Page 129 - Mechatronics with Experiments

CLOSED LOOP CONTROL 115
pure derivative control adds a zero at the origin, whereas the modified derivative control
adds a zero at the origin and a pole on the negative real axis to the loop transfer function.
Similarly, the Bode plot of pure derivative control has a gain that increases to infinity as
the frequency increases to infinity. Whereas, the gain of the modified derivative control has
®
a finite gain as frequency goes to infinity (Figure 2.49). The MATLAB code segment to
obtain the Bode plots of the pure derivative control and derivative with a first-order pole
control is given below.

Kd= 1.0 ;
N = 10.0 ;
num1 = [Kd 0] ; den1 = [1] ; sys1=tf(num1,den1) ; % Kds
num2 = [Kd 0] ; den2 = [Kd/N 1] ; sys2=tf(num2,den2) ; % 1/s
figure(1) ; grid on; % Bode Plots
subplot(1,2,1) ; bode(sys1); grid on;
subplot(1,2,2) ; bode(sys2); grid on;

The high frequency gain of the modified derivative control can be made almost zero
by adding a second first-order low pass filter in series with the first one.
1 1
u (s) = K s ⋅ ⋅ ⋅ e(s) (2.169)
d
D
(K ∕N )s + 1 (K ∕N )s + 1
1
D
D
1
Another way to implement a practical PD (proportional and derivative) control is to
implement it as a phase-lead compensator, that is
u(s) = (K s + K ) ⋅ e(s) (2.170)
D
p
(K s + K )
p
D
u(s) = ⋅ e(s) (2.171)
(T s + 1)
f
where the last equation is an approximation to a PD control. This type of filter is called a
phase-lead filter since for T < (K ∕K ), that is T = 0.1 ⋅ (K ∕K ), the phase of the filter
D
f
D
p
p
f
is positive, which is why it is called a phase-lead filter.
Another common modification to derivative control is to use it only on the feedback
signal (output sensor signal) instead of the error signal if the command signal has disconti-
nuities, that is step changes in the command signal where the derivative at the discontinuity
is infinite. With this modification, the transient response of the system is improved due to
avoided discontinuous and large variations in the control signal, such as
e(s) = r(s) − y(s) (2.172)
1
u (s) = K s ⋅ ⋅ (−y(s)) (2.173)
D
d
(K ∕N)s + 1
D
Other Practical Variations of PID Control Figure 2.51 shows variations of
the PID control algorithm in real-world applications. However, it should be noted that all of
these variations are not necessarily implemented in one given application. So far, we have
discussed the P-I-D (proportional, integral, and derivative) gains of the PID controller. In
addition we also discussed the integral anti-windup to deal with actuator saturation, and a
more practical application of derivative control to reduce effects of noise in the error signal.
In this figure we illustrate the following practical modifications to the PID controller.
1. Velocity and acceleration feedforward terms are used to improve the transient
response. The velocity feedforward term (K ) helps improve the overshoot and
vf
raise time by effectively commanding a larger control signal before the velocity error