We discuss the operation, design and implementation of the PD controller, Proporcional-plus-Diferencial. The PD controller is mainly used to improve the transient response of a control system.
Cascade Compensation - PD controller
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 poles and 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:
>> s=tf(‘s’);
>> G=1/((s+1)*(s+2)*(s+5));
>> rlocus(G);
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:
>> z=0.4;
>> sgrid(z,0);
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:
>> G1=23.7/((s+1)*(s+2)*(s+5));
>> sys1=feedback(G1,1);
>> step(sys1);
>> stepinfo(sys1)
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:
>> G2=((s+2))/((s+1)*(s+2)*(s+5));
>> rlocus(G2);
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:
>> z=0.4;
>> sgrid(z,0);
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:
>> G3=(51.2*(s+2))/((s+1)*(s+2)*(s+5));
>> sys2=feedback(G3,1);
>> step(sys2);
>> stepinfo(sys2)
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:
>>step(sys1, sys2)
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 an ideal derivative compensator to yield a 16% overshoot, with a threefold reduction in settling time.
Figure 10.
In construction…
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.
Source:
- Control Systems Engineering, Nise
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