---
title: "Characterizing control system by speed, accuracy, stability"
subject: "Control Systems"
chapter: 1
key: "semester_3/control_systems:speed-accuracy-stability"
source: https://www.bctengineer.com/q/semester_3/control_systems/speed-accuracy-stability
site: BCT Engineering
---

# Characterizing control system by speed, accuracy, stability

Control Systems · Chapter 1: Control System Background

- Asked in: 2070 Chaitra, 4 marks
- Video: [System Response Characteristics](https://www.youtube.com/watch?v=Euly5rHtex4) — Neso Academy
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_3/control_systems/speed-accuracy-stability
- Whole chapter: https://www.bctengineer.com/md/semester_3/control_systems/chapter-1

## Question

How can you characterize a control system in terms of (i) Speed (ii) Accuracy (iii) Stability? Explain.

## Short answer

Speed is judged by transient response (time constant, rise and settling times). Accuracy is the steady-state error, determined by system type and error constants Kp, Kv, Ka. Stability requires all closed-loop poles in the left-half plane; relative stability is measured by damping ratio, overshoot, gain and phase margins. These three conflict.

## Answer

A control system is judged by three basic requirements: how fast it responds, how accurately it follows the input, and whether it is stable.

#### (i) Speed
Speed is the quickness of the **transient response**. It is measured by:
- time constant $\tau$ (first order), settling time $t_s \approx 4\tau$;
- rise time $t_r$, peak time $t_p = \dfrac{\pi}{\omega_n\sqrt{1-\zeta^2}}$, settling time $t_s = \dfrac{4}{\zeta\omega_n}$ (2% band);
- bandwidth in frequency response (larger bandwidth = faster).

#### (ii) Accuracy
Accuracy is how closely the output follows the input in **steady state**, measured by steady-state error

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

with error constants $K_p$, $K_v$, $K_a$: step $e_{ss} = 1/(1+K_p)$, ramp $1/K_v$, parabola $1/K_a$. Higher type number or gain gives better accuracy.

#### (iii) Stability
A system is **stable** if every bounded input gives a bounded output (BIBO); for an LTI system all closed-loop poles must lie in the left half of the s-plane. Tests: Routh–Hurwitz, root locus, Bode, Nyquist. **Relative stability** (how far from instability) is measured by damping ratio, peak overshoot, gain margin and phase margin.

These three conflict: higher gain improves speed and accuracy but reduces damping and can cause instability, so design is a compromise.

## Check yourself

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

1. Which set of quantities measures the SPEED of a control system?
   A. Time constant, rise time, peak time, settling time and bandwidth
   B. Steady-state error and error constants
   C. Gain margin and phase margin
   D. Type number and Routh array
   Answer: A. Speed is the quickness of the transient response, measured by time-domain times and bandwidth. (Key point: Speed measures)

2. Accuracy is measured in steady state by
   A. the settling time of the response
   B. the steady-state error e_ss = lim s->0 of sE(s), via Kp, Kv, Ka
   C. the peak overshoot
   D. the location of the closed-loop poles only
   Answer: B. Accuracy is how closely the output follows the input once transients die out: the steady-state error. (Key point: Accuracy)

3. An LTI system is stable if
   A. any closed-loop pole lies on the right half-plane
   B. the gain is as large as possible
   C. all closed-loop poles lie in the left half of the s-plane
   D. the steady-state error is zero
   Answer: C. BIBO stability for an LTI system needs all poles in the left half-plane. (Key point: Stability condition)

## Related in Control Systems

- [Control type for constant-speed actuator](https://www.bctengineer.com/question.md?key=semester_3%2Fcontrol_systems%3Aconstant-speed-actuator) — Control Systems, chapter 1
- [Feedback effect in DC motor speed control](https://www.bctengineer.com/question.md?key=semester_3%2Fcontrol_systems%3Afeedback-dc-motor-speed) — Control Systems, chapter 1
- [Stability of 56/[s(s⁴+7s³+6s²+42s+8)] feedback system](https://www.bctengineer.com/question.md?key=semester_3%2Fcontrol_systems%3Astability-56-s5) — Control Systems, chapter 4
- [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 and its significance in engineering](https://www.bctengineer.com/question.md?key=semester_3%2Fcontrol_systems%3Awhat-is-cs-significance) — Control Systems, chapter 1

## Related in other subjects

- [Ward-Leonard system of speed control](https://www.bctengineer.com/question.md?key=semester_7%2Futilization_of_electrical_energy%3Award-leonard-speed-control) — Utilization of Electrical Energy, chapter 3
- [Order of accuracy, consistency and stability](https://www.bctengineer.com/question.md?key=semester_7%2Fapplied_computational_fluid_dynamics%3Aaccuracy-order-consistency-convergence) — Applied Computational Fluid Dynamics, chapter 3
- [Amplifier vs oscillator in terms of stability factor](https://www.bctengineer.com/question.md?key=semester_7%2Frf_and_microwave_engineering%3Aamplifier-oscillator-stability-factor) — RF and Microwave Engineering, chapter 6
- [Parking terms; speed-density and flow relations](https://www.bctengineer.com/question.md?key=semester_7%2Ftransportation_engineering_ii%3Aparking-terms-speed-density-relation) — Transportation Engineering II, chapter 1