---
title: "State model, controllability and observability of 2/(s³+6s²+11s+6)"
subject: "Control Systems"
chapter: 8
key: "semester_3/control_systems:ss-controllability-observability"
source: https://www.bctengineer.com/q/semester_3/control_systems/ss-controllability-observability
site: BCT Engineering
---

# State model, controllability and observability of 2/(s³+6s²+11s+6)

Control Systems · Chapter 8: State Space Analysis

- Asked in: 2069 Chaitra, 8 marks
- Video: [Controllability and Observability (Problems) - State Space Analysis - Control System](https://www.youtube.com/watch?v=Q7r29_Lv-7g) — Ekeeda
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_3/control_systems/ss-controllability-observability
- Whole chapter: https://www.bctengineer.com/md/semester_3/control_systems/chapter-8

## Question

A system has the transfer function Y(s)/U(s) = 2/(s³ + 6s² + 11s + 6). Find the state and output equation in matrix form and test the controllability and observability of the system.

## Short answer

Phase-variable form: A=[0 1 0;0 0 1;-6 -11 -6], B=[0;0;1], C=[2 0 0]. Controllability matrix Q_c = [B AB A²B] has rank 3 (det=-1). Observability matrix Q_o = [C; CA; CA²] has rank 3 (det=8). The system is completely controllable and observable.

## Answer

#### State model (phase-variable form)

Split the transfer function with an intermediate variable $X(s)$:

$$
\frac{X(s)}{U(s)} = \frac{1}{s^3 + 6s^2 + 11s + 6},\qquad Y(s) = 2X(s)
$$

So $\dddot{x} + 6\ddot{x} + 11\dot{x} + 6x = u$ and $y = 2x$. Choose $x_1 = x$, $x_2 = \dot{x}$, $x_3 = \ddot{x}$:

$$
\dot{x}_1 = x_2,\qquad \dot{x}_2 = x_3,\qquad \dot{x}_3 = -6x_1 - 11x_2 - 6x_3 + u,\qquad y = 2x_1
$$

$$
\begin{bmatrix} \dot{x}_1 \\ \dot{x}_2 \\ \dot{x}_3 \end{bmatrix} = \begin{bmatrix} 0 & 1 & 0 \\ 0 & 0 & 1 \\ -6 & -11 & -6 \end{bmatrix}\begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix} + \begin{bmatrix} 0 \\ 0 \\ 1 \end{bmatrix}u,\qquad y = \begin{bmatrix} 2 & 0 & 0 \end{bmatrix}\begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix}
$$

#### Controllability (Kalman test)
$Q_c = [\,B \;\; AB \;\; A^2B\,]$:

$$
AB = \begin{bmatrix} 0 \\ 1 \\ -6 \end{bmatrix},\qquad A^2B = A(AB) = \begin{bmatrix} 1 \\ -6 \\ -6 \cdot 0 - 11 + 36 \end{bmatrix} = \begin{bmatrix} 1 \\ -6 \\ 25 \end{bmatrix}
$$

$$
Q_c = \begin{bmatrix} 0 & 0 & 1 \\ 0 & 1 & -6 \\ 1 & -6 & 25 \end{bmatrix},\qquad |Q_c| = -1 \neq 0
$$

Rank $= 3 = n$, so the system is **completely state controllable**.

#### Observability
$Q_o = \begin{bmatrix} C \\ CA \\ CA^2 \end{bmatrix}$:

$$
C = [2\;\;0\;\;0],\qquad CA = [0\;\;2\;\;0],\qquad CA^2 = [0\;\;0\;\;2]
$$

$$
Q_o = \begin{bmatrix} 2 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 2 \end{bmatrix},\qquad |Q_o| = 8 \neq 0
$$

Rank $= 3$, so the system is **completely observable**.

This agrees with the transfer function: $s^3 + 6s^2 + 11s + 6 = (s+1)(s+2)(s+3)$ and the numerator is a constant, so there is no pole–zero cancellation.

**Answer:** the system is both controllable and observable.

## Check yourself

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

1. Which A matrix is obtained for s^3+6s^2+11s+6?
   A. Last row [-1 -2 -3]
   B. Last row [-6 -11 -6]
   C. Last row [6 11 6]
   D. Last row [-6 -6 -11]
   Answer: B. Last row is -a0, -a1, -a2. (Key point: State form)

2. Since Y = 2X, what is C?
   A. [0 0 2]
   B. [1 0 0]
   C. [2 0 0]
   D. [0 2 0]
   Answer: C. y = 2 x1. (Key point: Output matrix)

3. What condition proves complete controllability?
   A. Rank of [C; CA; CA^2] equals n
   B. |A| = 0
   C. Rank less than n
   D. Rank of [B AB A^2B] equals n
   Answer: D. Kalman test: Qc must have full rank. (Key point: Controllability test)

4. What is Qo here and what does it imply?
   A. 2I with determinant 8, so completely observable
   B. Singular, so unobservable
   C. Identity with determinant 1, so uncontrollable
   D. Zero matrix
   Answer: A. CA = [0 2 0], CA^2 = [0 0 2]. (Key point: Observability)

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