---
title: "True/false: proportional controller makes error zero"
subject: "Control Systems"
chapter: 7
key: "semester_3/control_systems:tf-proportional-sse-zero"
source: https://www.bctengineer.com/q/semester_3/control_systems/tf-proportional-sse-zero
site: BCT Engineering
---

# True/false: proportional controller makes error zero

Control Systems · Chapter 7: Performance Specifications and Compensation Design

- Asked in: 2068 Baisakh (old course), 1+3 marks
- Video: [PID Settings | Proportional-Only Control](https://www.youtube.com/watch?v=E780BPOjKwM) — RealPars
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_3/control_systems/tf-proportional-sse-zero
- Whole chapter: https://www.bctengineer.com/md/semester_3/control_systems/chapter-7

## Question

State whether the following statement is true or false and justify: Proportional controller makes the steady state error zero.

## Short answer

False. A proportional controller only reduces steady-state error; it cannot make it zero unless the plant already contains an integrator. Zero error requires integral action, which adds a pole at the origin.

## Answer

**False.** A proportional controller only **reduces** the steady-state error; it cannot make it zero (unless the plant already contains an integrator).

**Justification.** With $G_c(s) = K_p$ and a type-0 plant $G(s)$ in a unity feedback loop, the step error is

$$
e_{ss} = \lim_{s\to 0} \frac{s\cdot\frac{1}{s}}{1 + K_p G(s)} = \frac{1}{1 + K_p G(0)}
$$

Example: $G(s) = \dfrac{1}{s+1}$, so $G(0) = 1$.

| $K_p$ | $e_{ss} = 1/(1+K_p)$ |
|---|---|
| 1 | 0.5 |
| 9 | 0.1 |
| 99 | 0.01 |

The error falls as $K_p$ rises but becomes zero only for $K_p \to \infty$, which is not possible. Very high gain also makes higher-order systems oscillatory or unstable and saturates the actuator.

A P controller adds no pole at the origin, so it does not change the **system type**. Zero steady-state error needs integral action (PI or PID controller), which adds a pole at $s = 0$.

## Check yourself

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

1. A proportional controller makes the steady-state error zero. This statement is:
   A. False; it increases the error
   B. True for any plant
   C. False; it only reduces the error
   D. True only for step inputs
   Answer: C. A P controller only reduces steady-state error unless the plant already has an integrator. (Key point: True or false)

2. For a type-0 plant with unity feedback and P control, the step error is:
   A. 1/(Kp*G(0))
   B. 1/(1 + Kp*G(0))
   C. Kp*G(0)/(1 + Kp*G(0))
   D. 0
   Answer: B. ess = lim s(1/s)/(1+KpG(s)) = 1/(1+KpG(0)). (Key point: Error formula)

3. For G(s) = 1/(s+1), what is ess when Kp = 9?
   A. 0.01
   B. 0.5
   C. 0.9
   D. 0.1
   Answer: D. G(0) = 1, so ess = 1/(1+9) = 0.1. (Key point: Numbers)

4. What is needed for zero steady-state step error with a type-0 plant?
   A. Reducing the damping
   B. Integral action, which adds a pole at s = 0
   C. A very small Kp
   D. A derivative term
   Answer: B. Integral action (PI or PID) raises the type; only Kp tending to infinity would do it for P, which is impractical and destabilizing. (Key point: How to get zero error)

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