Given the state-space equations X(t) = [21]x(t) + [1] u(t) y(t) = [1 0]X(t) +u(t) (u(t)= 1,t≥ 0) If x(0) = [1], what is y(2)? [1]u(t)_and
Given the state-space equations X(t) = [21]x(t) + [1] u(t) y(t) = [1 0]X(t) +u(t) (u(t)= 1,t≥ 0) If x(0) = [1], what is y(2)? [1]u(t)_and
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Question
![### State-Space Equations Problem
**Given the state-space equations:**
\[ \dot{x}(t) = \begin{bmatrix} 2 & 1 \\ 0 & 1 \end{bmatrix} x(t) + \begin{bmatrix} 1 \\ 0 \end{bmatrix} u(t) \]
**and**
\[ y(t) = \begin{bmatrix} 1 & 0 \end{bmatrix} x(t) + u(t) \]
**where** \( u(t) = 1, t \geq 0 \).
**Initial conditions:**
\[ x(0) = \begin{bmatrix} 1 \\ 1 \end{bmatrix} \]
**Problem:**
What is \( y(2) \)?
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### Explanation:
This problem involves evaluating the state-space representation of a dynamic system. It requires the determination of the output \( y(t) \) at time \( t = 2 \) given initial conditions and a control input. The matrices describe the system's dynamics and how the input \( u(t) \) influences the states and output.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F71f6a4b8-527f-4030-b85a-828002701c6f%2F90957d1d-daa4-452c-af2b-1f3e00b8eba1%2Fk3wama4_processed.png&w=3840&q=75)
Transcribed Image Text:### State-Space Equations Problem
**Given the state-space equations:**
\[ \dot{x}(t) = \begin{bmatrix} 2 & 1 \\ 0 & 1 \end{bmatrix} x(t) + \begin{bmatrix} 1 \\ 0 \end{bmatrix} u(t) \]
**and**
\[ y(t) = \begin{bmatrix} 1 & 0 \end{bmatrix} x(t) + u(t) \]
**where** \( u(t) = 1, t \geq 0 \).
**Initial conditions:**
\[ x(0) = \begin{bmatrix} 1 \\ 1 \end{bmatrix} \]
**Problem:**
What is \( y(2) \)?
---
### Explanation:
This problem involves evaluating the state-space representation of a dynamic system. It requires the determination of the output \( y(t) \) at time \( t = 2 \) given initial conditions and a control input. The matrices describe the system's dynamics and how the input \( u(t) \) influences the states and output.
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