Page 125 - Mechatronics with Experiments

CLOSED LOOP CONTROL 111

figure(1); bode(num1,den1); title(’P control:
D(s) = Kp = 10.0’); grid on;
figure(2); bode(num2,den2); title(’D control:
D(s) = Kd s = 1.0 s’); grid on;
figure(3); bode(num3,den3); title(’I control:
D(s) = Ki/s = 1.0/s’);grid on;
figure(4); bode(num4,den4); title(’PD control:
D(s) = Kp+Kd s = 10.0+1.0 s’);grid on;
figure(5); bode(num5,den5); title(’PI control:
D(s) = Kp+Ki/s = 10.0+1.0/s’); grid on;
figure(6); bode(num6,den6); title(’PID :D(s)=(Kp+Kds)
(Kp+Ki/s)=(10.0+1.0s)(10+1.0/s)’);grid on;

Using frequency domain methods, the PID controller gains can be selected to shape
the Bode plots (frequency response) of the controller in order to give the desired shape
to the combined Bode plot of the controlled process and the controller. The desired shape
of the combined Bode plot, that is the frequency response of the loop transfer function
including the controller and the process, is primarily characterized by

1. stability: the gain margin (the magnitude of the plot at the frequency when the phase
◦
is −180 ), the phase margin (the difference of the phase of the frequency response
◦
from the −180 at the frequency that the magnitude crosses 0 dB), i.e., if the phase
◦
◦
of the frequency response at that frequency is −150 , the phase margin is +30 ),
2. speed of response (bandwidth): cross-over frequency (the frequency at which the
magnitude plot crosses 0 dB) which determines the speed of response (bandwidth)
of the closed loop system,
3. steady-state response and disturbance rejection: gain of the loop transfer function
at the low frequency range (the higher the low frequency gain is, the smaller the
steady-state error and the better the disturbance rejection against low frequency
disturbances),
4. noise rejection: gain of the loop transfer function at the high frequency range should
be low in order to reject high frequency noise.

2.12.7 Practical Implementation Issues of PID Control

Anti-Windup Integral Control When integral control is used in a control system
which has actuator saturation, the integral control can very adversely affect the transient
response and the stability of the closed loop system. Almost all practical control systems
have actuators with saturation. All physical actuation components have limits on the output
they can provide, that is a valve can be fully open, an electric motor can provide a known
maximum torque or force, an amplifier output voltage would be limited to a maximum
supply voltage. When an actuator saturates, the control output no longer changes, effectively
rendering an open loop control system. During that time, if the error sign does not change,
the output of the integrator continues to increase even though the actuator is saturated.
When the error sign changes, and the control signal sign should change, the integral control
component may prevent that due to its large contribution to the control signal that is a
result of the “accumulation” (integral) of the past errors. This is called integrator windup.It
results in poor transient response and possibly in stability problems. The solution is then to
add a component that will have an anti-windup function on the integrator. In digital control,
the easiest and most common method of implementing the integrator anti-windup is to stop