---
title: "Error coefficients for k(s+2)/[s²(s²+7s+12)]"
subject: "Control Systems"
chapter: 3
key: "semester_3/control_systems:error-coefficients-importance"
source: https://www.bctengineer.com/q/semester_3/control_systems/error-coefficients-importance
site: BCT Engineering
---

# Error coefficients for k(s+2)/[s²(s²+7s+12)]

Control Systems · Chapter 3: System Transfer Function and Responses

- Asked in: 2080 Baisakh, 6 marks
- Video: [#3 Error Coefficients & Steady State Error Calculation When Input is Given | Control System](https://www.youtube.com/watch?v=kZhCDjtXmlE) — ELECTRICAL HUB
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_3/control_systems/error-coefficients-importance
- Whole chapter: https://www.bctengineer.com/md/semester_3/control_systems/chapter-3

## Question

What is the importance of error coefficients in design of control system? For a unity feedback system having G(s) = k(s+2)/[s²(s²+7s+12)], determine (i) Type and order of the system (ii) Error coefficients and (iii) Steady state error for parabolic input r(t) = 2.5t².

## Short answer

Type 2, order 4; Kp=∞, Kv=∞, Ka=k/6; ess = 30/k for r(t)=2.5t².

## Answer

#### Importance of error coefficients

Static error coefficients ($K_p$, $K_v$, $K_a$) measure how well a closed-loop system follows standard test inputs in steady state. Their role in design:

- They give the steady-state error directly: step $e_{ss}=\frac{A}{1+K_p}$, ramp $e_{ss}=\frac{A}{K_v}$, parabola $e_{ss}=\frac{A}{K_a}$.
- They act as **design specifications**: e.g. "$K_v \ge 20$" fixes the minimum gain needed for a tracking system.
- They show the effect of the **type** of the system: adding an integrator (higher type) makes the error constant for that input infinite, so the error becomes zero.
- They show the trade-off between accuracy and stability: increasing gain increases $K_p, K_v, K_a$ (better accuracy) but usually reduces relative stability. Lag/PI compensators are designed mainly to raise error constants without disturbing the transient response.

#### (i) Type and order

$$
G(s) = \frac{k(s+2)}{s^2(s^2+7s+12)} = \frac{k(s+2)}{s^2(s+3)(s+4)}
$$

- Two poles at the origin: **Type 2**.
- Highest power of $s$ in the denominator is $s^4$: **Order 4**.

#### (ii) Error coefficients

$$
\begin{aligned}
K_p &= \lim_{s\to0}G(s) = \infty \\
K_v &= \lim_{s\to0}sG(s) = \infty \\
K_a &= \lim_{s\to0}s^2G(s) = \frac{k\times2}{12} = \frac{k}{6}
\end{aligned}
$$

#### (iii) Steady-state error for $r(t)=2.5t^2$

Standard parabolic input is $r(t)=\frac{A t^2}{2}$, so $\frac{A}{2}=2.5 \Rightarrow A=5$.

$$
e_{ss} = \frac{A}{K_a} = \frac{5}{k/6} = \frac{30}{k}
$$

**Answer:** Type 2, order 4; $K_p=\infty$, $K_v=\infty$, $K_a = k/6$; $e_{ss} = 30/k$.

**Note on stability:** the error formula is valid only if the closed loop is stable. The characteristic equation is $s^4+7s^3+12s^2+ks+2k=0$. Its Routh array has first column $1,\ 7,\ \frac{84-k}{7},\ \frac{k(k+14)}{k-84},\ 2k$, which always has a sign change for $k>0$. So this particular closed loop is unstable for every positive $k$; $30/k$ is the theoretical value that would apply once the system is stabilised (e.g. by a lead compensator).

## Check yourself

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

1. One key role of error coefficients in design is
   A. They give steady state error and act as design specs
   B. They give the settling time
   C. They give peak time
   D. They remove poles
   Answer: A. Specs like Kv >= 20 fix required gain. (Key point: Importance)

2. G=k(s+2)/[s^2(s+3)(s+4)] has
   A. Type 2, order 4
   B. Type 4, order 2
   C. Type 2, order 3
   D. Type 1, order 4
   Answer: A. Two poles at origin, highest denominator power s^4. (Key point: Type and order)

3. Ka = lim s^2 G(s) equals
   A. infinity
   B. k/6
   C. k/12
   D. k/2
   Answer: B. Ka = k*2/(3*4) = k/6. (Key point: Ka)

4. For r = 2.5t^2, A = 5, so ess equals
   A. 30/k
   B. 5/k
   C. 15/k
   D. 2.5/k
   Answer: A. ess = A/Ka = 5/(k/6) = 30/k. (Key point: Parabolic error)

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