---
title: "Closed loop speed control design conditions"
subject: "Embedded System"
chapter: 7
key: "semester_6/embedded_system:closed-loop-speed-control-conditions"
source: https://www.bctengineer.com/q/semester_6/embedded_system/closed-loop-speed-control-conditions
site: BCT Engineering
---

# Closed loop speed control design conditions

Embedded System · Chapter 7: Control System

- Asked in: 2075 Baisakh, 8 marks
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_6/embedded_system/closed-loop-speed-control-conditions
- Whole chapter: https://www.bctengineer.com/md/semester_6/embedded_system/chapter-7

## Question

Draw the block diagram of closed-loop control system for speed control of an automobile and explain the conditions for no unbound and no oscillation showing all the design steps.

## Short answer

Closed-loop speed control uses u=Kp·(r-v) with plant m·dv/dt=-b·v+u+d. Discrete pole p=1-(b+Kp)T/m. Bounded output requires |p|<1 → Kp < 2m/T - b. No oscillation requires 0≤p<1 → Kp ≤ m/T - b. Steady-state error = (b·r-d)/(b+Kp); slope effect reduced to d/(b+Kp). Choose Kp at no-oscillation limit, add integral for zero error.

## Answer

#### Block diagram

```text
 r (set speed)
   -->(+)--e--> [Controller] --u--> [Engine / throttle]
       ^ -        Kp                     |
       |                                 v   d (slope)
       |                              (+)<--
       |                                 |
       |                        [Car: 1/(m s + b)]
       |                                 |
       +----[Speed sensor]<--------------+--> v
```

#### Model and controller

m·dv/dt = -b·v + u + d, controller u = Kp·(r - v).

Sampling with period T: v[k+1] = a·v[k] + c·(u[k] + d[k]), a = 1 - b·T/m, c = T/m.

#### Design steps

1. Substitute the controller: v[k+1] = a·v[k] + c·Kp·(r - v[k]) + c·d[k].
2. Collect terms: v[k+1] = (a - c·Kp)·v[k] + c·Kp·r + c·d[k].
3. The closed-loop pole is p = a - c·Kp = 1 - (b + Kp)·T/m.

#### Condition for no unbounded output

The output stays bounded if |p| < 1, i.e. 0 < (b + Kp)·T/m < 2, so Kp < 2m/T - b.

#### Condition for no oscillation

The response is non-oscillatory if the pole is non-negative: 0 ≤ p < 1, which gives (b + Kp)·T/m ≤ 1, so Kp ≤ m/T - b.

So choose 0 < Kp ≤ m/T - b. For a larger gain (m/T - b < Kp < 2m/T - b) the speed is bounded but oscillates; beyond that it grows without bound.

#### Steady state

v_ss = (Kp·r + d)/(b + Kp). The error r - v_ss = (b·r - d)/(b + Kp) falls as Kp rises, and the effect of the slope is d/(b + Kp), better than d/b of open loop.

#### Final selection

Pick Kp as large as allowed by the no-oscillation limit to get small error, then add an integral term if zero steady-state error is needed. Verify by simulation.

## Check yourself

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

1. What is the closed-loop pole for the sampled P-controlled car?
   A. p = 1 - bT/m
   B. p = 1 + (b + Kp)T/m
   C. p = 1 - (b + Kp)T/m
   D. p = (b + Kp)T/m
   Answer: C. Substituting u = Kp(r - v) gives a - c*Kp. (Key point: Closed-loop pole)

2. What condition gives no oscillation?
   A. Kp <= m/T - b
   B. Kp < 2m/T - b
   C. Kp > m/T - b
   D. p < 0
   Answer: A. Non-oscillatory needs p >= 0. (Key point: No oscillation)

3. If m/T - b < Kp < 2m/T - b, what happens?
   A. Speed grows without bound
   B. Speed is smooth
   C. Steady error is zero
   D. Speed is bounded but oscillates
   Answer: D. The pole is negative but |p| < 1. (Key point: Between limits)

## Related in Embedded System

- [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 automobile cruise controller design](https://www.bctengineer.com/question.md?key=semester_6%2Fembedded_system%3Aopen-loop-cruise-controller) — Embedded System, chapter 7, asked 3×
- [Closed loop diagram; P, PI, PD, PID](https://www.bctengineer.com/question.md?key=semester_6%2Fembedded_system%3Aclosed-loop-block-diagram-pid-types) — 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×
- [PID control operation with block diagram](https://www.bctengineer.com/question.md?key=semester_6%2Fembedded_system%3Apid-operation-block-diagram) — Embedded System, chapter 7, asked 4×
- [Control evaluation steps; PD and PI design](https://www.bctengineer.com/question.md?key=semester_6%2Fembedded_system%3Acontrol-evaluation-steps-pd-pi) — Embedded System, chapter 7

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