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Derivative (Rate) Control

The final facet of PID control is the “D” term, which stands for derivative. This is a calculus concept like integral, except most people consider it easier to understand. Simply put, derivative is the expression of a variable’s rate-of-change with respect to another variable. Finding the derivative of a function (differentiation) is the inverse operation of integration. With integration, we calculated accumulated value of some variable’s product with time. With derivative, we calculate the ratio of a variable’s change per unit of time. Whereas integration is fundamentally a multiplicative operation (products), differentiation always involves division (ratios).

A controller with derivative (or rate) action looks at how fast the process variable changes per unit of time, and takes action proportional to that rate of change. In contrast to integral (reset) action which represents the “impatience” of the controller, derivative (rate) action represents the “cautious” side of the controller.

If the process variable starts to change at a high rate of speed, the job of derivative action is to move the control valve in such a direction as to counteract this rapid change, and thereby moderate the speed at which the process variable changes.

What this will do is make the controller “cautious” with regard to rapid changes in process variable. If the process variable is headed toward the setpoint value at a rapid rate, the derivative term of the equation will diminish the output signal, thus slowing tempering the control response and slowing the process variable’s approach toward setpoint. To use an automotive analogy, it is as if a driver, driving a very heavy vehicle, preemptively applies the brakes to slow the vehicle’s approach to an intersection, knowing that the vehicle doesn’t “stop on a dime.” The heavier the vehicle, the sooner a wise driver will apply the brakes, to avoid “overshoot” beyond the stop sign and into the intersection.

If we modify the controller equation to incorporate differentiation, it will look something like this:



  m = Controller output

  e = Error (difference between PV and SP)

  Kp = Proportional gain

  τi = Integral time constant (minutes)

  τd = Derivative time constant (minutes)

  t = Time

  b = Bias

The de/dt term of the equation expresses the rate of change of error (e) over time (t). The lower-case letter “d” symbols represent the calculus concept of differentials which may be thought of in this context as very tiny increments of the following variables. In other words, de/dt refers to the ratio of a very small change in error (de) over a very small increment of time (dt). On a graph, this isinterpreted as the slope of a curve at a specific point (slope being defined as rise over run).

It is also possible to build a controller with proportional and derivative actions, but lacking integral action. These are most commonly used in applications prone to wind-up, and where the elimination of offset is not critical:

Many PID controllers offer the option of calculating derivative response based on rates of change for the process variable (PV) only, rather than the error (PV SP or SP PV). This avoids huge “spikes” in the output of the controller if ever a human operator makes a sudden change in setpoint1. The mathematical expression for such a controller would look like this:


It should be mentioned that derivative mode should be used with caution. Since it acts on rates of change, derivative action will “go crazy” if it sees substantial noise in the PV signal. Even small amounts of noise possess extremely large rates of change (defined as percent PV change per minute of time) owing to the relatively high frequency of noise compared to the timescale of physical process changes.

Ziegler and Nichols, the engineers who wrote the ground-breaking paper entitled “Optimum Settings for Automatic Controllers” had these words to say regarding “pre-act” control (page 762 of the November 1942 Transactions of the A.S.M.E.):

The latest control effect made its appearance under the trade name “Pre-Act.” On some control applications, the addition of pre-act response made such a remarkable improvement that it appeared to be in embodiment of mythical “anticipatory” controllers. On other applications it appeared to be worse than useless. Only the difficulty of predicting the usefulness and adjustment of this response has kept it from being more widely used.


1It should not be assumed that such spikes are always undesirable. In processes characterized by long lag times, such a response may be quite helpful in overcoming that lag for the purpose of rapidly achieving new setpoint values. Slave (secondary) controllers in cascaded systems – where the controller receives its setpoint signal from the output of another (primary, or master) controller – may similarly benefit from derivative action calculated on error instead of just PV. As usual, the specific needs of the application dictate the ideal controller configuration.

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Comments (2)Add Comment
written by Scott Roberts, May 05, 2013
I would like to see basic "controller" web pages show how to implement D and I control with a capacitor and resistor, and even include hysteresis with 2 diodes. Without op amps. It's not practical except in the midst of a tight application like old TVs, but it's nice to show the simplest applications of theory. You can make then make mechanical and electronic circuit diagrams exactly the same, then show the parallel to the PID biology of a simplistic neuron model. P is for current state (reaction), I is for past state (memory), and D is future state (projection). Designing the PID is one level of intelligence, selecting what to measure and setting the goals is at a "higher" level, then changing the thing being controlled itself seems to be a higher level. Then changing if it even needs to be controlled is an even higher level. Not in a "FLOPS" type of intelligence, but some other vague sense.
Electrical and Mechanical Parallel
written by Scott Roberts, May 07, 2013
The best equations for mechanical and electrical parallels are based on treating meters like charges.


F=force, M=mass, v=velocity, k=spring force constant. So mass is the parallel of inductance. The ease of which a spring (electrostatic repulsion) is compressed (1/k) is the parallel of capacitance (electrostatic attraction).

To be predictive one therefore would measure the acceleration of charges (dI/dt) into an inductor. In the mechanical system you look at the acceleration of meters (dv/dt) "into a mass". My wording is backwards from traditional of "accelerating a mass" ("into meters" which are implied in the word "acceleration"). In that case I would say we are accelerating the inductor into charges to keep the parallel perfect.

To look at the past (reflective) you look at the acceleration of charges into a capacitor or the acceleration of meters into a spring.

Capacitors are a structure that determine how and how many OPPOSITE charges ATTRACT over a DISTANCE, increased as 1 dimension of the parallel plate structure gets SMALLER. The other two dimensions can get larger to increase C, for a given charge density.

Conversely, springs are a structure that determine how and how many SAME charges REPULSE over a distance. 1/k is increased by 1 dimension getting LONGER, and decreased by the other 2 dimensions for a given charge density.

Inductors are a structure for determining how and how many SAME charges RESIST over a TIME. As any of the 3 dimensions of the structure increases, inductance increases for a given charge density (not trivial to figure like a capacitor). Therefore, moving blindly ahead, a model of "mass" is: "A structure for determining how and how many OPPOSITE charges ATTRACT over a TIME, which increases with a decrease in any of the 3 dimensions of the structure, for a given charge density."

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