Successfully using derivative action in control systems can be a challenge. Understanding the frequency response analysis of closed-loop systems is crucial, but it may be too technical for most readers of this book. However, there are some tips that can be useful without going too much into detail.
The proper use of derivative action can be challenging. The general rule is to set it equal to 
(L/2), although some suggest using a divisor of six. The margin for error is small; using too little derivative renders it ineffective, while using too much can cause stability issues. The ratio of too much to too little is typically around 2 to 1 or 3 to 1. It’s not uncommon for the calibration of the derivative adjustment to be off by a factor of two with pneumatic controllers, making it challenging to use effectively. As a result, derivative action is seldom specified and is often turned off even when it is. Those who know how to use it correctly can become heroes, while others can find themselves struggling.
The following are the most important application points about the use of derivative action:
- The addition of derivative action to a control loop can generally decrease its natural period and lead to improved performance. However, the improvement is usually modest, typically in the 10 to 20% range, and in rare cases may reach 30%. Better tuning of a PI controller typically yields greater improvement than the addition of derivative action. The effect of derivative action can be observed in the step response depicted in Figure 7.1. It is apparent that if the normal step response exhibits curvature prior to the primary response, the derivative action can accentuate this curvature and effectively decrease the apparent dead time. As a result, the natural period is reduced from its original value.
Figure 7.1 [The effect of derivative action may be visualised as reducing the effective dead time.]
- Derivative action has the ability to stabilize a runaway process. In some rare cases, a process may be so difficult to control that manual control is not effective, similar to balancing a long stick on your finger. If you don’t move your finger, the stick will fall, indicating that the system is unstable and cannot remain stable. By closing the loop using the brain as a controller between the eyes and fingers, the stick can be kept upright. Although derivative action is not necessary for stabilizing a runaway process, it can be helpful, sometimes even significantly so.
- In batch processes, derivative action can assist in smoothly changing direction without overshooting or undershooting, particularly if the derivative and integral corners are interchanged. This process requires reducing the regular controller gain by the distance between the corners. However, there are more complexities involved in this process that are beyond the scope of this discussion.
- Once, we utilized derivative action to counterbalance a long pneumatic transmission delay in a batch process. In this process, a container was either filling or emptying, and the command to close the valve was dependent on the level and its derivative.
- Flow measurement has been achieved using derivative control by measuring the derivative of level or weight.
- The lead/lag logic block element typically consists of two parts: a lead element and a lag element. The lead element is the first part of the block and is used to introduce a phase shift in the input signal. It can be used to anticipate changes in the input signal and can help to improve the stability and performance of the control loop. In some cases, the lead element can also be used to provide a gain boost to the input signal.
Derivative action is not mathematically pure, as this results in a highly erratic controller output. Therefore, the practical implementation of derivative action involves constraining the high frequency gain (or short period gain), with typical values of 6 or 10. This limits the amplification of noise to the gain value (i.e., typically 6 or 10). The step response of the common implementation of proportional plus derivative action is relatively straightforward to visualize and is illustrated in figure 7.2. It consists of a spike with a gain of K, combined with a time constant decay of gain K-1 and time constant T.
Figure 7.2 [The step response of a proportional-plus-derivative controller will spike up at first and then decay back to a steady state value.]
A purely mathematical derivative function would respond to a pure step as an infinitely high, infinitely short pulse with an area proportional to the derivative time. However, in practice, implementing derivative action introduces noise to the controller output. To mitigate this, the high frequency gain is limited to values of 6 or 10. The step response to a typical implementation of proportional plus derivative action is a spike with gain K and a time constant decay of gain K-1 and time constant T.
The practical implementation of the derivative function aims to achieve the desired area under the response curve without going to infinite amplitude. The lower the amplitude, the slower the decay must be to achieve the same area. However, if the amplitude is too low, the decay is too long, and the desired benefit from derivative action is not achieved. In the above example, a larger K results in a smaller T for the same derivative time.
For digital controllers, the sampling nature presents special problems, particularly when attempting to get meaningful derivative action for short derivative times relative to the sampling rate. For instance, if a loop has a natural period of 16 seconds and a derivative time of two seconds, the controller needs to know about variations ten times as fast as the derivative time, or 0.2 seconds. To avoid confusion, a sampling system should sample the process five to ten times faster, or every 0.02 to 0.04 seconds (25 to 50 times per second).
Manufacturers of digital controllers do not disclose how they have addressed this issue, making it challenging to assess the benefits of derivative action in the field. While this does not imply that derivative action should not be used, the results, especially on relatively fast processes, may not live up to expectations. It is worth noting that using derivative action on processes with a natural period shorter than about a minute may not be effective if the controller is sampling the process 10 times per second. Therefore, it is advisable to experiment with derivative action to determine its effectiveness on a case-by-case basis.
Math / Algebra
The Laplace transform for derivative action is presented here.
(1) 
The response shown in figure 7.2 may be considered as made up of two parts, first the response of 
, and then the decay back. The algebra of this is:
(2) 