## Overview

PD control is a form of PID control where the integral term has been set to 0. Both the proportional and the derivative terms compound to create a fast response time and a control system capable of overcoming unexpected behavior. It is less expensive than typical PID loops while also being less complex and therefore easier to tune.

- Responds quickly to step changes
- Overshoot can be lessened by increasing dampening
- Handles processes with a time lag well

- Fast responding systems can become unstable
- Steady-state error remains
- At higher frequencies noise can be amplified, wearing out actuators prematurely
- Must be cautious with setpoint changes to avoid derivative kick

## PD Control in More Detail

A standard PD controller contains 2 terms:

- Proportional (P)
- Derivative (D)

The block diagram below shows a typical PD control system. The ouput of the system is compared to the desired setpoint . This resulting error is then split and fed into the proportional term and the derivitive term. The proportional gain, Kp, is multiplied this error to give the proportional term’s contribution to the change in the process. A similar process takes place for the derivitive term. The derivitave (or a close approximation) of the error is taken and then the result multiplied by the derivitive gain, Kd. Both terms are then summed together and the result is then sent as an affect on the process (ie closing a valve, tunring a motor, etc…). The speed at which this change is applied is based on the size of the previous term. The process output is then measured again and the results sent back and compared to the setpoint. This process continually repeats driving the process output towards the setpoint.

*Proportional Term*

The proportional term is the just the error (difference between setpoint and actual value) multiplied by the proportional gain. This can be thought of as a “jerk” towards the desired setpoint. A really high gain would cause the system to jerk passed the setpoint, or overshoot, then overshoot again on the way back. This allows the Proportional controller to be capable of a quick response but also introduces the possibility of instability. This is where the derivative comes on.

*Derivative Term*

The derivative term takes the derivative of the error value, and multiplies that by the derivative gain. The derivative acts to dampen the effect of the proportional path. As the system moves towards the setpoint, the error decreases. Since the derivative of a decreasing error is a negative number, it fights against the proportional effect and lessens its affect. If tuned properly, the derivative term will restrain the system from overshooting and provide a quick responding but stable system.

*PD Controller Mathematical Forms*

There are many ways to represent PD control mathematically. First, the proportional term and the derivative term both enter the second sum block. This gives the following mathematical equation:

In addition to this, it can sometimes be convenient to factor out an *ultimate gain*, leaving the equation in what is sometimes known as *standard form*. Here the *Kd *term has been replaced by *Td/Kp*. When *Kp *term is distrubuted the equation returns to the earlier form.

The control equation can also be written in terms of damping ratio and frequency. In this case, the Kp term is given by the square of the angular velocity and the Kd term is given by the derivative of Kp multiplied by a damping factor

## Damping Effects

By using a constant frequency and changing the damping coefficient, we can gain an intuitive understanding about how the system will respond.

#### Under Damped

In this first example the damping coefficient has been set to 0. This results in an instable system that would continue to resonate indefinitely. If we were to increase the coefficient slightly, but still keep the damping coefficient under 1, the system would eventually come to rest.

#### Critically Damped

In a critically damped system the damping coefficient is equal to 1. This is the ideal case. A critically damped system represents the quickest possible time to reach the new setpoint without overshooting.

#### Over Damped

An overdamped system is one where the damping coefficient is greater than 1. In this case the sytem will not overshoot and will reach the desired setpoint, however, it will not do it in the quickest way possible.

## Practical Examples of PD Controller

#### Temperature Control

A common use case for a PD Controller would be to control the temperature of a system. Due to the slow nature of temperature change, there will not be much external noise to disrupt the system. The lack of the integration term eliminates reset windup and prevents the system from being stuck in an unsteady state, while including the derivative term helps to increase stability while decreasing peak overshoot and settling time.

#### Avoiding Stick-Slip

Another use for PD control is to avoid *stick-slip*. Stick-slip occurs at very low velocities when friction causes an object to move intermittently. This intermittent movement often leads to the object overshooting it’s target. The standard solution for this problem has been to use high-gain PD control.

**When Not to Use a PD Controller**

The derivative control term does have its drawbacks. If the system contains a lot of noise, the derivative term will directly amplify it and this is never desired. In addition, the derivative term will have no effect on the steady-state performance. The lack of an integrating term doesn’t allow the error to build up over time and allow the system to correct itself. This results in a non-zero steady-state condition. In some cases this is okay but in others another control scheme may be warranted.