---
title: "Short note: PD and PI controller"
subject: "Control Systems"
chapter: 7
key: "semester_3/control_systems:short-pd-pi"
source: https://www.bctengineer.com/q/semester_3/control_systems/short-pd-pi
site: BCT Engineering
---

# Short note: PD and PI controller

Control Systems · Chapter 7: Performance Specifications and Compensation Design

- Asked in: 2068 Chaitra, 4 marks
- Video: [P, PI, PD, and PID Controllers: A Comparative Analysis in Control System Engineering](https://www.youtube.com/watch?v=OYPXiC7QjUM) — 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/short-pd-pi
- Whole chapter: https://www.bctengineer.com/md/semester_3/control_systems/chapter-7

## Question

Write a short note on PD and PI controller.

## Short answer

PD controller adds a zero, improving transient response, damping, and bandwidth, but amplifies high-frequency noise. PI controller adds a pole at the origin and a zero, increasing system type to eliminate steady-state error, but may slow response and cause integral wind-up.

## Answer

#### PD controller
The control signal is proportional to the error and its rate of change:

$$
u(t) = K_p e(t) + K_d \frac{de(t)}{dt}, \qquad G_c(s) = K_p + K_d s = K_d\left(s + \frac{K_p}{K_d}\right)
$$

- Adds a **zero** at $s = -K_p/K_d$; acts like a lead compensator (high-pass).
- Anticipates the error, so it adds damping: reduces overshoot, rise time and settling time.
- Improves stability and increases bandwidth.
- System type is unchanged, so steady-state error is not removed.
- Amplifies high-frequency noise because of the derivative action.

#### PI controller
The control signal is proportional to the error and its integral:

$$
u(t) = K_p e(t) + K_i \int e(t)\,dt, \qquad G_c(s) = K_p + \frac{K_i}{s} = \frac{K_p(s + K_i/K_p)}{s}
$$

- Adds a **pole at the origin** and a zero; acts like a lag compensator (low-pass).
- Raises the system type by one, so the steady-state error to a step (type-0 plant) becomes zero.
- Reduces bandwidth; the response becomes slower and may overshoot more.
- Can reduce relative stability if $K_i$ is too large; suffers integral wind-up.
- Filters high-frequency noise.

| Feature | PD | PI |
|---|---|---|
| Added singularity | Zero | Pole at origin + zero |
| Improves | Transient response | Steady-state accuracy |
| System type | Unchanged | Increased by 1 |
| Noise | Amplified | Filtered |

## Check yourself

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

1. What is the main effect of a PD controller?
   A. It raises the system type by one
   B. It improves the transient response by adding damping through a zero
   C. It reduces bandwidth to filter noise
   D. It removes steady-state error
   Answer: B. PD adds a zero and anticipates the error, reducing overshoot, rise and settling time; it acts like a lead compensator. (Key point: PD effect)

2. What does a PI controller do to the system type and why?
   A. Raises it by two, because it adds two poles
   B. Raises it by one, because it adds a pole at the origin
   C. Leaves it unchanged, because it adds only a zero
   D. Lowers it by one, because it adds a zero
   Answer: B. Ki/s contributes a pole at s = 0, so steady-state step error of a type-0 plant becomes zero. (Key point: PI effect on type)

3. Which statement about noise is correct?
   A. Both filter noise
   B. PI amplifies noise; PD filters it
   C. PD amplifies high-frequency noise; PI filters it
   D. Both amplify noise
   Answer: C. The derivative term raises high-frequency gain, while PI behaves like a low-pass (lag) network and reduces bandwidth. (Key point: Noise)

4. Why does PD not remove steady-state error?
   A. It leaves the system type unchanged
   B. It increases the damping to infinity
   C. It lowers the loop gain to zero
   D. It adds a pole at the origin
   Answer: A. PD adds only a zero; without an integrator the type and thus the error constants are unchanged. (Key point: PD and steady-state error)

## Related in Control Systems

- [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
- [Short note: Characteristics of PI and PD control](https://www.bctengineer.com/question.md?key=semester_3%2Fcontrol_systems%3Ashort-pi-pd-characteristics) — Control Systems, chapter 7
- [PD controller principle and working](https://www.bctengineer.com/question.md?key=semester_3%2Fcontrol_systems%3Apd-controller) — Control Systems, chapter 7, asked 2×
- [PI controller and its effect on time response](https://www.bctengineer.com/question.md?key=semester_3%2Fcontrol_systems%3Api-controller-role) — 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
- [Application of PI controller with example](https://www.bctengineer.com/question.md?key=semester_3%2Fcontrol_systems%3Api-application-example) — Control Systems, chapter 7

## Related in other subjects

- [Control evaluation steps; PD and PI design](https://www.bctengineer.com/question.md?key=semester_6%2Fembedded_system%3Acontrol-evaluation-steps-pd-pi) — Embedded System, chapter 7
- [Closed loop diagram; P, PI, PD, PID](https://www.bctengineer.com/question.md?key=semester_6%2Fembedded_system%3Aclosed-loop-block-diagram-pid-types) — Embedded System, chapter 7
- [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×
- [Response sequence with digital PI controller](https://www.bctengineer.com/question.md?key=semester_6%2Fdigital_control_system%3Aresponse-kronecker-delta-pi) — Digital Control System, chapter 3