---
title: "Pulse TF matrix (G = [3 1; −2 1], D = [1; 0])"
subject: "Digital Control System"
chapter: 5
key: "semester_6/digital_control_system:pulse-tf-matrix-g-3-1"
source: https://www.bctengineer.com/q/semester_6/digital_control_system/pulse-tf-matrix-g-3-1
site: BCT Engineering
---

# Pulse TF matrix (G = [3 1; −2 1], D = [1; 0])

Digital Control System · Chapter 5: Discrete Time State Equations

- Asked in: 2075 Bhadra, 8 marks; 2068 Magh, 8 marks (asked 2 times)
- 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-g-3-1
- Whole chapter: https://www.bctengineer.com/md/semester_6/digital_control_system/chapter-5

## Question

Determine pulse transfer function matrix for the system which is described by state space representation as: [x₁(k+1); x₂(k+1)]=[3 1; -2 1][x₁(k); x₂(k)]+[1; 1]u(k) and [y₁(k); y₂(k)]=[1 1; 2 1][x₁(k); x₂(k)]+[1; 0]u(k).

## Short answer

Pulse transfer function matrix is F(z) = C(zI-G)^{-1}H + D. For given matrices, F(z) = [z(z-2)/(z^2-4z+5); (3z-5)/(z^2-4z+5)]. Poles at z=2±j1 (|z|=√5>1), system unstable.

## 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 2×1.

#### Given
$$
\mathbf{G} = \begin{bmatrix} 3 & 1 \\ -2 & 1 \end{bmatrix},\quad \mathbf{H} = \begin{bmatrix} 1 \\ 1 \end{bmatrix},\quad \mathbf{C} = \begin{bmatrix} 1 & 1 \\ 2 & 1 \end{bmatrix},\quad \mathbf{D} = \begin{bmatrix} 1 \\ 0 \end{bmatrix}
$$

#### Step 1: Inverse of $z\mathbf{I}-\mathbf{G}$
$$
z\mathbf{I}-\mathbf{G} = \begin{bmatrix} z-3 & -1 \\ 2 & z-1 \end{bmatrix},\qquad |z\mathbf{I}-\mathbf{G}| = (z-3)(z-1) + 2 = z^2 - 4z + 5
$$
$$
(z\mathbf{I}-\mathbf{G})^{-1} = \frac{1}{z^2-4z+5}\begin{bmatrix} z-1 & 1 \\ -2 & z-3 \end{bmatrix}
$$

#### Step 2: Multiply by H
$$
(z\mathbf{I}-\mathbf{G})^{-1}\mathbf{H} = \frac{1}{z^2-4z+5}\begin{bmatrix} z \\ z-5 \end{bmatrix}
$$

#### Step 3: Multiply by C
$$
\mathbf{C}(z\mathbf{I}-\mathbf{G})^{-1}\mathbf{H} = \frac{1}{z^2-4z+5}\begin{bmatrix} z + (z-5) \\ 2z + (z-5) \end{bmatrix} = \frac{1}{z^2-4z+5}\begin{bmatrix} 2z-5 \\ 3z-5 \end{bmatrix}
$$

#### Step 4: Add D
$$
\begin{aligned}
F_1(z) &= \frac{2z-5}{z^2-4z+5} + 1 = \frac{z^2-2z}{z^2-4z+5}\\
F_2(z) &= \frac{3z-5}{z^2-4z+5}
\end{aligned}
$$

#### Result
$$
\mathbf{F}(z) = \begin{bmatrix} \dfrac{z(z-2)}{z^2-4z+5} \\[3mm] \dfrac{3z-5}{z^2-4z+5} \end{bmatrix}
$$
The poles are $z = 2 \pm j1$ ($|z| = \sqrt5 > 1$), so the system is unstable.

**Answer: $Y_1(z)/U(z) = \dfrac{z^2-2z}{z^2-4z+5}$, $Y_2(z)/U(z) = \dfrac{3z-5}{z^2-4z+5}$**

## Check yourself

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

1. Which formula gives the pulse transfer function matrix?
   A. F(z) = C(zI-G)^-1 H + D
   B. F(z) = (zI-G)^-1 H
   C. F(z) = C G H + D
   D. F(z) = C(zI+G)^-1 H
   Answer: A. It is the output matrix times the resolvent times H, plus the feed-through. (Key point: Formula)

2. What is |zI-G| for G = [3 1; -2 1]?
   A. z^2 + 4z + 5
   B. z^2 - 4z + 5
   C. z^2 - 4z + 1
   D. z^2 - 4z + 3
   Answer: B. (z-3)(z-1) - (-1)(2) = z^2 - 4z + 5. (Key point: Characteristic polynomial)

3. What does the non-zero D1 = 1 do to the first output?
   A. Multiplies it by 1
   B. Changes the poles
   C. Adds 1 to C(zI-G)^-1 H, giving (z^2-2z)/(z^2-4z+5)
   D. Is ignored
   Answer: C. (2z-5)/(den) + 1 = (z^2-2z)/(den); D only shifts numerator, not poles. (Key point: Effect of D)

4. Why is the system unstable?
   A. The poles lie at z = 0
   B. The poles are real and negative
   C. D is non-zero
   D. The poles 2 ± j1 have magnitude sqrt(5) > 1
   Answer: D. Discrete-time stability needs poles inside the unit circle. (Key point: Stability)

## Related in Digital Control System

- [Pulse transfer function matrix (G = [−4 1; −2 −1], D = [1; 0])](https://www.bctengineer.com/question.md?key=semester_6%2Fdigital_control_system%3Apulse-tf-matrix-d-nonzero) — Digital Control System, chapter 5, asked 7×
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