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//Example 4.8
//Equivalent discrete controller for DC motor speed control.
//------------------------------------------------------------------
//NOTE THAT-- The system response (continuous) to sampled control
//input depends on
//the sampling time set for continuous signal in SIMULATION.
//In this example we consider sampling period of 0.009 sec
//to represent continuous time signal.
//------------------------------------------------------------------
xdel(winsid())//close all graphics Windows
clear;
clc;
//------------------------------------------------------------------
// Continuous time system and controller
// System transfer function
s=%s;
num=[45 0];
den=[45 14 1]
Nms=poly(num,'s','coeff');
Dns=poly(den,'s','coeff');
Gp=syslin('c',Nms/Dns); //system transfer function
// Controller
numDa=[6 1];
denDa=[0 1]
Nms=poly(numDa,'s','coeff');
Dns=poly(denDa,'s','coeff');
sysD=syslin('c',1.4*Nms/Dns); //controller transfer function
//Closed loop responses
num=[1 0];
den=[1 0];
Nms=poly(num,'s','coeff');
Dns=poly(den,'s','coeff');
H=syslin('c',Nms/Dns)
sysDa=Gp*sysD/.H;
//step response and control input
t=0:0.009:5;
yt=csim('step',t,sysDa); //step response
figure(0)
plot2d(t,yt,1)
Gu=sysD/(1+Gp*sysD);
ut=csim('step',t,Gu); //control input
figure(1)
plot2d(t,ut,1)
//------------------------------------------------------------------
sys=tf2ss(Gp); //state space model of the system
con=tf2ss(sysD); //controller state space model
// discrete-time time system and controller
//Discretize the system and control with sampling time Ts=0.07
// using Bilinear Transform
Ts=0.07;
sysDd=cls2dls(sys,Ts); // discrete-time system state space model
conDd=cls2dls(con,Ts); // discrete-time controller state space model
//Pulse transfer function of system
Gpz=ss2tf(sysDd);
//Pulse transfer function of controller
Gcz=ss2tf(conDd);
//Closed loop response
Gz=Gpz*Gcz/(1+Gpz*Gcz)
//Control input pulse transfer function
Guz=Gcz/(1+Gpz*Gcz)
T=0:Ts:5;
r=ones(1,length(T));
yd=flts(r,Gz);............//Discrete respnse to discrete input
ud=flts(r,Guz); //Discrete Control input
//continuous response for digital input
t=0:0.009:5;
k=0;
for i=1:length(yd)
for j=1:8
if (k+j)>length(t) then
break
else
YD(1,k+j)=yd(i);
end
end
k=k+j;
end
yt=csim(1-YD,t,Gp*sysD);
scf(0)
plot2d(t,yt,5);
scf(1)
plot2d2(T,ud,5);
//------------------------------------------------------------------
//Discretize the system and control with sampling time Ts=0.035
// using Bilinear Transform
Ts=0.035;
sysDd=cls2dls(sys,Ts); // discrete-time system state space model
conDd=cls2dls(con,Ts); // discrete-time controller state space model
Gpz=ss2tf(sysDd); //Pulse transfer function of system
Gcz=ss2tf(conDd); //Pulse transfer function of controller
//Closed loop response
Gz=Gpz*Gcz/(1+Gpz*Gcz)
//Control input pulse transfer function
Guz=Gcz/(1+Gpz*Gcz)
T=0:Ts:5;
r=ones(1,length(T));
yd=flts(r,Gz);............//Discrete respnse to discrete input
ud=flts(r,Guz); //Discrete Control input
t=0:0.009:5;
k=0;
for i=1:length(yd)
for j=1:4
if (k+j)>length(t) then
break
else
YD(1,k+j)=yd(i);
end
end
k=k+j;
end
yt=csim(1-YD,t,Gp*sysD);
scf(0)
plot2d(t,yt,2);
scf(1)
plot2d2(T,ud,2);
scf(0)
//Title, labels and grid to the figure
exec .\fig_settings.sci; //custom script to set the figure properties
title('Comparision plots of Speed-control system with continuous...
and discrete controllers','fontsize',3)
xlabel('Time t (sec.)','fontsize',2)
hl=legend(['Continuous time','Discrete-time, Ts=0.07 s'...
,'Discrete-time, Ts=0.035 s'],4);
scf(1)
//Title, labels and grid to the figure
exec .\fig_settings.sci; //custom script to set the figure properties
title('Comparision plots of Speed-control system with continuous...
and discrete controllers','fontsize',3)
xlabel('Time t (sec.)','fontsize',2)
hl=legend(['Continuous time','Discrete-time, Ts=0.07 s',...
'Discrete-time, Ts=0.035 s']);
//------------------------------------------------------------------
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