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 Control System Design

 Introduction

Maple has tools for linear control system design in the DynamicSystems package. You can

 • Work with transfer functions, state space models, or differential equations
 • Linearize systems
 • Analyze the controllability, observability, phase and gain margin, and more
 • Generate control plots, including Bode, root-locus and Nyquist plots
 • Work symbolically or numerically

In this example, we will calculate the controllability matrix of a model of a DC motor, and generate a root-locus plot.

 DC Motor System

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 Controllability Matrix and Root-Locus Plot

 > $\mathrm{eq_num}≔\mathrm{eval}\left(\mathrm{eq_sym},\mathrm{params}\right):\phantom{\rule[-0.0ex]{0.0em}{0.0ex}}\phantom{\rule[-0.0ex]{0.0em}{0.0ex}}$
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 > $\mathrm{ControllabilityMatrix}\left(\mathrm{sys_num}\right)$
 $\left[\begin{array}{ccc}{2}& {-4}& \frac{{1999}}{{250}}\\ {0}& {0}& \frac{{1}}{{5}}\\ {0}& \frac{{1}}{{5}}& {-}\frac{{3}}{{5}}\end{array}\right]$ (1)
 > $\mathrm{RootLocusPlot}\left(\mathrm{sys_num}\right)$

 Symbolic Controllability Matrix

You can also work symbolically, and maintain the parameter relationships present in the original equation system. Here, for example, we generate a symbolic controllability matrix.

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 > $\mathrm{ControllabilityMatrix}\left(\mathrm{sys_sym}\right)$
 $\left[\begin{array}{ccc}\frac{{1}}{{\mathrm{L}}}& {-}\frac{{\mathrm{R}}}{{{\mathrm{L}}}^{{2}}}& \frac{{{\mathrm{R}}}^{{2}}}{{{\mathrm{L}}}^{{3}}}{-}\frac{{{\mathrm{K}}}^{{2}}}{{{\mathrm{L}}}^{{2}}{}{\mathrm{J}}}\\ {0}& {0}& \frac{{\mathrm{K}}}{{\mathrm{J}}{}{\mathrm{L}}}\\ {0}& \frac{{\mathrm{K}}}{{\mathrm{J}}{}{\mathrm{L}}}& {-}\frac{{\mathrm{K}}{}{\mathrm{R}}}{{\mathrm{J}}{}{{\mathrm{L}}}^{{2}}}{-}\frac{{\mathrm{b}}{}{\mathrm{K}}}{{{\mathrm{J}}}^{{2}}{}{\mathrm{L}}}\end{array}\right]$ (2)
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 Applications