PD Control
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## 1 Introduction to Proportional-Derivative (PD) Control

Proportional-Derivative control is useful for fast response controllers that do not need a steady-state error of 0. Proportional controllers are fast. Derivative controllers are fast. The two together is very fast. Below is a review.

### 1.1 Proportional Action[1]

Proportional action provides an instantaneous response to the control error. This is useful for improving the response of a stable system but cannot control an unstable system by itself. Additionally, the gain is the same for all frequencies leaving the system with a nonzero steady-state error.

### 1.2 Derivative Action[2]

Derivative action acts on the derivative or rate of change of the control error. This provides a fast response, as opposed to the integral action, but cannot accomodate constant errors (i.e. the derivative of a constant, nonzero error is 0). Derivatives have a phase of +90 degrees leading to an anticipatory or predictive repsonse. However, derivative control will produce large control signals in response to high frequency control errors such as set point changes (step command) and measurement noise.

In order to use derivative control the transfer functions must be proper. This often requires a pole to be added to the controller (this pole is not present in the equations below).

## 2 Proportional-Derivative (PD) Control

Proportional-Derivative or PD control combines proportional control and derivative control in parallel.

 $LaTeX: C_{PD}\left(s\right)=K_P\left(1+T_{d}s\right)$ PD[3]

where

$LaTeX: K_P$ is the PID control gain and
$LaTeX: T_{d}$ is the derivative gain or derivative time.

## 3 Design Notes

With proportional controller alone the shape of the open loop transfer function will be the same as the plant but the overall magnitude of the plant will be higher. With derivative control the open loop transfer function above the frequency fo the derivative (zero) will have a +20 dB/decade slope. The phase will gain +90 deg above the zero as well.

Integral control drives the system to a steady-state error of zero by averaging the noise and disturbances. Thus only correcting for the slower error signals such as a step command. Proportional control will follow the nosie and amplify it by the magnitude of the controller. Derivative control will amlify noise by following the difference between 2 noisy error signals.

Keep these things in mind when designing a PD control.

## 4 References

• Goodwin, G. C., Graebe, S. F., and Salgado, M. E. 2000 Control System Design. 1st. Prentice Hall PTR. ISBN 0139586539
• Franklin, G. F., Emami-Naeini, A., and Powell, J. D. 1993 Feedback Control of Dynamic Systems. 3rd. Addison-Wesley Longman Publishing Co., Inc. ISBN 0201527472

### 4.1 Notes

1. Goodwin et all, pg. 160
2. Goodwin et all, pg. 161
3. extrapolation from Franklin et all, pg. 185, Eqn. 4.31