---
title: "LTI system and closed loop in modern systems"
subject: "Control Systems"
chapter: 1
key: "semester_3/control_systems:lti-define-cl-pronounced"
source: https://www.bctengineer.com/q/semester_3/control_systems/lti-define-cl-pronounced
site: BCT Engineering
---

# LTI system and closed loop in modern systems

Control Systems · Chapter 1: Control System Background

- Asked in: 2072 Chaitra, 4 marks
- Video: [Control Systems Lectures - LTI Systems](https://www.youtube.com/watch?v=3eDDTFcSC_Y) — Brian Douglas
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_3/control_systems/lti-define-cl-pronounced
- Whole chapter: https://www.bctengineer.com/md/semester_3/control_systems/chapter-1

## Question

Define linear time invariant system. Justify the statement "modern complex systems are more pronounced with closed loop control system".

## Short answer

An LTI system obeys superposition and has constant parameters, described by linear constant-coefficient differential equations. Modern complex systems use closed-loop control because it reduces parameter sensitivity, rejects disturbances, speeds response, stabilizes unstable plants, and enables automation.

## Answer

#### Linear time-invariant (LTI) system
A system is **linear** if it obeys the **principle of superposition** (additivity and homogeneity): if $r_1 \to c_1$ and $r_2 \to c_2$, then $a r_1 + b r_2 \to a c_1 + b c_2$. It is **time-invariant** if its parameters do not change with time, so a delayed input gives an equally delayed output: $r(t-T) \to c(t-T)$.

An LTI system is described by a linear differential equation with **constant coefficients**, e.g.

$$
a_2\ddot c + a_1\dot c + a_0c = b_0r
$$

and can be represented by a transfer function $G(s) = C(s)/R(s)$. Example: an RLC circuit with fixed R, L, C.

#### "Modern complex systems are more pronounced with closed-loop control"
Modern systems (power grids, aircraft, robots, process plants) have many interacting parts, changing loads and strict accuracy needs. Closed-loop control suits them because:

1. **Low sensitivity:** $S_G^T = \dfrac{1}{1+GH}$, so ageing and parameter changes hardly affect the output.
2. **Disturbance rejection:** load and noise effects are cut by $(1+GH)$.
3. **Faster response:** time constant $\tau/(1+K)$.
4. **Stabilization:** inherently unstable systems (fighter aircraft, maglev, rockets) can be operated.
5. **Automation:** cheap sensors and digital controllers make feedback easy and allow operation without humans.

So the statement is justified; open loop is used only where the system is simple and well known (washing-machine timer, traffic timer).

## Check yourself

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

1. A system is linear if it obeys
   A. Principle of superposition
   B. Only time invariance
   C. Zero input
   D. Fixed output
   Answer: A. Additivity and homogeneity: a r1 + b r2 gives a c1 + b c2. (Key point: Linearity)

2. Time-invariant means a delayed input r(t-T) produces
   A. c(t)
   B. c(t-T)
   C. c(t+T)
   D. zero
   Answer: B. Parameters do not change with time, so output is equally delayed. (Key point: Time invariance)

3. An LTI system is described by a differential equation with
   A. Time-varying coefficients
   B. Constant coefficients
   C. Nonlinear terms
   D. No coefficients
   Answer: B. Linear constant-coefficient ODEs map to a transfer function. (Key point: LTI description)

4. Why is closed loop favoured in modern complex systems?
   A. Low sensitivity, disturbance rejection and stabilization
   B. It needs no sensors
   C. It is always cheaper
   D. It removes all dynamics
   Answer: A. Feedback gives accuracy despite parameter change and makes unstable plants workable. (Key point: Closed loop for complex systems)

## Related in Control Systems

- [Why closed loop suits modern complex systems](https://www.bctengineer.com/question.md?key=semester_3%2Fcontrol_systems%3Adefine-cs-cl-vs-ol-complex) — Control Systems, chapter 1, asked 2×
- [Define control system and closed loop block roles](https://www.bctengineer.com/question.md?key=semester_3%2Fcontrol_systems%3Adefine-cs-bd-role) — Control Systems, chapter 1, asked 5×
- [Closed loop control system block diagram explained](https://www.bctengineer.com/question.md?key=semester_3%2Fcontrol_systems%3Awhat-is-cs-bd-closed-loop) — Control Systems, chapter 1, asked 4×
- [Control system block diagram and closed loop advantages](https://www.bctengineer.com/question.md?key=semester_3%2Fcontrol_systems%3Awhat-is-cs-bd-cl-advantages) — Control Systems, chapter 1
- [Control system importance and closed loop components](https://www.bctengineer.com/question.md?key=semester_3%2Fcontrol_systems%3Adefine-cs-importance-components) — Control Systems, chapter 1
- [Traffic light system of Kathmandu: open or closed loop](https://www.bctengineer.com/question.md?key=semester_3%2Fcontrol_systems%3Atraffic-light-justify) — Control Systems, chapter 1

## Related in other subjects

- [Steps to design closed loop control system](https://www.bctengineer.com/question.md?key=semester_6%2Fembedded_system%3Aclosed-loop-design-steps) — Embedded System, chapter 7
- [Open loop versus closed loop control](https://www.bctengineer.com/question.md?key=semester_6%2Fembedded_system%3Aopen-vs-closed-loop) — Embedded System, chapter 7, asked 6×
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- [Closed loop speed control design conditions](https://www.bctengineer.com/question.md?key=semester_6%2Fembedded_system%3Aclosed-loop-speed-control-conditions) — Embedded System, chapter 7