---
title: "Root locus of k(s−1)/[s(s−1)]"
subject: "Control Systems"
chapter: 5
key: "semester_3/control_systems:rl-k-s-1-s-s"
source: https://www.bctengineer.com/q/semester_3/control_systems/rl-k-s-1-s-s
site: BCT Engineering
---

# Root locus of k(s−1)/[s(s−1)]

Control Systems · Chapter 5: Root Locus Technique

- Asked in: 2075 Chaitra, 6 marks
- Video: [root locus in control system](https://www.youtube.com/watch?v=MIVkzOiFvhI) — Smart Engineer
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_3/control_systems/rl-k-s-1-s-s
- Whole chapter: https://www.bctengineer.com/md/semester_3/control_systems/chapter-5

## Question

Sketch root locus plot for the system with open loop transfer function G(s)H(s) = k(s−1)/[s(s−1)].

## Short answer

The pole-zero cancellation at s=+1 is not allowed. One closed-loop pole remains fixed at s=+1 for all k, making the system unstable. The other branch moves from the origin along the negative real axis to -∞.

## Answer

$$
G(s)H(s) = \frac{k(s-1)}{s(s-1)}
$$

The factor $(s-1)$ appears in both numerator and denominator. It must **not** be cancelled before forming the characteristic equation, because the cancelled pole at $s = +1$ is still a real mode of the system.

#### Characteristic equation
$$
\begin{aligned}
1 + G(s)H(s) &= 0 \\
s(s-1) + k(s-1) &= 0 \\
(s-1)(s+k) &= 0
\end{aligned}
$$

Closed-loop poles: $s = +1$ (for every $k$) and $s = -k$.

#### Root locus construction
- Open-loop poles: $0$ and $+1$. Open-loop zero: $+1$.
- At $s = +1$ the pole and zero coincide, so a branch "starts and ends" at the same point. One closed-loop pole stays fixed at $s = +1$ for all $k$.
- The reduced part $k/s$ has a pole at $0$ and no finite zero. Its real-axis locus is the whole negative real axis, with asymptote angle $180^\circ$.
- That branch starts at $s = 0$ ($k = 0$) and moves left along the negative real axis, with the pole at $s = -k$ (e.g. $-5$ for $k = 5$). There is no breakaway point and no $j\omega$ crossing.

```text
               jw
               |
 <=============x-----⊗------> σ
  s = -k       0    +1
 (moves left)       (fixed pole,
                    pole-zero overlap)
```

#### Comments
- If the pair is cancelled first, one gets $G = k/s$, which seems stable for all $k > 0$. That conclusion is **wrong**.
- The closed-loop pole at $s = +1$ lies in the right half-plane for every $k$. The system is therefore **unstable for all values of $k$**. Its output contains a growing mode $e^{t}$ that the gain cannot move.
- Pole-zero cancellation in the right half-plane is never used to stabilise a system. Exact cancellation is impossible in practice, and the hidden unstable mode still exists internally.

**Answer:** The root locus is a fixed point at $s = +1$ plus a branch from $0$ to $-\infty$ on the negative real axis. The system is unstable for all $k$.

## Check yourself

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

1. Why must (s-1) not be cancelled before forming the characteristic equation?
   A. It is a repeated factor
   B. The cancelled pole at +1 is still a real mode of the closed-loop system
   C. Cancellation changes K
   D. The zero is at the origin
   Answer: B. The pole at +1 remains a closed-loop pole, so cancelling hides an unstable mode. (Key point: Do not cancel RHP pole-zero)

2. The characteristic equation (s-1)(s+k) = 0 gives poles at:
   A. s = 1 and s = k
   B. s = -1 and s = -k
   C. s = 1 and s = -k
   D. s = 0 and s = -k
   Answer: C. One pole stays fixed at +1 for all k, the other moves along the negative axis at -k. (Key point: Closed-loop poles)

3. What is the stability of the system?
   A. Stable for all k > 0
   B. Unstable for all k
   C. Stable only for k < 1
   D. Marginal for all k
   Answer: B. The fixed pole at +1 lies in the RHP for every k, giving a growing e^t mode. (Key point: Stability conclusion)

## Close variants set in other exams

- [Root locus of K(s+1)/[(s²+2s+2)(s²+2s+5)]](https://www.bctengineer.com/question.md?key=semester_3%2Fcontrol_systems%3Arl-k-s1-s2-2s-2) — Control Systems, chapter 5, asked 3×

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