---
title: "Controllable canonical form of (z+1)/(z²+1.3z+0.4)"
subject: "Digital Control System"
chapter: 5
key: "semester_6/digital_control_system:ccf-z1-1-3z"
source: https://www.bctengineer.com/q/semester_6/digital_control_system/ccf-z1-1-3z
site: BCT Engineering
---

# Controllable canonical form of (z+1)/(z²+1.3z+0.4)

Digital Control System · Chapter 5: Discrete Time State Equations

- Asked in: 2079 Chaitra, 8 marks
- Video: [Examples On Controllable Canonical Form And Observable Canonical Form | GATE Control System](https://www.youtube.com/watch?v=5TNpxBrULzA) — Ekeeda GATE & ESE
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_6/digital_control_system/ccf-z1-1-3z
- Whole chapter: https://www.bctengineer.com/md/semester_6/digital_control_system/chapter-5

## Question

Consider the following system: Y(z)/U(z)=(z+1)/(z²+1.3z+0.4). Obtain the state space representation of the above pulse transfer function in the controllable canonical form.

## Short answer

With denominator coefficients a1=1.3, a2=0.4 and numerator coefficients b0=0, b1=1, b2=1, introduce an intermediate variable q(k). The controllable canonical form is x(k+1)=[0 1; −0.4 −1.3]x(k)+[0;1]u(k), y=[1 1]x(k).

## Answer

In the controllable canonical form the state variables are successive time shifts of an intermediate variable $q(k)$. $\mathbf{G}$ is a companion matrix with the denominator coefficients in its last row, and $\mathbf{H} = [0\ \ 1]^T$.

#### Coefficients
$$
\frac{Y(z)}{U(z)} = \frac{z+1}{z^2+1.3z+0.4} = \frac{0\cdot z^2 + 1\cdot z + 1}{z^2 + 1.3z + 0.4}
$$
$a_1 = 1.3$, $a_2 = 0.4$, $b_0 = 0$, $b_1 = 1$, $b_2 = 1$.

#### Derivation
Introduce $Q(z)$ such that
$$
\frac{Q(z)}{U(z)} = \frac{1}{z^2+1.3z+0.4},\qquad Y(z) = (z+1)Q(z)
$$
In the time domain, $q(k+2) + 1.3q(k+1) + 0.4q(k) = u(k)$ and $y(k) = q(k+1) + q(k)$.
Choose $x_1(k) = q(k)$ and $x_2(k) = q(k+1)$:
$$
\begin{aligned}
x_1(k+1) &= x_2(k)\\
x_2(k+1) &= -0.4x_1(k) - 1.3x_2(k) + u(k)\\
y(k) &= x_1(k) + x_2(k)
\end{aligned}
$$

#### Controllable canonical form
$$
\begin{bmatrix} x_1(k+1) \\ x_2(k+1) \end{bmatrix} = \begin{bmatrix} 0 & 1 \\ -0.4 & -1.3 \end{bmatrix}\begin{bmatrix} x_1(k) \\ x_2(k) \end{bmatrix} + \begin{bmatrix} 0 \\ 1 \end{bmatrix}u(k)
$$
$$
y(k) = \begin{bmatrix} 1 & 1 \end{bmatrix}\begin{bmatrix} x_1(k) \\ x_2(k) \end{bmatrix}
$$

#### Check
$$
\mathbf{C}(z\mathbf{I}-\mathbf{G})^{-1}\mathbf{H} = \frac{\begin{bmatrix} 1 & 1 \end{bmatrix}\begin{bmatrix} z+1.3 & 1 \\ -0.4 & z \end{bmatrix}\begin{bmatrix} 0 \\ 1 \end{bmatrix}}{z^2+1.3z+0.4} = \frac{1 + z}{z^2+1.3z+0.4}
$$

## Check yourself

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

1. What are a1, a2, b0, b1, b2?
   A. 0.4, 1.3, 0, 1, 1
   B. 1.3, 0.4, 1, 1, 1
   C. 1, 1, 0, 1.3, 0.4
   D. 1.3, 0.4, 0, 1, 1
   Answer: D. Read directly from (z+1)/(z^2+1.3z+0.4). (Key point: Coefficients)

2. How are the states chosen?
   A. x1 = q(k), x2 = q(k+1)
   B. x1 = y(k), x2 = u(k)
   C. x1 = q(k+1), x2 = q(k+2)
   D. x1 = u(k), x2 = u(k+1)
   Answer: A. Successive shifts of the intermediate variable q. (Key point: State choice)

3. What are G and H?
   A. G = [0 1; 0.4 1.3], H = [1; 0]
   B. G = [0 1; -0.4 -1.3], H = [0; 1]
   C. G = [0 1; -1.3 -0.4], H = [0; 1]
   D. G = [-0.4 -1.3; 1 0], H = [1; 0]
   Answer: B. The last row is [-a2 -a1]. (Key point: G and H)

4. What is C?
   A. [0 1]
   B. [1 0]
   C. [1 1]
   D. [0.4 1.3]
   Answer: C. y = q(k+1) + q(k) = x2 + x1. (Key point: Output equation)

## Close variants set in other exams

- [Controllable and observable forms of (z+1)/(z²+1.3z+0.4)](https://www.bctengineer.com/question.md?key=semester_6%2Fdigital_control_system%3Accf-ocf-z1-1-3z) — Digital Control System, chapter 5, asked 14×
- [Diagonal canonical form of (z+1)/(z²+1.3z+0.4)](https://www.bctengineer.com/question.md?key=semester_6%2Fdigital_control_system%3Adiagonal-form-z1-1-3z) — Digital Control System, chapter 5, asked 6×
- [Controllable canonical form of (z⁻¹+2z⁻²)/(1+4z⁻¹+3z⁻²)](https://www.bctengineer.com/question.md?key=semester_6%2Fdigital_control_system%3Accf-z1-2z2-4z1-3z2) — Digital Control System, chapter 5, asked 15×
- [Controllable canonical form of (4z³+3z²+5z+4)/(2z³+5z²+2z+3)](https://www.bctengineer.com/question.md?key=semester_6%2Fdigital_control_system%3Accf-4z3-3z2) — Digital Control System, chapter 5, asked 14×
- [Controllable canonical form of a third-order system](https://www.bctengineer.com/question.md?key=semester_6%2Fdigital_control_system%3Accf-1-3z-5z-4z) — Digital Control System, chapter 5, asked 14×
- [Controllable and observable forms of a 0.368/0.264 system](https://www.bctengineer.com/question.md?key=semester_6%2Fdigital_control_system%3Accf-ocf-0-368-0-264) — Digital Control System, chapter 5, asked 14×
- [Controllable and observable forms (denominator 0.36z⁻²)](https://www.bctengineer.com/question.md?key=semester_6%2Fdigital_control_system%3Accf-ocf-0-36-variant) — Digital Control System, chapter 5, asked 14×
- [Why state representation; controllable canonical form](https://www.bctengineer.com/question.md?key=semester_6%2Fdigital_control_system%3Astate-representation-why-ccf) — Digital Control System, chapter 5, asked 2×

## Related in Digital Control System

- [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×
- [Controllable canonical form of (3z²+7z+15)/(z²+2z+4)](https://www.bctengineer.com/question.md?key=semester_6%2Fdigital_control_system%3Accf-3z2-7z-15) — Digital Control System, chapter 5
- [Observable canonical form of (0.3z⁻¹+0.6z⁻²)/(1−0.2z⁻¹+0.5z⁻²)](https://www.bctengineer.com/question.md?key=semester_6%2Fdigital_control_system%3Aocf-0-3-0-6) — Digital Control System, chapter 5, asked 13×
- [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×
- [Observable canonical form of (4z²+3z+5)/(5z²+z−4)](https://www.bctengineer.com/question.md?key=semester_6%2Fdigital_control_system%3Aocf-4z2-3z-5) — Digital Control System, chapter 5