Let's test stability of the open-loop transfer function
[
G(s)=\frac{-8}{s^4 + s^3 + 2s^2 + 2s + 3}.
]
(The factor (-8) only scales output — it does **not** move poles — so pole locations are the root

Question

Let's test stability of the open-loop transfer function
[
G(s)=\frac{-8}{s^4 + s^3 + 2s^2 + 2s + 3}.
]
(The factor (-8) only scales output — it does **not** move poles — so pole locations are the roots of the denominator polynomial)
[
P(s)=s^4 + s^3 + 2s^2 + 2s + 3.
]

### 1) Build the Routh array

Coefficients (highest → lowest): (1,;1,;2,;2,;3).

Routh table rows:

[
\begin{array}{c|ccc}
s^4 & 1 & 2 & 3[4pt]
s^3 & 1 & 2 & 0[4pt]
s^2 & \dfrac{1\cdot2 - 1\cdot1}{1}=1 & \dfrac{1\cdot3 - 1\cdot0}{1}=3 & [8pt]
s^1 & \dfrac{1\cdot2 - 1\cdot3}{1} = -1 & 0 & [8pt]
s^0 & 3 & &
\end{array}
]

First column (top → bottom): (1,;1,;1,;-1,;3).

### 2) Count sign changes in the first column

Sign sequence: (+;+;+;-;+) — there are **two** sign changes:

* from (+) to (-) (one),
* from (-) to (+) (second).

By Routh–Hurwitz, the number of sign changes = number of poles in the **right half plane (RHP)**. So there are **2 RHP poles**.

### 3) Conclusion about stability and pole distribution

* The open-loop system is **unstable** (because it has RHP poles).
* Number of poles: total degree 4 → 4 poles split as:

* **Right half plane (Re > 0): 2 poles**
  * **Left half plane (Re < 0): 2 poles**
  *

VIPSolutions · Accepted Answer

# Stability Analysis of the Open-Loop Transfer Function

Given transfer function:

$$
G(s)=\frac{-8}{s^4 + s^3 + 2s^2 + 2s + 3}
$$

The characteristic polynomial:

$$
P(s) = s^4 + s^3 + 2s^2 + 2s + 3
$$

*Note:* The factor $-8$ only scales the, not the pole locations. Poles are roots of the denominator polynomial.

---

## 1) Build the Routh array

Coefficients (highest to lowest): $1, 1, 2, 2, 3$.

### Routh table rows:

| $s^4$ | 1 | 2 | 3 |
|---------|---|---|---|
| $s^3$ | 1 | 2 |  |
| $s^2$ | $\dfrac{1 \times 2 - 1 \times 1}{1} = 1$ | $\dfrac{1 \times 3 - 1 \times }{1} = 3$ |   |
| $s^1$ | $\dfrac{1 \times 2 - 1 \times 3}{1} = -1$ |  |   |
| $s^$ | 3 |   |   |

First column (top → bottom): $\boxed{1,\; 1,\; 1,\; -1,\; 3}$.

---

## 2) Count sign changes in the first column

Sequence of signs: $+\;+\;+\;-\;+$

Number of sign changes:

- From $+$ to $-$: 1
- From $-$ to $+$: 1

**Total sign changes:** 2

By Routh–Hurwitz criterion, the number of sign changes equals the number of poles in the **right half plane (RHP)**.

**Result:** There are **2 RHP poles**.

---

## 3) Conclusion about stability and pole distribution

- The open-loop system is **unstable** because it has RHP poles.
- Total poles: degree 4 polynomial, so:

- **Right half plane (Re > ): 2 poles**
  - **Left half plane (Re < ): 2 poles**
  - **Imaginary axis (jω):  poles**

---

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