---
title: "Working of PI controller"
subject: "Control Systems"
chapter: 7
key: "semester_3/control_systems:pi-controller-working"
source: https://www.bctengineer.com/q/semester_3/control_systems/pi-controller-working
site: BCT Engineering
---

# Working of PI controller

Control Systems · Chapter 7: Performance Specifications and Compensation Design

- Asked in: 2074 Chaitra, 4 marks
- Video: [PI Controller Explained: Basics, Block Diagram, Transfer Function, Pros, and Cons](https://www.youtube.com/watch?v=ucppTVy9jRs) — 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/pi-controller-working
- Whole chapter: https://www.bctengineer.com/md/semester_3/control_systems/chapter-7

## Question

Discuss working of PI controller.

## Short answer

A PI controller combines proportional action with integral action. The integral term accumulates error over time, eliminating steady-state error. It adds a pole at the origin and a zero; this may increase overshoot and reduce relative stability due to phase lag.

## Answer

A **PI controller** combines proportional action with integral action. Its output is

$$
u(t)=K_pe(t)+K_i\int_0^te(\tau)\,d\tau=K_p\left[e(t)+\frac{1}{T_i}\int_0^te\,d\tau\right]
$$

$$
G_c(s)=K_p+\frac{K_i}{s}=\frac{K_p\left(s+\frac{1}{T_i}\right)}{s},\quad T_i=\frac{K_p}{K_i}
$$

```text
          +--->[  Kp  ]---+
 e(t) ----|               (+)---> u(t)
          +--->[ Ki/s ]---+
```

#### Working

1. **P part** reacts at once to the present error, giving fast correction.
2. **I part** adds up (integrates) the error over time. As long as any error remains, the integral keeps growing and keeps changing the output.
3. Therefore the system can settle only when the error is **exactly zero**. A constant controller output is then held by the integrator alone; this removes the offset that a pure P controller leaves.

For a step error $E$, the output is $u(t)=K_pE+K_iEt$: an instant jump followed by a ramp.

#### Effects

- Adds a **pole at the origin** (type +1) and a **zero at $-1/T_i$**.
- **Steady-state error** is eliminated for a step (type 0 plant) or ramp (type 1 plant).
- Adds phase lag, so **overshoot increases and relative stability decreases**; the zero is placed near the origin to keep this small.
- Acts like a low-pass filter (similar to a lag compensator); noise is not amplified.
- Problem: **integral wind-up** when the actuator saturates.

**Example:** in a water-level control with P only, the level settles 2 cm below the set point; adding integral action slowly raises the valve opening until the level reaches the set point exactly.

## Check yourself

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

1. What is the PI controller transfer function?
   A. Kp + Ki/s = Kp(s + 1/Ti)/s
   B. Kp + Ki s
   C. Kp(1 + Ti s)
   D. Kp/(1 + Ti s)
   Answer: A. It has a pole at the origin and a zero at -1/Ti. (Key point: Transfer function)

2. Why does PI action remove steady-state offset?
   A. The P part grows without limit
   B. The integral keeps changing the output as long as any error remains, so settling needs exactly zero error
   C. The I part reacts to the rate of change of error
   D. It lowers the plant gain to zero
   Answer: B. A constant output is held by the integrator alone only when the error is zero. (Key point: Why offset vanishes)

3. For a step error E, what is the controller output?
   A. Only a ramp
   B. Only a step Kp*E
   C. An instant jump Kp*E followed by a ramp Ki*E*t
   D. An impulse
   Answer: C. P gives the jump; the integral of a constant gives a ramp. (Key point: Response to step error)

4. What is the main drawback of PI control?
   A. It amplifies high-frequency noise
   B. It increases steady-state error
   C. It reduces system type
   D. It adds phase lag, increasing overshoot and reducing relative stability; also integral wind-up when the actuator saturates
   Answer: D. The pole at the origin adds lag; the zero is placed near the origin to limit this. (Key point: Side effects)

## Related in Control Systems

- [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
- [Application of PI controller with example](https://www.bctengineer.com/question.md?key=semester_3%2Fcontrol_systems%3Api-application-example) — Control Systems, chapter 7
- [Short note: PD and PI controller](https://www.bctengineer.com/question.md?key=semester_3%2Fcontrol_systems%3Ashort-pd-pi) — 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×
- [Range of PI gains Kp, KI for stability](https://www.bctengineer.com/question.md?key=semester_3%2Fcontrol_systems%3Api-kp-ki-range-stable) — Control Systems, chapter 4

## Related in other subjects

- [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
- [Digital PI design for plant with 2 s dead time](https://www.bctengineer.com/question.md?key=semester_6%2Fdigital_control_system%3Api-design-dead-time-2s) — Digital Control System, chapter 4, asked 3×
- [Digital PI design, plant e^(−5s)/(s+0.4), Kv](https://www.bctengineer.com/question.md?key=semester_6%2Fdigital_control_system%3Api-design-dead-time-5s-kv) — Digital Control System, chapter 4, asked 2×
- [Digital PI design, plant e^(−2s)/(s+2)](https://www.bctengineer.com/question.md?key=semester_6%2Fdigital_control_system%3Api-design-dead-time-s2) — Digital Control System, chapter 4, asked 2×