Given a state space model\[\begin{bmatrix}\dot{x}_1 \\\dot{x}_2 \\\dot{x}_3\end{bmatrix}=\begin{bmatrix}1 & 0 & 0 \\0 & 0 & 1 \\0 & -1 & -2 \end{bmatrix}\begin{bmatrix}x_1 \\x_2 \\x_3\end{bmatrix}+\begin{bmatrix}1 \\0 \\1\end{bmatrix}u\]\[ y = \begin{bmatrix}1 & 1 & 0\end{bmatrix}\begin{bmatrix}x_1 \\x_2 \\x_3\end{bmatrix}\]with input \( u \) and output \( y \).a). Derive the transfer function representation.b). Derive the differential equations representation.c). Compute the response \( y(t) \) with step control input \( u(t) = 1(t) \) and zero initial condition.d). Compute the state response \( x(t) \) with control input \( u(t) = 1(t) \) and initial condition \( x(0) = [1 \ 1 \ 0]^T \).

Answered: Image /qna-images/answer/e5bb95bc-aec1-4cb6-80b8-fc2396368eea.jpg

Given a state space model [1 0 1 + 0 u -1 [1 1 0] |#2 - input u and output y. Derive the transfer function representation. Derive the differential equations representation. Compute the response y(t) with step control input u(t) = 1(t) and zero initial condition.

Given a state space model [1 0 1 + 0 u -1 [1 1 0] |#2 - input u and output y. Derive the transfer function representation. Derive the differential equations representation. Compute the response y(t) with step control input u(t) = 1(t) and zero initial condition.

Elements Of Electromagnetics

7th Edition

ISBN:9780190698614

Author:Sadiku, Matthew N. O.

Publisher:Sadiku, Matthew N. O.

ChapterMA: Math Assessment

Section: Chapter Questions

Problem 1.1MA

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Concept explainers

Free undamped vibrations

Whenever an object/system displaces in either side of its equilibrium position with the help of an external/internal means, then the object/system moves regularly or randomly on both sides of its equilibrium position. This motion is called vibration. Generally, there are three types of vibration: longitudinal vibration, transverse vibration, and torsional vibration.

Question

Given a state space model

\[
\begin{bmatrix}
\dot{x}_1 \\
\dot{x}_2 \\
\dot{x}_3
\end{bmatrix}
=
\begin{bmatrix}
1 & 0 & 0 \\
0 & 0 & 1 \\
0 & -1 & -2
\end{bmatrix}
\begin{bmatrix}
x_1 \\
x_2 \\
x_3
\end{bmatrix}
+
\begin{bmatrix}
1 \\
0 \\
1
\end{bmatrix}
u
\]

\[
y = \begin{bmatrix}
1 & 1 & 0
\end{bmatrix}
\begin{bmatrix}
x_1 \\
x_2 \\
x_3
\end{bmatrix}
\]

with input \( u \) and output \( y \).

a). Derive the transfer function representation.

b). Derive the differential equations representation.

c). Compute the response \( y(t) \) with step control input \( u(t) = 1(t) \) and zero initial condition.

d). Compute the state response \( x(t) \) with control input \( u(t) = 1(t) \) and initial condition \( x(0) = [1 \ 1 \ 0]^T \).

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