EXPERIMENT NO 12
IMPLEMENT PI AND PD CONTROLLERS.
Program: PI CONTROLLERS
To manually calculate values and plot a graph of Time vs Output, follow
these steps:
1. Understand the formula:
Your PI controller updates the system output with:
output(i+1)=output(i)+dt×(control signal−output(i))
Where:
control signal = P + I
P = Kp * error(i)
I = Ki * integral
2. Manually compute values:
Using your system parameters:
Set initial output as 0
Calculate error (setpoint - output)
Compute P and I terms
Update output with the given equation
Let’s compute for a few first steps manually:
(a)Time=0.0
Error=setpoint-output=10-0=10
P=kp*error=(1*10)=10
Integral initial=0
I=ki*integral=0.1*0=0
Control signal=P+I=10+0=10
Initial output=0
(b)Time=0.1
Error=setpoint-output=10-0=10
P=kp*error=(1*10)=10
Integral =integral+error*dt=0+10*0.1=1
I=ki*integral=0.1*1=0.1
Control signal=P+I=10+0.1=10.1
Output(next)=output+dt(control signal-output)=0+(0.1*(10.1-0)=1.01
Tabular column:
Time P (Kp * I (Ki * Control Signal Output
Error Integral
(s) Error) Integral) (P + I) (next)
0.0 10.0 10.0 0.0 0.0 10.0 0.0
0.1 10.0 10.0 1.0 0.1 10.1 1.01
0.2 9.0 9.0 1.9 0.19 9.19 1.91
0.3 8.09 8.09 2.71 0.27 8.36 2.74
0.4 7.26 7.26 3.43 0.34 7.6 3.22
0.5 6.78 6.78 4.108 0.4108 7.19 3.617
. . . . . . .
. . . . . . .
1 5.41 5.41 7.02 0.702 6.112 4.74
2 …… ……. …… …….. ……… 5.70
Continue these steps until t = 20 sec to obtain all values.
3. Plot the graph:
Once you compute values, plot Time vs Output using any graphing tool (Excel,
MATLAB, or manually on paper). If using MATLAB, your given plot() command will
automatically create the graph.
PD CONTROLLERS
To manually calculate values and plot Time vs Output for your PD
Controller, follow these steps:
1. Understand the Update Formula:
Your PD controller updates the system output using:
output(i+1)=output(i)+dt×(control_signal−output(i))
where:
control_signal = P + D
Proportional Term → P = Kp * error(i)
Derivative Term → D =( Kd * (error(i) - prev_error)) / dt
2. Manually Compute Values:
Start with:
Initial output = 0
Calculate error (setpoint - output)
Compute P and D terms
Update output using the equation
Here’s how the first few steps will look:
(a)Time=0.0
Error=setpoint-output=10-0=10
P=kp*error=(1*10)=10
Prev_error =error
Pre_error=10
D=(kd*(error-prev_error))/dt=(0.1*(10-10))/0.1=0
Control signal=P+D=10+0=10
Initial output=0
(b)Time=0.1
Error=setpoint-output=10-0=10
P=kp*error=(1*10)=10
D=(kd*(error-prev_error))/dt=(0.1*(10-10))/0.1=0
Control signal=P+I=10+0=10
Output(next)=output+dt(control signal-output)=0+(0.1*(10-0)=1
(c)Time=0.2
Error=setpoint-output(next)=10-1=9
P=kp*error=(1*9)=9
D=(kd*(error-prev_error))/dt=(0.1*(9-10))/0.1=-1.0
Control signal=P+I=9-1=8
Output(next)=output+dt(control signal-output)=1+(0.1*(8-1)=1.7
Tabular column
Control
Time P (Kp * D ( Kd * (error(i) - Output
Error Signal
(s) Error) prev_error) )/ dt) (next)
(P+D)
0.0 10.0 10.0 0.0 10.0 0.0
0.1 10.0 10.0 0.0 10.0 1.0
0.2 9.0 9.0 -1.0 8.0 1.7
0.3 8.2 8.2 -0.8 7.4 2.5
Continue calculating these values manually until t = 20 sec.
3. Plot the Graph:
Once you compute Time vs Output values:
Use Excel, MATLAB, or graph paper to plot the points
MATLAB’s given plot() command will automatically generate the
graph from the computed values
EXPERIMENT NO 13
IMPLEMENT A PID CONTROLLER AND HENCE REALIZE
AN ERROR DETECTOR
To manually calculate and plot Time vs Output and Time vs Error,
follow these steps:
1. Understanding the Update Formula
Your PID controller updates the system output with:
output(i+1)=output(i)+dt×(control_signal−output(i))
Where:
Control Signal → P + I + D
Proportional Term → P = Kp * error(i)
Integral Term → I = Ki * integral
Derivative Term → D = Kd * derivative
2. Manually Compute Values
Start with:
Initial output = 0
Compute error (setpoint - output)
Calculate P, I, and D terms
Update output using the equation
Here are the first few steps manually:
(a)Time=0.0
Error=setpoint-output(initial)=10-0=10
P=kp*error=(1*10)=10
I=ki*integral=0.1*0=0
D=(kd*(error-prev_error))/dt=(0.01*(10-10))/0.1=0
Control signal=P+I+D=10+0+0=10
Initial output=0
(b)Time=0.1
Error=setpoint-output(initial)=10-0=10
P=kp*error=(1*10)=10
Integral =integral+error*dt=0+10*0.1=1
I=ki*integral=0.1*1=0.1
D=(kd*(error-prev_error))/dt=(0.01*(10-10))/0.1=0
Control signal=P+I+D=10+1+0=11
Output(next)=output+dt(control signal-output)=0+(0.1*(11-0)=1.1
(c)Time=0.2
Error=setpoint-output(next)=10-1.1=8.9
P=kp*error=(1*8.9)=8.9
Integral =integral(above value)+error*dt=1+8.9*0.1=1.89
I=ki*integral=0.1*1.89=0.189
D=(kd*(error-prev_error))/dt=(0.01*(8.9-10))/0.1=-0.11
Control signal=P+I+D=8.9+0.189-0.11=8.97
Output(next)=output+dt(control signal-output)=1.1+(0.1*(8.97-1.1)=1.88
(c)Time=0.3
Error=setpoint-output(next)=10-1.88=8.12
P=kp*error=(1*8.12)=8.12
Integral =integral(above value)+error*dt=1.89+8.12*0.1=2.702
I=ki*integral=0.1*2.702=0.2702
D=(kd*(error-prev_error))/dt=(0.01*(8.12-8.9))/0.1=-0.078
Control signal=P+I+D=8.12+0.2702-0.078=8.31
Output(next)=output+dt(control signal-output)=1.88+(0.1*(8.31-1.88)=2.52
Time P (Kp * I (Ki * D (Kd * Control Output
Error
(s) Error) Integral) Derivative) Signal (next)
0.0 10.0 10.0 0.0 0.0 10.0 0.0
0.1 10.0 10.0 1.0 0.0 11.0 1.1
0.2 8.9 8.9 1.89 -0.11 8.97 1.88
0.3 8.12 8.12 2.702 -0.078 8.31 2.52
Repeat these calculations until t = 20 sec to get full values.
3. Plot the Graphs
Once you compute values:
1. For Time vs Output
o Plot Time on the x-axis and Output on the y-axis.
o Use a blue line ('b') for output.
2. For Time vs Error
o Plot Time on the x-axis and Error on the y-axis.
o Use a green line ('g') for error.
EXPERIMENT NO 14
DEMONSTRATE THE EFFECT OF PI, PD AND PID
CONTROLLER ON THE SYSTEM RESPONSE
n=[1];
d=[1 10 20];
kp_pd=500;
kd_pd=10;
c_pd=tf([kp_pd kd_pd],1);
p=tf(n,d);
t1=feedback(c_pd*p,1);
t=0:0.01:2;
subplot(3,1,1);
step(t1,t);
grid;
title('pd controller');
kp_pi=30;
kd_pi=70;
c_pi=tf([kp_pi kd_pi],[1 0]);
p=tf(n,d);
t2=feedback(c_pi*p,1);
subplot(3,1,2);
step(t2,t);
grid;
title('pi controller')
kp_pid=500;
ki_pid=400;
kd_pid=50;
c_pid=tf([kd_pid kp_pid ki_pid],[1 0]);
p=tf(n,d);
t3=feedback(c_pid*p,1);
subplot(3,1,3);
step(t3,t);
grid;
title('pid controller');
figure(2);
step(t1,t2,t3,'r---');
title('comparision of all the response');
legend({'pd controller','pi controller','pid controller'},'Location','best');
legend('boxoff');
grid;
Output:
System Transfer Function:
G(s)=1/s^2+10s+20
1. PD Controller:
Kp=500 or K_p = 500,
Kd=10 or K_d = 10
Controller transfer function:
CPD(s)=Kp+Kds
=500+10s
Open-loop transfer function:
TOL(s)=CPD(s)⋅G(s)
=(10s+500)/s^2+10s+20
Closed-loop transfer function:
TCL(s)=TOL(s)/1+TOL(s)
=10s+500/s^2+20s+(520)
2. PI Controller:
Kp=30 or K_p = 30,
Ki=70 or K_i = 70
Controller transfer function:
CPI(s)=Kp+Kis=30+70s
Open-loop transfer function:
TOL(s)=CPI(s)⋅G(s)
=(30s+70)/s(s^2+10s+20)
Closed-loop transfer function:
TCL(s)=TOL(s)/1+TOL(s)
=(30s+70)/s^3+10s^2+50s+70
3. PID Controller:
Kp=500 or K_p = 500, Ki=400 or K_i = 400, Kd=50 or K_d = 50
Controller transfer function:
CPID(s)=Kp+Kis+Kds
=500+400s+50s
Open-loop transfer function:
TOL(s)=CPID(s)⋅G(s)
=(50s^2+500s+400)/s(s^2+10s+20)
Closed-loop transfer function:
TCL(s)=TOL(s)/1+TOL(s)
=(50s^2+500s+400)/s^3+60s2+520s+400
