---
title: "Measures to track the reference input"
subject: "Control Systems"
chapter: 1
key: "semester_3/control_systems:track-reference-measures"
source: https://www.bctengineer.com/q/semester_3/control_systems/track-reference-measures
site: BCT Engineering
---

# Measures to track the reference input

Control Systems · Chapter 1: Control System Background

- Asked in: 2075 Chaitra, 4 marks; 2078 Kartik, 4 marks (asked 2 times)
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_3/control_systems/track-reference-measures
- Whole chapter: https://www.bctengineer.com/md/semester_3/control_systems/chapter-1

## Question

There exists a permanent mismatch to track the reference by the actual system. Suggest a way out to follow your system as per reference and justify.

## Short answer

Use a closed-loop system with integral action (PI or PID controller) to raise the system type. An integrator makes the position error constant infinite for step inputs, eliminating the steady-state error permanently because the integral term accumulates any error and forces the output to match the reference.

## Answer

A permanent mismatch between the reference and the output is a **steady-state error** $e_{ss}$. It can be removed by **using a closed-loop system with integral action** (PI or PID controller), i.e. by raising the **type number** of the loop.

#### Justification
For unity feedback, $E(s) = \dfrac{R(s)}{1+G(s)}$ and

$$
e_{ss} = \lim_{s\to0} sE(s)
$$

For a unit step, $e_{ss} = \dfrac{1}{1+K_p}$, where the position error constant is $K_p = \lim_{s\to0}G(s)$.

- **Type-0 system**, e.g. $G(s) = \dfrac{K}{\tau s+1}$: $K_p = K$, so $e_{ss} = \dfrac{1}{1+K} \ne 0$. The mismatch is permanent.
- Raising $K$ reduces the error (e.g. $K = 99$ gives 1%) but never makes it zero, and high gain can make higher-order systems oscillatory or unstable.
- **Add an integrator** (PI controller $G_c(s) = K_c + K_i/s$): the loop becomes type 1 and the position constant $K_p \to \infty$, so

$$
e_{ss} = \frac{1}{1+\infty} = 0
$$

The integrator keeps changing its output as long as any error exists, so the output is forced to follow the reference exactly.

#### Other measures
- If the system is open loop, convert it to **closed loop** with a sensor so that the error is measured and corrected.
- For ramp references, use a type-2 loop (double integration) or a lag compensator to raise $K_v$.
- Calibrate the sensor and feedback element: an error in $H$ appears directly in the output, since $C/R \approx 1/H$.

## Check yourself

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

1. A permanent mismatch between reference and output is called
   A. Overshoot
   B. Steady-state error
   C. Rise time
   D. Damping
   Answer: B. A persistent difference after transients die out is the steady-state error e_ss. (Key point: Meaning of permanent mismatch)

2. What is the proper way to remove the steady-state error for a step reference?
   A. Only reduce the gain
   B. Add an integrator (PI controller) to raise the type number
   C. Remove the feedback
   D. Use a smaller sensor
   Answer: B. An integrator makes the loop type 1, so Kp tends to infinity and e_ss = 1/(1+Kp) = 0. (Key point: Remedy)

3. For the type-0 system G(s) = K/(tau s + 1) with unity feedback, the unit-step steady-state error is
   A. 0
   B. K
   C. 1/(1+K)
   D. 1/K
   Answer: C. Kp = K, so e_ss = 1/(1+Kp) = 1/(1+K), which is never zero. (Key point: Type-0 error)

4. Why does an integrator drive the steady-state error to zero?
   A. It reduces the plant gain
   B. Its output keeps changing as long as any error exists
   C. It removes the sensor
   D. It makes the system open loop
   Answer: B. The integrator output keeps changing until the error is zero, forcing the output to follow the reference exactly. (Key point: Why integrator works)

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- [Block diagram reduction with G4 on input and H1–H2](https://www.bctengineer.com/question.md?key=semester_3%2Fcontrol_systems%3Abdr-g1-g4-h1h2-2074c) — Control Systems, chapter 3, asked 2×
- [How feedback rejects disturbance input](https://www.bctengineer.com/question.md?key=semester_3%2Fcontrol_systems%3Afeedback-reject-disturbance) — Control Systems, chapter 1
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