---
title: "PI controller and its effect on time response"
subject: "Control Systems"
chapter: 7
key: "semester_3/control_systems:pi-controller-role"
source: https://www.bctengineer.com/q/semester_3/control_systems/pi-controller-role
site: BCT Engineering
---

# PI controller and its effect on time response

Control Systems · Chapter 7: Performance Specifications and Compensation Design

- Asked in: 2081 Baisakh, 4 marks; 2081 Baisakh, 4 marks
- Video: [Proportional Integral PI Controller](https://www.youtube.com/watch?v=lVQxEwxVqgk) — TutorialsPoint
- 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-role
- Whole chapter: https://www.bctengineer.com/md/semester_3/control_systems/chapter-7

## Question

Discuss briefly about PI controller and its effect on time response.

## Short answer

A PI controller adds a pole at the origin and a zero, increasing system type to eliminate steady-state error. It increases order, reduces relative stability, and tends to increase overshoot and settling time while improving low-frequency accuracy.

## Answer

A **PI (proportional–integral) controller** produces an output proportional to the error plus the integral of the error:

$$
u(t)=K_pe(t)+K_i\int_0^te(t)\,dt,\qquad G_c(s)=K_p+\frac{K_i}{s}=\frac{K_p\left(s+\frac{K_i}{K_p}\right)}{s}
$$

It adds a **pole at the origin** and a **zero at $s=-K_i/K_p$** to the open-loop transfer function.

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

#### Effect on time response

Consider a plant $G(s)=\dfrac{\omega_n^2}{s(s+2\zeta\omega_n)}$ with PI control. The open loop becomes

$$
G_c(s)G(s)=\frac{\omega_n^2(K_ps+K_i)}{s^2(s+2\zeta\omega_n)}
$$

1. **System type increases by one** (type 1 to type 2). Step error stays zero and the ramp steady-state error becomes zero ($K_v=\infty$). This is the main benefit: **steady-state accuracy improves**.
2. **Order increases** (2nd to 3rd order). The characteristic equation $s^3+2\zeta\omega_ns^2+K_p\omega_n^2s+K_i\omega_n^2=0$ is stable only if $2\zeta\omega_nK_p>K_i$; too large $K_i$ causes instability.
3. **Transient response:** the integrator adds phase lag, so damping decreases. Overshoot increases and settling time becomes longer; rise time usually decreases a little.
4. The PI controller acts as a **low-pass filter** (like a lag compensator), so it reduces high-frequency noise.

| Property | Effect of PI |
|---|---|
| Steady-state error | eliminated / reduced |
| Type, order | both increase by 1 |
| Overshoot | increases |
| Settling time | increases |
| Relative stability | reduces |
| Noise | attenuated |

**Example:** a speed-control loop with only P control settles with a small offset; adding the integral term removes the offset, but the speed may overshoot more before settling.

## Check yourself

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

1. What does a PI controller Gc(s) = Kp + Ki/s add to the open-loop transfer function?
   A. A pole at the origin and a zero at -Ki/Kp
   B. A zero at the origin and a pole at -Ki/Kp
   C. Two poles at the origin
   D. A zero at -Kp/Ki only
   Answer: A. Gc = Kp(s + Ki/Kp)/s has a pole at s = 0 and a zero at s = -Ki/Kp. (Key point: PI structure)

2. What is the main benefit of PI control on the time response?
   A. Reduced rise time and noise gain
   B. System type rises by one, so steady-state error (e.g. ramp error of a type-1 plant) becomes zero
   C. Reduced order of the system
   D. Greater damping and lower overshoot
   Answer: B. The integrator raises the type by one, improving steady-state accuracy. (Key point: Steady-state effect)

3. How does PI control typically affect the transient response?
   A. Overshoot decreases and settling time shortens
   B. Damping decreases, so overshoot and settling time increase and large Ki can destabilise
   C. No change to transient response
   D. Order of the system drops by one
   Answer: B. The integrator adds phase lag and raises the order, lowering damping; too large Ki leads to instability. (Key point: Transient drawback)

## Related in Control Systems

- [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
- [Short note: PD and PI controller](https://www.bctengineer.com/question.md?key=semester_3%2Fcontrol_systems%3Ashort-pd-pi) — Control Systems, chapter 7
- [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 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×
- [Effect of feedback on gain and speed of response](https://www.bctengineer.com/question.md?key=semester_3%2Fcontrol_systems%3Afeedback-gain-speed) — Control Systems, chapter 1

## 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, plant e^(−5s)/(s+0.4), step response](https://www.bctengineer.com/question.md?key=semester_6%2Fdigital_control_system%3Api-design-dead-time-5s-step) — Digital Control System, chapter 4
- [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×
- [Controller design for peak time ≤ 1.2 s (ζ = 0.5)](https://www.bctengineer.com/question.md?key=semester_6%2Fdigital_control_system%3Adesign-peak-time-zeta-0-5) — Digital Control System, chapter 4, asked 2×