---
title: "Pulse transfer function matrix (G = [−4 1; −2 −1], D = [1; 0])"
subject: "Digital Control System"
chapter: 5
key: "semester_6/digital_control_system:pulse-tf-matrix-d-nonzero"
source: https://www.bctengineer.com/q/semester_6/digital_control_system/pulse-tf-matrix-d-nonzero
site: BCT Engineering
---

# Pulse transfer function matrix (G = [−4 1; −2 −1], D = [1; 0])

Digital Control System · Chapter 5: Discrete Time State Equations

- Asked in: 2068 Jestha, 8 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_6/digital_control_system/pulse-tf-matrix-d-nonzero
- Whole chapter: https://www.bctengineer.com/md/semester_6/digital_control_system/chapter-5

## Question

Determine pulse transfer function matrix for the following state space representation: x(k+1)=Gx(k)+Hu(k), y(k)=Cx(k)+Du(k), where G=[-4 1; -2 -1], H=[1; 1], C=[1 1; 2 1], D=[1; 0].

## Short answer

Compute F(z)=C(zI−G)^(−1)H+D. The mode at z=−2 cancels, and D1=1 gives direct feed-through. The pulse transfer function matrix is Y1(z)/U(z) = (z+5)/(z+3), Y2(z)/U(z) = 3/(z+3).

## Answer

The pulse transfer function matrix is $\mathbf{F}(z) = \mathbf{C}(z\mathbf{I}-\mathbf{G})^{-1}\mathbf{H} + \mathbf{D}$. There is one input and two outputs, so $\mathbf{F}(z)$ is a 2×1 matrix.

#### Step 1: $z\mathbf{I}-\mathbf{G}$ and its inverse
$$
z\mathbf{I}-\mathbf{G} = \begin{bmatrix} z+4 & -1 \\ 2 & z+1 \end{bmatrix}
$$
$$
|z\mathbf{I}-\mathbf{G}| = (z+4)(z+1) + 2 = z^2 + 5z + 6 = (z+2)(z+3)
$$
$$
(z\mathbf{I}-\mathbf{G})^{-1} = \frac{1}{(z+2)(z+3)}\begin{bmatrix} z+1 & 1 \\ -2 & z+4 \end{bmatrix}
$$

#### Step 2: Multiply by H
$$
(z\mathbf{I}-\mathbf{G})^{-1}\mathbf{H} = \frac{1}{(z+2)(z+3)}\begin{bmatrix} z+1+1 \\ -2+z+4 \end{bmatrix} = \frac{1}{(z+2)(z+3)}\begin{bmatrix} z+2 \\ z+2 \end{bmatrix} = \begin{bmatrix} \dfrac{1}{z+3} \\[2mm] \dfrac{1}{z+3} \end{bmatrix}
$$

#### Step 3: Multiply by C
$$
\mathbf{C}(z\mathbf{I}-\mathbf{G})^{-1}\mathbf{H} = \begin{bmatrix} 1 & 1 \\ 2 & 1 \end{bmatrix}\begin{bmatrix} \dfrac{1}{z+3} \\[2mm] \dfrac{1}{z+3} \end{bmatrix} = \begin{bmatrix} \dfrac{2}{z+3} \\[2mm] \dfrac{3}{z+3} \end{bmatrix}
$$

#### Step 4: Add D
$$
\mathbf{F}(z) = \begin{bmatrix} \dfrac{2}{z+3} + 1 \\[2mm] \dfrac{3}{z+3} + 0 \end{bmatrix} = \begin{bmatrix} \dfrac{z+5}{z+3} \\[2mm] \dfrac{3}{z+3} \end{bmatrix}
$$

#### Remarks
- The factor $(z+2)$ cancels: the mode at $z = -2$ is not excited by the input, so it does not appear in $\mathbf{F}(z)$.
- The non-zero $D_1 = 1$ makes $Y_1/U$ have equal numerator and denominator degree (direct feed-through).

**Answer: $\dfrac{Y_1(z)}{U(z)} = \dfrac{z+5}{z+3}$, $\dfrac{Y_2(z)}{U(z)} = \dfrac{3}{z+3}$**

## Check yourself

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

1. What size is F(z)?
   A. 1 x 2
   B. 2 x 2
   C. 3 x 1
   D. 2 x 1
   Answer: D. Two outputs and one input. (Key point: Matrix size)

2. What is (zI-G)^-1 H?
   A. [1/(z+3); 1/(z+3)]
   B. [1/(z+2); 1/(z+2)]
   C. [(z+2); (z+2)]
   D. [1/(z+3); 0]
   Answer: A. Numerators z+2 cancel one factor of the determinant. (Key point: Intermediate result)

3. How does D = [1; 0] change the result?
   A. Adds 1 to both outputs
   B. Adds 1 to the first output only, giving (z+5)/(z+3)
   C. Changes the poles
   D. Has no effect
   Answer: B. 2/(z+3) + 1 = (z+5)/(z+3). (Key point: Role of D)

4. What does the cancelled (z+2) factor mean?
   A. The system is unstable at z = -2
   B. The mode at -3 is lost
   C. The mode at -2 is not excited by the input
   D. D is wrong
   Answer: C. A pole-zero cancellation hides an uncontrollable mode. (Key point: Cancellation)

## Close variants set in other exams

- [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 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×

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