Page 111 - Mechatronics with Experiments

CLOSED LOOP CONTROL 97
The type of the loop transfer function can also be immediately determined from the slope
of the magnitude curve as frequency goes to zero or from the phase plot at low frequencies.
The robustness specification deals with the sensitivity of the system. The most impor-
tant advantage of feedback control over open loop control is that the feedback improves
the robustness of the system performance against the variations in process dynamics and
disturbances. The closed loop system should not only be stable and have good response
quality for the nominal parameters of the operating conditions, but also should stay stable
and have good response quality despite the real-world imperfections.
In summary, the correlation between the time-domain specifications and frequency
domain behavior is as follows

Good stability means a large gain margin and phase margin. In order to have a rea-

sonably good P.M., the slope of the magnitude curve should be about −20 dB/decade
around the cross-over frequency.
Larger loop gain at low frequencies results in lower steady-state errors, and good

disturbance rejection against low frequency disturbances.
Low loop gain and a fast decaying rate at the high frequency region increase the

ability to reject the effect of high frequency noise.
Overall, the stability, steady-state error, and robustness characteristics of a CLS is well
represented in the frequency response of the loop transfer function, whereas the transient
response is not represented with the same accuracy. The s-domain pole-zero representation
of a CLS correlates to the transient response behavior well, but does not give information
about disturbance and sensor noise rejection ability. Therefore, frequency response (i.e.,
Bode plots) and s-domain methods (i.e., root locus method) complement each other in
the graphical information they display regarding the control system characteristics (i.e.,
transient and steady-state response).

2.12 BASIC FEEDBACK CONTROL TYPES

Figure 2.40 shows the three basic feedback control actions: proportional, integral, and
derivative control actions. Figure 2.41 shows the input–output behavior of these control
types. In practical terms, proportional control action is generated based on the current
error, the integral control action is generated based on the past error, and the derivative
control action is generated based on the anticipated future error. The integral of the error
can be interpreted as the past information about it. The derivative of the error can be
interpreted as as a measure of future error to come. Assume that the error signal entering
the control blocks has a trapeziodal form. The control actions generated by the proportional,
integral, and derivative actions are shown in Figure 2.41. Proportional - integral - derivative
(PID) control has control decision blocks which take into account the past, current, and
future error. In a way, it covers all the history of error. Therefore, most practical feedback
controllers are either a form of the PID controller or have the properties of a PID controller.
The block diagram of a textbook standard PID controller is shown in Figure 2.42. The
control algorithm can be expressed in both the continuous (analog) time domain (which
can be implemented using op-amps) and in the discrete (digital) time domain (which can be
implemented using a digital computer in software). At any given time t, the control signal
u(t) is determined as function,

t
u(t) = K e(t) + K e( )d + K ̇ e(t) (2.130)
p I ∫ D
0