---
title: "Transfer function and stability from [0 −1; −2 −3]"
subject: "Control Systems"
chapter: 8
key: "semester_3/control_systems:ss-tf-0-1-2-3"
source: https://www.bctengineer.com/q/semester_3/control_systems/ss-tf-0-1-2-3
site: BCT Engineering
---

# Transfer function and stability from [0 −1; −2 −3]

Control Systems · Chapter 8: State Space Analysis

- Asked in: 2082 Baisakh, 5+1 marks
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_3/control_systems/ss-tf-0-1-2-3
- Whole chapter: https://www.bctengineer.com/md/semester_3/control_systems/chapter-8

## Question

Determine transfer function for the system whose state space representation is given by: [ẋ1; ẋ2] = [0 −1; −2 −3][x1; x2] + [1; 2]u, y = [1 0][x1; x2]. Also comment on the stability of the system.

## Short answer

The transfer function is Y(s)/U(s) = (s+1)/(s^2+3s-2). One pole at s = 0.562 lies in the right half-plane, so the system is unstable.

## Answer

For $\dot{x} = Ax + Bu$, $y = Cx + Du$:

$$
\frac{Y(s)}{U(s)} = C(sI - A)^{-1}B + D = \frac{C\,\operatorname{adj}(sI-A)\,B}{|sI-A|} + D
$$

$$
A = \begin{bmatrix} 0 & -1 \\ -2 & -3 \end{bmatrix},\quad B = \begin{bmatrix} 1 \\ 2 \end{bmatrix},\quad C = \begin{bmatrix} 1 & 0 \end{bmatrix}
$$

#### Step 1: $(sI-A)$ and determinant

$$
sI - A = \begin{bmatrix} s & 1 \\ 2 & s+3 \end{bmatrix},\qquad |sI-A| = s(s+3) - 2 = s^2 + 3s - 2
$$

#### Step 2: Inverse

$$
(sI-A)^{-1} = \frac{1}{s^2+3s-2}\begin{bmatrix} s+3 & -1 \\ -2 & s \end{bmatrix}
$$

#### Step 3: Transfer function

$$
\begin{aligned}
(sI-A)^{-1}B &= \frac{1}{s^2+3s-2}\begin{bmatrix} (s+3) - 2 \\ -2 + 2s \end{bmatrix} = \frac{1}{s^2+3s-2}\begin{bmatrix} s+1 \\ 2s-2 \end{bmatrix} \\
\frac{Y(s)}{U(s)} &= \begin{bmatrix} 1 & 0 \end{bmatrix}\frac{1}{s^2+3s-2}\begin{bmatrix} s+1 \\ 2s-2 \end{bmatrix} = \frac{s+1}{s^2+3s-2}
\end{aligned}
$$

#### Stability
Characteristic equation $s^2 + 3s - 2 = 0$:

$$
s = \frac{-3 \pm \sqrt{9 + 8}}{2} = \frac{-3 \pm 4.123}{2} = 0.562,\ -3.562
$$

One pole lies in the **right half** of the s-plane ($s = +0.562$). Also the coefficients of the characteristic polynomial do not all have the same sign, which already fails the necessary condition for stability. So the system is **unstable**; its response contains a growing term $e^{0.562t}$.

**Answer:** $\dfrac{Y(s)}{U(s)} = \dfrac{s+1}{s^2+3s-2}$; unstable.

## Check yourself

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

1. For A = [0 -1; -2 -3], what is |sI - A|?
   A. s^2 + 3s + 2
   B. s^2 + 3s - 2
   C. s^2 - 3s - 2
   D. s^2 + 3s + 1
   Answer: B. sI - A = [s 1; 2 s+3], so determinant = s(s+3) - 2 = s^2 + 3s - 2. (Key point: Determinant)

2. With B = [1;2], C = [1 0], what is Y/U?
   A. (2s-2)/(s^2 + 3s - 2)
   B. 1/(s^2 + 3s - 2)
   C. (s+1)/(s^2 + 3s - 2)
   D. (s+1)/(s^2 + 3s + 2)
   Answer: C. adj(sI-A)B = [s+1; 2s-2], and C picks the first element, s+1. (Key point: Transfer function)

3. What is the stability of this system?
   A. Stable, since the zero is at -1
   B. Stable, since the sum of poles is negative
   C. Marginally stable, since one pole is near zero
   D. Unstable, since a pole lies at s = +0.562
   Answer: D. The roots of s^2 + 3s - 2 are 0.562 and -3.562; a right half-plane pole means instability. (Key point: Stability decision)

4. What shows at once that s^2 + 3s - 2 cannot be stable?
   A. Its coefficients do not all have the same sign
   B. Its degree is two
   C. It has a constant term
   D. Its leading coefficient is 1
   Answer: A. For stability all coefficients of the characteristic polynomial must be positive. (Key point: Quick necessary condition)

## Related in Control Systems

- [Vertical two-mass system transfer function X1/F](https://www.bctengineer.com/question.md?key=semester_3%2Fcontrol_systems%3Atf-vertical-m1-m2-k1-b1) — Control Systems, chapter 2
- [Two-mass system X1/F, X2/F with F-V analogy](https://www.bctengineer.com/question.md?key=semester_3%2Fcontrol_systems%3Afv-tf-m1-m2-k1b1-k2b2) — Control Systems, chapter 2, asked 3×
- [Transfer function and stability from [1 0; 1 1]](https://www.bctengineer.com/question.md?key=semester_3%2Fcontrol_systems%3Ass-tf-1-0-1-1) — Control Systems, chapter 8
- [Free body diagram and X1/F, X2/F of two-mass system](https://www.bctengineer.com/question.md?key=semester_3%2Fcontrol_systems%3Afbd-m1-m2-k1k2-b1b2) — Control Systems, chapter 2
- [Transfer function from state equations, output x1](https://www.bctengineer.com/question.md?key=semester_3%2Fcontrol_systems%3Atf-from-de-3x1-x2) — Control Systems, chapter 8, asked 2×
- [Transfer function from state equations, output x2](https://www.bctengineer.com/question.md?key=semester_3%2Fcontrol_systems%3Atf-from-de-3x1-x2-y) — Control Systems, chapter 8, asked 2×

## Related in other subjects

- [Pulse transfer function from state space form](https://www.bctengineer.com/question.md?key=semester_6%2Fdigital_control_system%3Aderive-pulse-tf-state-space) — Digital Control System, chapter 5, asked 3×
- [Find x3[n] from X3=X1X2, x1 5-point, x2 4-point](https://www.bctengineer.com/question.md?key=semester_7%2Fdigital_signal_analysis_and_processing%3Adft-product-x3-73s) — Digital Signal Analysis and Processing, chapter 7, asked 3×
- [Practical factors in sampling; sampling interval for X1(t)X2(t)](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Asampling-practical-x1x2) — Communication Systems, chapter 4
- [Pulse transfer function matrix (G = [−4 1; −2 −1], D = [1; 0])](https://www.bctengineer.com/question.md?key=semester_6%2Fdigital_control_system%3Apulse-tf-matrix-d-nonzero) — Digital Control System, chapter 5, asked 7×