---
title: "PD controller principle and working"
subject: "Control Systems"
chapter: 7
key: "semester_3/control_systems:pd-controller"
source: https://www.bctengineer.com/q/semester_3/control_systems/pd-controller
site: BCT Engineering
---

# PD controller principle and working

Control Systems · Chapter 7: Performance Specifications and Compensation Design

- Asked in: 2076 Chaitra, 4 marks; 2080 Bhadra, 4 marks (asked 2 times)
- Video: [PD Controller Explained: Basics, Block Diagram, Transfer Function, Pros, and Cons](https://www.youtube.com/watch?v=Yc0VDIrsNnA) — Engineering Funda
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_3/control_systems/pd-controller
- Whole chapter: https://www.bctengineer.com/md/semester_3/control_systems/chapter-7

## Question

Discuss principle and working performance of PD controller.

## Short answer

A PD controller adds a derivative term Kd de/dt to the proportional action. It provides anticipatory phase lead, increasing damping ratio, reducing overshoot and settling time, and improving stability, while steady-state error remains unchanged. Noise amplification is a drawback.

## Answer

A **PD (proportional–derivative) controller** produces a control signal proportional to the error plus the rate of change of the error:

$$
u(t)=K_pe(t)+K_d\frac{de(t)}{dt},\qquad G_c(s)=K_p+K_ds=K_p(1+T_ds),\ T_d=\frac{K_d}{K_p}
$$

```text
          +-->[ Kp ]-----+
 e(t) ----|              (+)---> u(t)
          +-->[ Kd d/dt ]+
```

#### Principle

- The P part acts on the **present** error.
- The D part acts on the **trend** of the error. When the error is falling fast (output rushing toward the set point), $de/dt$ is negative and the D term reduces the drive **before** the output overshoots. So D gives anticipatory action and extra damping.
- It adds a **zero at $s=-K_p/K_d$** to the open-loop transfer function, i.e. phase lead.

#### Working performance (second-order system)

For $G(s)=\dfrac{\omega_n^2}{s(s+2\zeta\omega_n)}$ with PD control (taking $K_p=1$):

$$
\frac{C(s)}{R(s)}=\frac{\omega_n^2(1+T_ds)}{s^2+(2\zeta\omega_n+\omega_n^2T_d)s+\omega_n^2}
$$

$$
\zeta'=\zeta+\frac{\omega_nT_d}{2}
$$

| Property | Effect of PD |
|---|---|
| Damping ratio | increases |
| Peak overshoot | decreases |
| Rise time, settling time | decrease (faster) |
| Type of system | unchanged |
| Steady-state error | unchanged (for given $K_p$) |
| Stability margin | improves |
| Noise | amplified (drawback) |

**Example:** with $\omega_n=2$ rad/s and $\zeta=0.25$, choosing $T_d=0.3$ s gives $\zeta'=0.25+0.3=0.55$, cutting the step overshoot from about 44% to about 16% (the added zero keeps it a little above the 12.6% of a plain ζ = 0.55 system).

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. What does the derivative part of a PD controller act on?
   A. The accumulated past error
   B. The present error only
   C. The rate of change (trend) of the error, giving anticipatory damping
   D. The integral of the output
   Answer: C. D acts on the error trend, reducing drive before the output overshoots. (Key point: Principle)

2. What does PD control add to the open-loop transfer function?
   A. A pole at the origin
   B. A zero at s = -Kp/Kd, giving phase lead
   C. A pole at -Kp/Kd
   D. Nothing, only a gain
   Answer: B. Gc = Kp(1 + Td s) has a zero at -Kp/Kd, which provides phase lead. (Key point: Effect on open loop)

3. For a second-order system with PD control, which set of effects is correct?
   A. Damping decreases, overshoot increases, steady-state error removed
   B. Damping ratio becomes zeta + wn Td/2, overshoot and settling time fall, type unchanged
   C. Type increases by one, overshoot rises
   D. Noise is attenuated and damping falls
   Answer: B. zeta' = zeta + wn Td/2 reduces overshoot and settling time, but type and steady-state error are unchanged and noise is amplified. (Key point: Performance)

## Related in Control Systems

- [Short note: PD and PI controller](https://www.bctengineer.com/question.md?key=semester_3%2Fcontrol_systems%3Ashort-pd-pi) — Control Systems, chapter 7
- [Working of PI controller](https://www.bctengineer.com/question.md?key=semester_3%2Fcontrol_systems%3Api-controller-working) — Control Systems, chapter 7
- [P, I, D controllers and role of PI, PD](https://www.bctengineer.com/question.md?key=semester_3%2Fcontrol_systems%3Apid-pi-pd-role) — Control Systems, chapter 7
- [Derivative feedback controller: block diagram and effect](https://www.bctengineer.com/question.md?key=semester_3%2Fcontrol_systems%3Aderivative-feedback-bd-effect) — Control Systems, chapter 7, asked 2×

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