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Control System Design

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Introduction

 

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

 

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Work with transfer functions, state space models, or differential equations

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Linearize systems

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Analyze the controllability, observability, phase and gain margin, and more

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Generate control plots, including Bode, root-locus and Nyquist plots

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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

 

restart: withDynamicSystems:

eq_symLi.t+R⋅it=vt−K⋅θ.t,Jθ..t+b⋅θ.t+Ks⋅θt=K⋅it :

paramsJ=0.1, b=0.1, K=0.01, R=1, L=0.5, Ks = 1:

 

Controllability Matrix and Root-Locus Plot

 

eq_numevaleq_sym,params:

sys_num  StateSpaceeq_num, inputvariable=vt, outputvariable=thetat,it:

ControllabilityMatrixsys_num

2−41999250001501535

(1)

RootLocusPlotsys_num

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.

sys_sym  StateSpaceeq_sym, inputvariable=vt, outputvariable=thetat,it:

ControllabilityMatrixsys_sym

1LRL2R2L3K2L2J00KJL0KJLKRJL2bKJ2L

(2)

Applications

Robot Arm Code Generation

LQR Controller for Inverted Pendulum