---
title: "Transfer function from state equations, output x1"
subject: "Control Systems"
chapter: 8
key: "semester_3/control_systems:tf-from-de-3x1-x2"
source: https://www.bctengineer.com/q/semester_3/control_systems/tf-from-de-3x1-x2
site: BCT Engineering
---

# Transfer function from state equations, output x1

Control Systems · Chapter 8: State Space Analysis

- Asked in: 2073 Shrawan, 6 marks
- Video: [Transfer Function Derivation from State Model Statement Problem No  4 - Control Systems](https://www.youtube.com/watch?v=n_spUERHzP8) — Ekeeda
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_3/control_systems/tf-from-de-3x1-x2
- Whole chapter: https://www.bctengineer.com/md/semester_3/control_systems/chapter-8

## Question

The differential equations related to a system are dx1/dt = −3x1 + x2 and dx2/dt = −2x1 + u for t > 0. Its output equation is given by y = x1. Derive the transfer function of the system with these differential equations and output equation.

## Short answer

From state model with A=[-3 1; -2 0], B=[0; 1], C=[1 0], the transfer function is Y(s)/U(s) = 1/(s^2+3s+2).

## Answer

#### Step 1: Write the state model
$$
\dot{x}_1 = -3x_1 + x_2, \qquad \dot{x}_2 = -2x_1 + u, \qquad y = x_1
$$

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

#### Step 2: Transfer function $C(sI-A)^{-1}B$

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

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

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

$$
\frac{Y(s)}{U(s)} = \frac{1}{s^2 + 3s + 2} = \frac{1}{(s+1)(s+2)}
$$

#### Check by direct Laplace transform (zero initial conditions)

$$
\begin{aligned}
sX_1 &= -3X_1 + X_2 \;\Rightarrow\; X_2 = (s+3)X_1 \\
sX_2 &= -2X_1 + U \;\Rightarrow\; s(s+3)X_1 + 2X_1 = U \\
\frac{Y}{U} &= \frac{X_1}{U} = \frac{1}{s^2+3s+2}
\end{aligned}
$$

Poles at $s = -1, -2$: the system is stable and overdamped.

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

## Check yourself

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

1. What are A and B from x1' = -3x1 + x2, x2' = -2x1 + u?
   A. A = [-3 1; 0 -2], B = [1; 0]
   B. A = [-3 1; -2 0], B = [0; 1]
   C. A = [-3 -2; 1 0], B = [0; 1]
   D. A = [-3 1; -2 0], B = [1; 0]
   Answer: B. Rows of A come from the coefficients in each equation; u appears only in the second. (Key point: State model)

2. What is |sI - A|?
   A. s^2 + 3s - 2
   B. s^2 - 3s + 2
   C. s^2 + 3s + 2
   D. s^2 + 2s + 3
   Answer: C. s(s+3) + 2. (Key point: Determinant)

3. What is Y/U?
   A. 1/(s^2 + 3s - 2)
   B. (s+3)/(s^2 + 3s + 2)
   C. s/(s^2 + 3s + 2)
   D. 1/((s+1)(s+2))
   Answer: D. X2 = (s+3)X1 and s(s+3)X1 + 2X1 = U. (Key point: Result)

4. What do the poles -1 and -2 imply?
   A. Stable and overdamped
   B. Unstable
   C. Marginally stable
   D. Underdamped
   Answer: A. Both are real and negative. (Key point: Stability)

## Close variants set in other exams

- [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×

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