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Lab 6

This document provides instructions for a homework assignment on state-space design methods. Students are asked to analyze 4 different systems by: (1) calculating the transfer function and poles; (2) writing the state-space representation; (3) computing the eigenvalues of A; (4) comparing poles and eigenvalues; (5) plotting the step response; and (6) designing a closed-loop controller to place the closed poles at specific locations. Matlab commands are provided but students are instructed to complete parts of the analysis by hand without using Matlab.

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Zahin Tazwar 1921486642
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52 views1 page

Lab 6

This document provides instructions for a homework assignment on state-space design methods. Students are asked to analyze 4 different systems by: (1) calculating the transfer function and poles; (2) writing the state-space representation; (3) computing the eigenvalues of A; (4) comparing poles and eigenvalues; (5) plotting the step response; and (6) designing a closed-loop controller to place the closed poles at specific locations. Matlab commands are provided but students are instructed to complete parts of the analysis by hand without using Matlab.

Uploaded by

Zahin Tazwar 1921486642
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
Available Formats
Download as PDF, TXT or read online on Scribd
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EEE342

Homework 6: State-space design method


Due: 11 June, 2023 (In class or by 2.00pm in SAC1019)

Consider the following systems:

1. 𝑧̈ + 5 𝑧̇ + 6 𝑧 = 𝑢
2. 𝑧̈ + 4 𝑧̇ + 5 𝑧 = 𝑢
3. 𝑧̈ +4𝑧 =𝑢
4. 𝑧̈ + 𝑧̇ − 2 𝑧 = 𝑢

For each system do the following:

!(#)
a) Calculate the transfer function %(#). What are the poles of the system?

b) Put the system in the state-space form:

𝑥̇ = 𝐴𝑥 + 𝐵𝑢
𝑦 = 𝐶𝑥

Define the state vector as 𝑥 = [𝑥& 𝑥' ]( where, 𝑥& = 𝑧, 𝑥' = 𝑧̇ , 𝑦 = 𝑥&

What are the A, B, C matrices for the system?

c) Compute the eigenvalues of the A matrix. Do this by hand. Do not use Matlab. [In
Matlab, you can use the ‘eig’ command to calculate the eigenvalues of a matrix].

d) Are the poles of the system and eigenvalues of the A matrix the same or
different?

e) Plot the step response.

[You can use the following Matlab commands:

sys = ss(A,B,C,0);
step(sys);
]

f) Now we want to construct the closed-loop system 𝑥̇ = (𝐴 − 𝐵𝑘) 𝑥 using the


feedback control law 𝑢 = −𝑘𝑥.

Find the gain vector 𝑘 so that the closed poles are located at [-1 -2]. Do this by hand.
Do not use Matlab. [In Matlab, you can use the command ‘place’ or ‘acker’ to
calculate the gain vector 𝑘.]

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