---
title: "What derivative controller is and why useful"
subject: "Control Systems"
chapter: 7
key: "semester_3/control_systems:derivative-controller-useful"
source: https://www.bctengineer.com/q/semester_3/control_systems/derivative-controller-useful
site: BCT Engineering
---

# What derivative controller is and why useful

Control Systems · Chapter 7: Performance Specifications and Compensation Design

- Asked in: 2076 Asoj, 4 marks
- Video: [Derivative Controller Explained: Basics, Block Diagram, Transfer Function, Pros, and Cons](https://www.youtube.com/watch?v=COzesIxBZV0) — 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/derivative-controller-useful
- Whole chapter: https://www.bctengineer.com/md/semester_3/control_systems/chapter-7

## Question

What is derivative controller? How and why it can be useful?

## Short answer

A derivative controller outputs a signal proportional to the error's rate of change. It provides anticipatory action, adds damping, reduces overshoot and settling time, and improves stability, but amplifies noise.

## Answer

A **derivative controller** gives an output proportional to the **rate of change of the error**:

$$
u(t)=K_d\frac{de(t)}{dt},\qquad G_c(s)=K_ds
$$

In practice it is used with proportional action as a **PD controller**, $G_c(s)=K_p+K_ds$.

#### How it works

```text
 error e(t)        D output
   ^  /\              ^ +
   | /  \             |--+  (rising error)
   |/    \______      |  |   +-- (falling)
   +----------->      +--+---+--->
```

- When the error is **rising**, D output is positive and pushes harder.
- When the error is **falling quickly** (output approaching the set point fast), D output is negative and **reduces the drive before overshoot occurs**.
- For a constant error, D output is zero.

So it acts on the **future trend** of the error: it is an anticipatory controller.

#### Why it is useful

1. **Adds damping:** for a second-order plant with PD control, $\zeta'=\zeta+\frac{K_d\omega_n}{2}$ (with $K_p=1$), so **overshoot falls**.
2. **Faster settling:** $t_s=4/(\zeta'\omega_n)$ decreases.
3. **Improves stability:** the added zero gives phase lead and raises the phase margin, like a lead compensator.
4. **Quick response to sudden disturbances**, because it reacts to the rate of change rather than waiting for the error to grow.

#### Limitations

- Has no effect on steady-state error (zero output for constant error), so it is never used alone.
- Amplifies high-frequency noise ($|G_c|=K_d\omega$).

**Example:** in a DC motor position servo, PD control lets the shaft reach the target angle quickly with little overshoot; P alone with high gain would oscillate.

## Check yourself

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

1. What is the output of a derivative controller?
   A. Proportional to the rate of change of error, u = Kd de/dt
   B. Proportional to the area under the error
   C. Proportional to the error itself
   D. Proportional to the square of the error
   Answer: A. Gc(s) = Kd s, so the output depends on how fast the error changes. (Key point: Definition)

2. When the error is falling quickly, what does D action do?
   A. Pushes harder to reach the set point sooner
   B. Gives negative output that reduces the drive before overshoot occurs
   C. Does nothing until the error is zero
   D. Switches the controller off permanently
   Answer: B. It acts on the trend of the error, applying braking early. (Key point: Anticipation)

3. Why is a derivative controller never used alone?
   A. It makes the system unstable at any gain
   B. It needs an integrator in the plant
   C. For a constant error its output is zero, so it cannot reduce steady-state error
   D. It reduces damping
   Answer: C. D action only responds to changes, so a constant offset is not corrected. (Key point: Steady-state)

4. For PD control of a second-order plant (Kp = 1), what are the benefits?
   A. wn increases by Kd, so the response is faster with more overshoot
   B. Type of system increases by one
   C. Noise is filtered out
   D. zeta' = zeta + Kd*wn/2, so overshoot and settling time fall and phase margin rises
   Answer: D. The added damping lowers overshoot and settling time; the zero gives phase lead. (Key point: Benefits)

## Related in Control Systems

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- [Derivative feedback in second order time response](https://www.bctengineer.com/question.md?key=semester_3%2Fcontrol_systems%3Aderivative-feedback-second-order) — Control Systems, chapter 7
- [Derivative feedback for ramp input](https://www.bctengineer.com/question.md?key=semester_3%2Fcontrol_systems%3Aderivative-feedback-ramp) — Control Systems, chapter 7

## Related in other subjects

- [Digital PD controller design for plant 1/s²](https://www.bctengineer.com/question.md?key=semester_6%2Fdigital_control_system%3Apd-design-double-integrator) — Digital Control System, chapter 4, asked 2×