Page 81 - Mechatronics with Experiments

CLOSED LOOP CONTROL 67
2.4 PERFORMANCE SPECIFICATIONS FOR
CONTROL SYSTEMS

The performance desired from a control system can be described under three groups:
1. response quality:
(a) transient response,
(b) steady-state response,
2. stability,
3. robustness of the system stability and response quality against various uncertainties
such as disturbances, process dynamic variations, sensor noise.
The main advantage of feedback control over open loop control is its ability to reduce the
effect of disturbances and process dynamic variations on the quality of system response. In
other words, the main advantage of feedback control is the robustness it provides against
various uncertainties.
A basic feedback control system and typical uncertainties associated with it are shown
in Figure 2.14b. As we discussed in the previous section, the total response of the system
due to command, disturbance, and sensor noise (for H = 1 case) is
DG G DG
y = r + w − v
1 + DG 1 + DG 1 + DG
The goal of the control is to make y(t) equal to r(t). Therefore DG ≫ 1 should be in general.
If DG ≫ G, the effect of disturbance, w, is reduced. However, the sensor noise directly
contributes to the output, y. In order to track r and reject disturbance, w, we want DG ≫ 1
(large), but in order to reject sensor noise we want DG ≪ 1 (small). This is the basic
dillema of feedback control design. A compromise is reached by the following engineering
judgment: disturbance, w(t) is generally of low frequency content, whereas sensor noise
v(t) is high frequency content. Therefore, if we design a controller such that DG ≫ 1
around the low frequency region to reject disturbances, and DG ≪ 1 around the high
frequency region to reject sensor noise, the closed loop system has good robustness against
uncertainties.
The robustness of the closed loop system (CLS) is closely related to the gain of
loop transfer function as a function of frequency (Figure 2.16). Therefore the robustness
properties are best conveyed in the frequency domain. In general, a loop transfer function
should have a large gain at low frequency in order to reject low frequency disturbances
and slow variations in process dynamics, and low loop gain at high frequency in order to
reject sensor noise. The s-plane pole-zero representation of a transfer function does not
convey gain information. Hence, robustness properties are not well conveyed by the s-plane
pole-zero structure of the transfer function.
Stability requirements are equally well described in the s-plane as well as frequency
domain. In the s-plane, all the CLS poles must be on the left-hand plane. In the frequency
domain, the gain margin and phase margin must be large enough to provide a sufficient
stability margin. The desired relative stability margin from a CLS can be expressed either
in terms of gain and phase margin in frequency domain, or in terms of the distance of CLS
poles from the imaginary axis in the s-plane.
Finally, the response quality must be specified. The response of a dynamic system
can be divided into two parts: (i) transient response part, (ii) steady-state response part.
Transient response is the immediate response of the system when it is commanded
for new desired output. The steady-state response is the response of the system after a