---
title: "State model from 10(s²+2s)/(s³+5s²+8s+15)"
subject: "Control Systems"
chapter: 8
key: "semester_3/control_systems:ss-from-tf-10s2-2s"
source: https://www.bctengineer.com/q/semester_3/control_systems/ss-from-tf-10s2-2s
site: BCT Engineering
---

# State model from 10(s²+2s)/(s³+5s²+8s+15)

Control Systems · Chapter 8: State Space Analysis

- Asked in: 2081 Baisakh, 8 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-from-tf-10s2-2s
- Whole chapter: https://www.bctengineer.com/md/semester_3/control_systems/chapter-8

## Question

A system is described by the transfer function Y(s)/U(s) = 10(s² + 2s)/(s³ + 5s² + 8s + 15). Find its state and output equation in matrix form.

## Short answer

Phase-variable realization: state equation ̇x = [0 1 0; 0 0 1; -15 -8 -5]x + [0;0;1]u, output equation y = [0 20 10]x.

## Answer

A state model can be written directly from the transfer function in **phase variable (controllable canonical) form**. The denominator gives the system matrix and the numerator gives the output matrix.

#### Given transfer function
$$
\frac{Y(s)}{U(s)} = \frac{10s^2 + 20s}{s^3 + 5s^2 + 8s + 15}
$$
The numerator order (2) is less than the denominator order (3), so there is no direct term ($D = 0$).

#### Step 1: Introduce an intermediate variable
Let
$$
\frac{Y(s)}{U(s)} = \frac{Y(s)}{X(s)}\cdot\frac{X(s)}{U(s)}, \quad \frac{X(s)}{U(s)} = \frac{1}{s^3 + 5s^2 + 8s + 15}, \quad \frac{Y(s)}{X(s)} = 10s^2 + 20s
$$
So
$$
\dddot{x} + 5\ddot{x} + 8\dot{x} + 15x = u, \qquad y = 10\ddot{x} + 20\dot{x}
$$

#### Step 2: Choose state variables
$$
x_1 = x, \quad x_2 = \dot{x}, \quad x_3 = \ddot{x}
$$
Then
$$
\begin{aligned}
\dot{x}_1 &= x_2 \\
\dot{x}_2 &= x_3 \\
\dot{x}_3 &= -15x_1 - 8x_2 - 5x_3 + u
\end{aligned}
$$

#### Step 3: Output equation
$$
y = 10\ddot{x} + 20\dot{x} = 0\cdot x_1 + 20x_2 + 10x_3
$$

#### State and output equations in matrix form
$$
\begin{bmatrix}\dot{x}_1\\ \dot{x}_2\\ \dot{x}_3\end{bmatrix} =
\begin{bmatrix}0 & 1 & 0\\ 0 & 0 & 1\\ -15 & -8 & -5\end{bmatrix}
\begin{bmatrix}x_1\\ x_2\\ x_3\end{bmatrix} +
\begin{bmatrix}0\\ 0\\ 1\end{bmatrix} u
$$
$$
y = \begin{bmatrix}0 & 20 & 10\end{bmatrix}\begin{bmatrix}x_1\\ x_2\\ x_3\end{bmatrix}
$$

So
$$
A = \begin{bmatrix}0 & 1 & 0\\ 0 & 0 & 1\\ -15 & -8 & -5\end{bmatrix},\;
B = \begin{bmatrix}0\\ 0\\ 1\end{bmatrix},\;
C = \begin{bmatrix}0 & 20 & 10\end{bmatrix},\; D = 0
$$

#### Block diagram of the realization
```text
u -->(+)--> 1/s --x3--> 1/s --x2--> 1/s --x1
      ^ -5x3 -8x2 -15x1 (fed back)
y = 20*x2 + 10*x3
```

#### Check
Using $G(s) = C(sI - A)^{-1}B$, the last column of $(sI-A)^{-1}$ is $\frac{1}{\Delta(s)}[1\;\; s\;\; s^2]^T$ with $\Delta(s) = s^3 + 5s^2 + 8s + 15$. Hence
$$
G(s) = \frac{0\cdot 1 + 20s + 10s^2}{s^3 + 5s^2 + 8s + 15} = \frac{10(s^2 + 2s)}{s^3 + 5s^2 + 8s + 15}
$$
which is the given transfer function.

**Answer:** $\dot{x} = Ax + Bu$, $y = Cx$ with $A$, $B$, $C$ as above (phase variable form). The state model is not unique; other forms (observable canonical, Jordan) give the same transfer function.

## Check yourself

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

1. Why is D = 0 for Y/U = 10(s^2+2s)/(s^3+5s^2+8s+15)?
   A. Because the numerator has a zero at s = 0
   B. Because the numerator order (2) is less than the denominator order (3)
   C. Because the system is unstable
   D. Because the denominator has no constant term
   Answer: B. A direct feed-through term appears only when numerator and denominator orders are equal; here the transfer function is strictly proper. (Key point: No direct term)

2. What differential equation does X(s)/U(s) = 1/(s^3+5s^2+8s+15) give?
   A. x''' + 5x'' + 8x' + 15x = u
   B. x''' + 5x'' + 8x' + 15x = 10u
   C. 15x''' + 8x'' + 5x' + x = u
   D. x''' - 5x'' - 8x' - 15x = u
   Answer: A. Cross-multiplying gives (s^3+5s^2+8s+15)X = U, i.e. x''' + 5x'' + 8x' + 15x = u. (Key point: Intermediate variable)

3. What is the last row of A in the phase-variable form?
   A. [-5 -8 -15]
   B. [15 8 5]
   C. [-15 -8 -5]
   D. [0 20 10]
   Answer: C. x3' = -15x1 - 8x2 - 5x3 + u, so the last row is the negated denominator coefficients [-15 -8 -5]. (Key point: State matrix A)

4. With x1 = x, x2 = x', x3 = x'', what is C for y = 10x'' + 20x'?
   A. [10 20 0]
   B. [0 20 10]
   C. [20 10 0]
   D. [0 10 20]
   Answer: B. y = 0*x1 + 20*x2 + 10*x3, so C = [0 20 10]. (Key point: Output matrix C)

## Related in Control Systems

- [State model from 20(4s+2)/(s³+5s²+8s+2)](https://www.bctengineer.com/question.md?key=semester_3%2Fcontrol_systems%3Ass-from-tf-20-4s2) — Control Systems, chapter 8, asked 2×
- [State space advantages; state model of (2s²+3s+1)/(s³+5s²+6s+7)](https://www.bctengineer.com/question.md?key=semester_3%2Fcontrol_systems%3Ass-adv-tf-2s2-3s-1) — Control Systems, chapter 8
- [Root locus of K(s²−2s+5)/(s²+1.5s−1)](https://www.bctengineer.com/question.md?key=semester_3%2Fcontrol_systems%3Arl-k-s2-2s5-s2-15s) — Control Systems, chapter 5, asked 6×
- [Transfer matrix and stability of MIMO state model](https://www.bctengineer.com/question.md?key=semester_3%2Fcontrol_systems%3Ass-mimo-tf-stability) — Control Systems, chapter 8
- [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×
- [Bode plot of 48(1+s)/[s²(1+3s)(1+0.5s)(2+0.2s)]](https://www.bctengineer.com/question.md?key=semester_3%2Fcontrol_systems%3Abode-48-1s-s2-1-3s) — Control Systems, chapter 6

## Related in other subjects

- [Pulse transfer function matrix of a general companion form](https://www.bctengineer.com/question.md?key=semester_6%2Fdigital_control_system%3Apulse-tf-matrix-general-companion) — Digital Control System, chapter 5, asked 7×
- [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×
- [Bode plot: 64(S+2)/(S(S²+0.5S)(S²+3.2S+64))](https://www.bctengineer.com/question.md?key=semester_3%2Felectrical_circuit_theory%3Abode-64s2-s-s2-05s-s2) — Electrical Circuit Theory, chapter 5, asked 4×
- [Pulse transfer function matrix (G = [−4 1; −2 −1], D = 0)](https://www.bctengineer.com/question.md?key=semester_6%2Fdigital_control_system%3Apulse-tf-matrix-g-4-1) — Digital Control System, chapter 5, asked 7×