Program: 12To simulate a second order system and study the effect of
(a) P
(b) PI
(c) PD
(d) PID controller on the step response.
AIM: -
To implement P, PI, PD and PID controller on the step response of a feedback control system
APPARATUS: - Matlab software
THEORY: -
An automatic controller and basic control unit: An automatic controller compares the actual
value of the plant output with the reference input determines the deviation(error) and produces a
control signal that will reduce the error to zero or to a small value.
Fig.5.1 Block diagram representation
The manner in which the automatic controller produces the control signal is called the control action.
Controllers may be classified according to their control action as follows:
Proportion controller: (P Controller)
Here the controller will produce the control signal proportional to the error signal
The relationship b/w the o/p of the controller u(t) and actuating error signal e(t) is
u(t)=Kpe(t) ............ (1) or U(s)=KpE(S) or Kp=U(s)/E(s) where Kp= proportional gain
the proportional controller essentially an amplifier with an adjustable gain, the erroe signal E(s) is get
multiplied or amplified by the gain Kp to give controlled signal U(s)
Kp
Where R(s)-> reference signal
E(s)-> error signal
U(s)-> control signal
B(s)-> feed back signal
Integral controller: An integral controller will produce control signal proportional to the integral of the
error signal
An integral controller will produce a control signal proportional to the integral of the error
signal
u(t)= Ki ∫e(t)dt…......... (2)
where Ki is an adjustable constant
The Transfer function of the integral controller is
𝑈(𝑠) 𝐾𝐼
=
𝐸(𝑠) 𝑠
Proportional plus integral controller (PI Controller):
Kp+Ki/S
The PI controller produces control signal consisting of two terms
i) proportional to the actuating error signal
ii) proportional to the integral of actuating error signal
u(t)= Kpe(t)+ Ki ∫e(t)dt….......... (3)
where Ki is an adjustable constant called integral gain and Kp--> proportional gain
Proportional plus Derivative controller (PD Controller):
Kp+KdS
The PD controller produces control signal consisting of two terms
i) proportional to the actuating error signal
ii) proportional to the derivative of actuating error signal
Thus the control action of a PD Controller can be defined as
u(t)= Kpe(t)+Kd d/dt(e(t))
and its TF is
𝑈(𝑠)
= 𝐾𝑃 + 𝐾𝑑𝑠
𝐸(𝑠)
PD controller is used to improve the transient response of the system
Proportional plus integral plus Derivative controller(PID Controller):
Kp+Ki/S+KdS
The combination of proportional control action, integral control action, and derivative control action is
termed as PID controller
u(t)= Kpe(t)+ Ki ∫e(t)dt + Kd/dt(e(t))
PI-Controller:
Kp
Plant
Ki/S
Simulation
(i) Effect of P controller on response of a system
Consider the OLTF
1
𝐺𝐻 = 𝑠2 + 0.8𝑠
Program:
clc; hold off;
kv=[1 2 10];
t=0:0.1:15;
for i=1:3
kp=kv(i);
num= [kp];
den= [1 0.8 kp]
H=tf(num,den)
step(H,t);
hold on
end
legend('kp=1','kp=2','kp=10');
(ii) Effect of PI controller on response of a system
% Kp=1;
% G=1/s(s+1.4), H=1, Kp+Ki/s
clc; hold off;
kv=[0.1 0.2 0.3];
t=0:0.1:15;
for i=1:3
kp=1,
ki=kv(i);
num=[kp ki];
den= [1 1.4 kp ki];
H=tf(num,den)
step(H,t);
hold on
end
legend('ki=0.1','ki=0.2','ki=0.3');
(i) Effect of PD controller on response of a system
% Kp=1;
% G=1/s(s+0.8), H=1, Kp+Kd.s
clc; hold off;
kv=[0.1 0.2 0.3];
t=0:0.1:15;
for i=1:3
kp=1,
kd=kv(i);
num=[kd kp];
den= [1 0.8+kd kp];
H=tf(num,den)
step(H,t);
hold on
end
legend('kd=0.1','kd=0.5','kd=1.0');
Conclusion:
Settling
Parameter Rise time Overshoot S.S. Error Stability
Time
Small
Kp Decreases Increases Decreases Worse
change
Significant
Ki Decreases Increases Increases Worse
Decrease
Minor Minor Minor If Kd is small,
Kd No Change
Decrease Decrease Decrease Better
