What Is Derivative Controller?
Derivative controller (also known as D controller) refers to a simple derivative action (or differentiator) in series with a constant gain. Mathematically, the output of derivative controller equals the derivative of its input multiplied by a constant. The reason for the importance of derivative controller in control theory and industrial applications is that it can speed up the response of plant to sudden changes in setpoint.
For example, consider the feedback control system for room temperature control as shown in Figure 1. In this example it is assumed that a heater with adjustable power is embedded in a room and the controller has automatically adjusted its power such that the room temperature at steady state equals the desired temperature which equals 20 degrees Celsius.
Now, if we suddenly increase the desired temperature (or setpoint) from 20 to 30 degrees Celsius, a step error with amplitude 10 appears at the input of controller. Clearly, we expect that in response to this step change in the error, the controller significantly increases the power of heater in order to increase the temperature as fast as possible.
Now ask yourself what mathematical operator should be inside the controller box in Figure 1 such that a step change in its input leads to a very big change at its output. Yes, the answer is the differential operator (or differentiator). Ideally, the derivative of step (or Heaviside) function is the impulse, also known as the Dirac delta function.
Roughly speaking, in practice the derivative of step function is something whose amplitude is extremely large. It concludes that the function of derivative controller is speeding up the response of closed loop system to step (or sudden sharp changes) in setpoint and making the reactions of feedback control system to such discontinuous changes very fast.
Figure 1: feedback control system for room temperature control
Although it may look interesting at the first glimpse, derivative controller is never used alone in real world applications. Very few applications of derivative controller in combination with P/PI controller can be found in the literature (in other words, rarely in practice the controller is in the form of PD or PID).
One reason for the lack of interest to derivative controller – either alone or in combination with proportional/integral terms – is that it can severely amplify of noise. Note that in in practice there is a sensor in the feedback path of Figure 1 which is also the main source of thermal noise (also known as measurement noise). This noise enters into the controller and in case of having a derivative operator inside it, the noise is amplified and appears at the output of controller, which is drastically undesired. This noise can even lead to instability in practice.
The derivative controller can also studied in the frequency domain. The transfer function of an ideal derivative controller equals \(s\), where \(s\) stands for the Laplace variable. Since the ideal derivative controller is actually a noncausal system, it cannot be used in online real world applications.
The practical derivative controller is realized by putting the ideal derivative controller in series with a lowpass filter. This lowpass filter has two functions; first, making the noncausal system causal, and second, reducing noise amplification. Hence, the transfer function of practical derivative controller equals \(s/(1+sN)\) where \(1/N\) is the bandwidth of the lowpass filter. The value of \(N\) must be chosen very carefully.
More precisely, assigning an unnecessarily small value to \(N\) leads to noise amplification. On the other hand, assigning an unnecessarily large value to \(N\) decreases the bandwidth of closed loop control system, which results in slow response (recall that closed loop bandwidth is in proportion with the rise time of the step response of the feedback system). Considering the fact that the bandwidth of open loop and closed loop systems are usually very close, a good choice for \(N\) (reciprocal of the BW of the lowpass filter in series with derivatie controller) is something slightly smaller than the desire closed loop BW.
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Written by: Farshad Merrikh Bayat