In this article, we discuss the PD controller and Lead Compensation, two ways to improve the transient response of a feedback control system by using cascade compensation. Typically, the objective is to design a response that has a desirable percent overshoot and a shorter settling time than the uncompensated system.
Improving Transient Response - Compensation
We have seen before that setting the gain at a particular value on the root locus yields the transient response dictated by the poles at that point on the root locus. Thus, we are limited to those responses that exist along the root locus. (See Sketching Root Locus with Matlab – Control Systems)
Unfortunately, most of the time the overshoot specification for designing control systems exceed the posibilities of the current root locus. What can we do then?
Rather than change the existing system, we augment, or compensate, the system with additional poles and zeros, so that the compensated system has a root locus that goes through the desired pole location for some value of gain. One of the advantages of compensating a system in this way is that additional poles and zeros can be added at the low-power end of the system before the plant. We should evaluate the transient response through simulation after the design is complete to be sure the requirements have been met.
There are two configurations of compensation mostly used in control systems design: cascade compensation and feedback compensation. These methods are modeled in Figure 1 and Figure 2:
With cascade compensation, the compensating network, G1(s), is placed at the low-power end of the forward path in cascade with the plant, Figure 1.
With feedback compensation, the compensator, H1(s), is placed in the feedback path, Figure 2.
Both methods change the open-loop poles and zeros, thereby creating a new root locus that goes through the desired closed-loop pole location.
Cascade Compensation - PD controller
As we said before, sometimes poles and zeros must be added in the forward path to produce a new open-loop function whose root locus goes through the design point on the s-plane, in order to meet design requirements. One way to speed up the original system that generally works is to add a single zero to the forward path.
This zero can be represented by a cascade compensator whose transfer function Gc(s) is:
This function, the sum of a differentiator s and a pure gain Zc, is called an ideal derivative compensation, or Proportional-Derivative PD controller. In summary, transient responses unattainable by a simple gain adjustment (proportional controller) can be obtained by augmenting the system’s zeros with an ideal derivative controller.
Let´s use the Root Locus of Figure 3 to find out how a PD controller works. There, we have the Root Locus of a control system which forward transfer function G(s) with unitary feedback is:
If K=1, the commands in Matlab would be:
Suppose that we want to operate the system of Figure 3 with a damping ratio ξ=0.4. Figure 4 shows that we can get this damping ratio with a proportional compensator, setting the gain K=23.7:
Use right click to select the damping:
Figure 5 shows the Step Response of the closed-loop system for Kp=23.7 and ξ=0.4, and the values of the main parameters:
Figure 5. Step response of the closed-loop uncompensated system
Suppose now that we want to mantain the damping ratio ξ=0.4, improving rise time and settling time, making the system faster. That would be imposible using only a proportional controller because we are limited by the Root Locus according to Figures 3 and 4.
The uncompensated system of Figure 3 could becomes a compensated system by the addition of a compensating zero at -2, in Figure 6, using a cascade compensator whose transfer function Gc(s) is:
Figure 6. Root Locus for the compensated system.
Figure 7 shows that we can get a damping ratio ξ=0.4. setting the gain K=51.2:
Use right click to select the damping:
Figure 7. Location in the RL of ξ=0.4
Figure 8 shows the Step Response of the closed-loop system for Kp=51.2 and ξ=0.4, and the values of the main parameters:
Figure 8. Step response of the closed-loop compensated system
Mantaining the same damping ratio ξ=0.4, Rise Time has improved (from 0.6841 s to 0.1955 s) and Settling Time has improved (from 3.7471 s to 1.1218 s). However, Overshoot has increased (from 23.3070 to 25.3568) and also the Peak has increased (from 0.8672 to 1.1420). Figure 9 compares graphically both of the responses, before and after the PD compensation:
Figure 9. Step response of Compensated Vs. Uncompensated System.
Figure 9 also shows that the final value is closer to the reference value (1), so the steady-state error has improved with PD compensation (from 0.297 to 0.088). However, readers must not assume that, in general, improvement in transient response always yields an improvement in steady-state error.
Now that we have seen what PD compensation can do, we are ready to design our own PD compensator to meet a transient response specification.
1) Given the system of Figure 10, design PD compensator to yield a 16% overshoot, with a threefold reduction in settling time.
How do we implement the PD controller?
The PD compensator used to improve the transient response is implemented with a proportional-plus-derivative (PD) controller. In Figure 11 the transfer function of the controller is:
Figure 11. Implementation of Proportional-plus-Derivative (PD) controller.
Just as the active ideal integral compensator can be approximated with a passive lag
network, an active ideal derivative compensator can be approximated with a passive
lead compensator. When passive networks are used, a single zero cannot be
produced; rather, a compensator zero and a pole result. However, if the pole is
farther from the imaginary axis than the zero, the angular contribution of the
compensator is still positive and thus approximates an equivalent single zero. In
other words, the angular contribution of the compensator pole subtracts from the
angular contribution of the zero but does not preclude the use of the compensator to improve transient response, since the net angular contribution is positive, just as for a single PD controller zero.
The advantages of a passive lead network over an active PD controller are that
(1) no additional power supplies are required and (2) noise due to differentiation is
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