Introductory Circuit Analysis (13th Edition)
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ISBN: 9780133923605
Author: Robert L. Boylestad
Publisher: PEARSON
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A system is described by the differential equation (see attached).
a)What is the order of the system. How many poles does the system's transfer function have. How many states are needed to describe the system completely.
b) Determine the system's transfer function, Y(s)/U(s) (the poles of the system are at ―1, ― 2, and ―4);
c) Determine matrices A, B, C, and D to describe the system in state-space form x' = Ax + Bu, y = Cx + Du.
![The given differential equation is:
\[
\frac{d^3 y}{dt^3} + 7 \frac{d^2 y}{dt^2} + 14 \frac{dy}{dt} + 8y = 3 \frac{d^3 u}{dt^3} + 21 \frac{d^2 u}{dt^2} + 47 \frac{du}{dt} + 36u
\]
This is a third-order linear differential equation involving the functions \( y(t) \) and \( u(t) \) and their derivatives. The equation relates the derivatives of these functions up to the third degree with various coefficients:
- On the left-hand side, we have the third, second, and first derivatives of \( y \) as well as \( y \) itself, each multiplied by their respective coefficients: 1, 7, 14, and 8.
- On the right-hand side, the third, second, and first derivatives of \( u \) and \( u \) itself are multiplied by 3, 21, 47, and 36, respectively.
This equation could be used to model dynamic systems in engineering or physics where both \( y(t) \) and \( u(t) \) are important and interact with each other in complex ways.](https://content.bartleby.com/qna-images/question/2587a113-5ca6-415a-ac64-2fd0e756bc9a/5f73b1a7-e0b2-4b7e-929a-20f3e25db017/3gd4zu_processed.png)
Transcribed Image Text:The given differential equation is:
\[
\frac{d^3 y}{dt^3} + 7 \frac{d^2 y}{dt^2} + 14 \frac{dy}{dt} + 8y = 3 \frac{d^3 u}{dt^3} + 21 \frac{d^2 u}{dt^2} + 47 \frac{du}{dt} + 36u
\]
This is a third-order linear differential equation involving the functions \( y(t) \) and \( u(t) \) and their derivatives. The equation relates the derivatives of these functions up to the third degree with various coefficients:
- On the left-hand side, we have the third, second, and first derivatives of \( y \) as well as \( y \) itself, each multiplied by their respective coefficients: 1, 7, 14, and 8.
- On the right-hand side, the third, second, and first derivatives of \( u \) and \( u \) itself are multiplied by 3, 21, 47, and 36, respectively.
This equation could be used to model dynamic systems in engineering or physics where both \( y(t) \) and \( u(t) \) are important and interact with each other in complex ways.
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