This is a part of a review I'm studying, NOT a graded assignment, please do not reject. Thank you!Am I doing this problem right? I would like someone to verify I'm doing this correctly, thank you! A control system is represented by two state variables, and its state description matrices are:\[ A = \begin{bmatrix} 0 & 1 \\ -4 & -5 \end{bmatrix}, \quad B = \begin{bmatrix} 0 \\ 1 \end{bmatrix}, \quad C = \begin{bmatrix} 1 & 2 \end{bmatrix}, \quad q(0) = \begin{bmatrix} 2 \\ 1 \end{bmatrix}, \quad u(t) = 5e^{-3t}u(t) \]### Tasks:a) Find \[ y_{zi}(t) \], i.e., \[ y_{zi}(t) = Ce^{At}q(0) \]b) Write the **“expected”** response form for \[ y_{zs}(t) \] **(DO NOT HAVE TO CALCULATE \[ y_{zs}(t) \])**### Solution:#### a)\[ y_{zi}(t) = Ce^{At}q(0) \]#### b)- The system is represented in a standard form, which translates to the form: \[ \frac{b(s)}{s^2 + a_1 s + a_0} \]- Based on matrix values, an equivalent equation is derived: \[ \frac{12s + 1}{s^2 + 4s + 5} \]- To find the roots, solve: \[ (s+a-i)(s+a+i) \]- These roots are complex conjugates, representing an underdamped system.- **Response form:** \[ y_{zs}(t) = e^{\omega t} [\sin(\omega_n t) + \cos t] \]This depicts the expected behavior of the system's output over time given the complex conjugate roots, indicating oscillatory behavior typical in underdamped systems.

The state-matrices of a control system are given as:The initial condition and input are To…

Answered: a) b) A control system is represented…

Introductory Circuit Analysis (13th Edition)

13th Edition

ISBN:9780133923605

Author:Robert L. Boylestad

Publisher:Robert L. Boylestad

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This is a part of a review I'm studying, NOT a graded assignment, please do not reject. Thank you!

Am I doing this problem right? I would like someone to verify I'm doing this correctly, thank you!

A control system is represented by two state variables, and its state description matrices are:

\[ A = \begin{bmatrix} 0 & 1 \\ -4 & -5 \end{bmatrix}, \quad B = \begin{bmatrix} 0 \\ 1 \end{bmatrix}, \quad C = \begin{bmatrix} 1 & 2 \end{bmatrix}, \quad q(0) = \begin{bmatrix} 2 \\ 1 \end{bmatrix}, \quad u(t) = 5e^{-3t}u(t) \]

### Tasks:
a) Find \[ y_{zi}(t) \], i.e., \[ y_{zi}(t) = Ce^{At}q(0) \]

b) Write the **“expected”** response form for \[ y_{zs}(t) \] **(DO NOT HAVE TO CALCULATE \[ y_{zs}(t) \])**

### Solution:

#### a)
\[ y_{zi}(t) = Ce^{At}q(0) \]

#### b)
- The system is represented in a standard form, which translates to the form:

\[ \frac{b(s)}{s^2 + a_1 s + a_0} \]

- Based on matrix values, an equivalent equation is derived:

\[ \frac{12s + 1}{s^2 + 4s + 5} \]

- To find the roots, solve:

\[ (s+a-i)(s+a+i) \]

- These roots are complex conjugates, representing an underdamped system.

- **Response form:**

\[ y_{zs}(t) = e^{\omega t} [\sin(\omega_n t) + \cos t] \]

This depicts the expected behavior of the system's output over time given the complex conjugate roots, indicating oscillatory behavior typical in underdamped systems.

Expert Solution

Step 1: Summarize the given information.

The state-matrices of a control system are given as:

$A equals open square brackets table row 0 1 row cell negative 4 end cell cell negative 5 end cell end table close square brackets comma space B equals open square brackets table row 0 row 1 end table close square brackets comma space C equals open square brackets table row 1 2 end table close square brackets.$

The initial condition and input are

$q open parentheses 0 to the power of minus close parentheses equals open square brackets table row 2 row 1 end table close square brackets comma space u open parentheses t close parentheses equals 5 e to the power of negative 2 t end exponent straight u open parentheses t close parentheses.$

To determine:

zero-input response $y_{z i} (t),$
general form of zero-state response $y_{z s} (t) .$

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