Download Properties of Feedback Control Systems: Tracking, Disturbance Rejection, and Robustness and more Study notes Control Systems in PDF only on Docsity!
ECE 486 PROPERTIES OF FEEDBACK Fall 08
Reading: FPE, Section 4.
Recall the basic feedback control loop that we looked at earlier:
Figure 1: Feedback Control.
In the above figure, P (s) is the plant, C(s) is the controller, r is the reference signal, y is the output of the system, and d is a disturbance affecting the control loop (e.g., wind on an airplane, noise in a sensor, faults in an electrical grid, etc.).
There are three main properties that a good control system should have:
- Tracking. The output of the system should behave like the reference signal. This property is studied by examining the transfer function from the reference input to the output.
- Disturbance Rejection. The disturbances should affect the output as little as possible. This property is studied by examining the transfer function from the disturbance to the output.
- Robustness. The output should track the reference signal even if the plant model is not exactly known, or changes slightly. This property is studied by examining the sensitivity of the transfer function to perturbations in the plant, as we show below.
Sensitivity. Let Try(s) denote the transfer function of the control system from the reference r to the output y. Now suppose that we allow the plant P (s) to change by a small amount δP (s) to become the new plant P¯ (s) = P (s) + δP (s). This will cause the transfer function Try(s) to also change by a small amount δTry (s), to become T¯ry (s) = Try(s) + δTry (s). The question is: how does δTry (s) compare to δP (s)? More specifically, the sensitivity is defined as the fractional (or percentage) change in the transfer function as related to the fractional change in the plant model:
S(s) =
δTry (s) Try (s) δP (s) P (s)
δTry (s) δP (s)
P (s) Try(s)
Note that for small perturbations δP (s) and δTry (s), the expression δT δPry (^ (ss)) is the derivative of Try (s) with
respect to P (s). In order to have good robustness, we want the sensitivity to be as small as possible.
We will analyze the tracking, disturbance rejection and robustness properties for both a feedforward control configuration and a feedback control configuration, and see why feedback is an important concept for control system design.
1 Feedforward Control
Recall from the first lecture that feedforward control does not make use of measurements of the system output in order to supply the input. The block diagram for feedforward control is shown in Fig. 2.
Figure 2: Feedforward Control.
We can examine how well feedforward control satisfies the properties listed above.
- Tracking. The transfer function from the reference to the output is obtained by assuming that the disturbance is not present (i.e., take d = 0), and is given by
Try(s) =
Y (s) R(s)
= P (s)C(s).
We can potentially get perfect tracking by choosing C(s) = P (s)−^1 , in which case we have Y (s) = R(s). Note, however, that “inverting” the plant in this way may not always be possible in prac- tice. Setting the implementation details aside for now, we note that feedforward control can potentially provide perfect tracking.
- Disturbance Rejection. The transfer function from the disturbance to the output is obtained by assuming that the reference signal is not present (i.e., r = 0), and is given by
Tdy (s) =
Y (s) D(s)
= P (s).
The effect of the disturbance on the output is determined completely by the plant. For example, if the disturbance is a step input, the steady state effect of this disturbance will be given by the DC gain of the plant P (0). Depending on the system, this could be quite bad. Furthermore, the feedforward control scheme provides us with no way to reduce this influence. Thus, feedforward control is bad at rejecting disturbances.
- Robustness. To examine how robust feedforward control is to variations in the plant, we examine the sensitivity S(s) =
δTry (s) δP (s)
P (s) Try(s)
Since Try(s) = P (s)C(s), we have
S(s) = C(s)
P (s) P (s)C(s)
We can summarize the above findings as follows.
Feedback control is good for tracking, good for disturbance rejection and good for robustness.
In addition to the above benefits, we will see later that feedback control can also stabilize an unstable plant (whereas feedforward control cannot). Also, it is worth noting that although we used high gain control to show that feedback provides good tracking, disturbance rejection and robustness, it is not the only option (and sometimes not even the best option). In practice, high gains are hard to implement in hardware (due to saturation effects and physical limitations). Furthermore, in some cases, high gain can even destabilize a stable system (we will see this when we study root-locus methods). We will be studying ways to design more sophisticated feedback controllers to get around these problems.
