IEE Colloquium |
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Place: |
Savoy Place, London |
| Digest No: | 97/380 |
| Summary: | Since Kharitonov published his seminal paper in 1978 \cite{DE_14_2086}, there has
been a lot of work considering the analysis of robustness in uncertain systems based on Kharitonov's approach \cite{B.BARMISH.NEWTO,CSM_9_7}. However,
interest in the synthesis approach has been very limited. This paper presents the
development of a toolbox for the synthesis of Robust Eigenvalue Assignment Controllers using a
Kharitonov based approach. For a given multi-input multi-output system \begin{equation} \begin{array}{rcl} \dot{x} &=& A(\mbox{\bf{q}}) x + B(\mbox{\bf{q}}) u \\ y &=& C(\mbox{\bf{q}}) x \end{array} \label{eqsys} \end{equation} with the parametric uncertainty vector {\bf q}, S\o ylemez and Munro \cite{PIEE_soylemez} have defined a cost function which increses as the perturbation from the nominal desired eigenvalues increases. Moreover, this cost function is always less than one if a pre-defined performance criteria is satisfied and greater than one otherwise. Munro and S\o ylemez \cite{PC_96_1332} have also demonstrated the use of symbolic algebra tools in this context, exploiting a newly found formula for the general solution of the state-feedback eigenvalue assignment problem \cite{IJ_soylemez}. A toolbox for finding permisible robust solutions to the pole assignment synthesis problem and for analysing these solutions has been developed as a result of the above mentioned work. Here, the definition of this problem is given using a symbolic algebra language (namely $Mathematica^{TM}$), and a parameterised general solution for the feedback compensators that assign the nominal eigenvalues of the closed-loop system is automatically derived. The resulting closed-loop system characteristic polynomial depends on the parametric uncertainties ($\mbox{\bf{q}}$) and on some of the free controller variables ($\mbox{\bf{k}}$), along with the frequency variable ($s$). This polynomial is then converted to C$++$ code and transferred to an optimization routine, where with the help of a GA procedure an acceptable robust solution is obtained. An interesting point with respect to the optimization algorithm is that instead of calculating the cost function for each and every point in the ball defined by the perturbation vector, only a few points need to be checked. This, essentially reduces the synthesis problem to an analysis problem. The combined use of symbolic algebra and C$++$ brings together the power of symbolic algebra and efficiency (ie speed) \& object-oriented programming capabilities of C$++$. Therefore, it is possible to obtain solutions in a very short time. The results produced by the optimization program are then passed to $Mathematica$ for analysis purposes. It is then possible to see the pole-spread, robust root locus or time-response of the multivariable uncertain system being considered. |
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