---
title: "Transfer function, poles and zeros from 3×3 state model"
subject: "Control Systems"
chapter: 8
key: "semester_3/control_systems:ss-tf-poles-zeros-3x3"
source: https://www.bctengineer.com/q/semester_3/control_systems/ss-tf-poles-zeros-3x3
site: BCT Engineering
---

# Transfer function, poles and zeros from 3×3 state model

Control Systems · Chapter 8: State Space Analysis

- Asked in: 2074 Asoj, 8 marks
- Video: [Finding Poles and Zeros using State-space Methods](https://www.youtube.com/watch?v=FPPWH797nFk) — richard pates
- 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-poles-zeros-3x3
- Whole chapter: https://www.bctengineer.com/md/semester_3/control_systems/chapter-8

## Question

Given state equation and output equation, find transfer function Y(s)/U(s) and determine the poles and zeros. Ẋ = [0 1 0; −1 −1 0; 1 0 0]X + [0; 1; 0]u and y = [0 0 1]X

## Short answer

Compute transfer function as 1/(s^3+s^2+s). Poles: s=0, s=-0.5±j0.866. No finite zeros exist.

## Answer

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

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

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

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

$$
sI - A = \begin{bmatrix} s & -1 & 0 \\ 1 & s+1 & 0 \\ -1 & 0 & s \end{bmatrix}
$$

Expanding along the third column:

$$
|sI - A| = s\begin{vmatrix} s & -1 \\ 1 & s+1 \end{vmatrix} = s\left[s(s+1) + 1\right] = s^3 + s^2 + s
$$

#### Step 2: Needed adjoint elements
$C$ picks row 3 and $B$ picks column 2, so only $\operatorname{adj}(sI-A)_{32}$ is needed. It is the cofactor of element (2,3):

$$
\operatorname{adj}_{32} = (-1)^{2+3}\begin{vmatrix} s & -1 \\ -1 & 0 \end{vmatrix} = -\left[(s)(0) - (-1)(-1)\right] = 1
$$

(The full column $\operatorname{adj}(sI-A)B = [\,s,\; s^2,\; 1\,]^T$.)

#### Step 3: Transfer function

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

#### Poles and zeros
- **Poles:** $s(s^2 + s + 1) = 0$:

$$
s = 0, \qquad s = \frac{-1 \pm \sqrt{1-4}}{2} = -0.5 \pm j0.866
$$

- **Zeros:** the numerator is a constant, so there are **no finite zeros** (three zeros at infinity).

The pole at the origin makes the system marginally stable (type 1); the complex pair has $\omega_n = 1$ rad/s and $\zeta = 0.5$.

**Answer:** $\dfrac{Y(s)}{U(s)} = \dfrac{1}{s(s^2+s+1)}$; poles $0,\ -0.5 \pm j0.866$; no finite zeros.

## Check yourself

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

1. What is |sI - A| for this A?
   A. s^3 + s^2 + 1
   B. s^3 + s^2 + s
   C. s^3 - s^2 + s
   D. s^2 + s + 1
   Answer: B. Expanding along the third column gives s[s(s+1)+1]. (Key point: Determinant)

2. What is Y/U?
   A. s/(s^3 + s^2 + s)
   B. 1/(s^2 + s + 1)
   C. 1/(s(s^2 + s + 1))
   D. (s+1)/(s^3 + s^2 + s)
   Answer: C. The needed adjoint element is 1, so the numerator is constant. (Key point: Transfer function)

3. What are the poles?
   A. 0 and -1, -1
   B. -1 and +/- j
   C. 0.5 +/- j0.866 and 0
   D. 0 and -0.5 +/- j0.866
   Answer: D. s^2 + s + 1 has roots -0.5 +/- j0.866. (Key point: Poles)

4. How many finite zeros exist?
   A. None, because the numerator is constant
   B. One at s = 0
   C. Two
   D. Three
   Answer: A. A constant numerator has no finite zeros. (Key point: Zeros)

## Close variants set in other exams

- [Transfer function from third-order state model](https://www.bctengineer.com/question.md?key=semester_3%2Fcontrol_systems%3Ass-tf-3x3-1-2-3) — Control Systems, chapter 8, asked 3×

## Related in Control Systems

- [Poles and zeros from state model](https://www.bctengineer.com/question.md?key=semester_3%2Fcontrol_systems%3Ass-zeros-poles-5-6-1) — Control Systems, chapter 8
- [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×
- [Transfer function from state model [0 1; −5 −6]](https://www.bctengineer.com/question.md?key=semester_3%2Fcontrol_systems%3Ass-tf-0-1-5-6) — Control Systems, chapter 8, asked 3×
- [Transfer function from symbolic mass-spring-damper state model](https://www.bctengineer.com/question.md?key=semester_3%2Fcontrol_systems%3Ass-tf-mbk-symbolic) — Control Systems, chapter 8
- [Transfer function from state model [0 −1; −2 −3]](https://www.bctengineer.com/question.md?key=semester_3%2Fcontrol_systems%3Ass-tf-0-1-2-2) — Control Systems, chapter 8

## Related in other subjects

- [Difference equation example; poles/zeros response](https://www.bctengineer.com/question.md?key=semester_7%2Fdigital_signal_analysis_and_processing%3Apole-zero-define-difference-eq) — Digital Signal Analysis and Processing, chapter 3
- [Poles and zeros of a network function](https://www.bctengineer.com/question.md?key=semester_3%2Felectrical_circuit_theory%3Apoles-and-zeros-significance) — Electrical Circuit Theory, chapter 5
- [Butterworth order, poles and transfer function](https://www.bctengineer.com/question.md?key=semester_5%2Ffilter_design%3Abutterworth-order-0-5db-20db-2000) — Filter Design, chapter 2
- [Inverse Chebyshev characteristics, order, poles and zeros](https://www.bctengineer.com/question.md?key=semester_5%2Ffilter_design%3Ainverse-chebyshev-poles-zeros) — Filter Design, chapter 2, asked 2×