Introduction to Modern Control system

June 10, 2022

,Dedication

To my father Abebe Beyene, my mother Zenebech Degefu, my brothers Eshetu
Abebe, Negash Abebe and Habtamu Abebe and my friends.




Family is the beginning and the end.
Daniel Abebe




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,Acknowledgment

First of all, my hearty gratitude goes to the almighty God. Then I am highly
grateful to my colleagues Mr.Dita Tilahun, Mr.Workagegn Tatek, Mr.Alemie As-
sefa, Mr.Getsh Fekadie, Mr.Addisu Shimelis and Mr.Getaneh Dagnachew for their
comment’s regarding content of the lecture note and Asst.Prof.Mikias Hailu for his
constructive idea during the preparation of this lecture note. Finally my honorable
gratitude goes to my friend Mr.Yonas Asres who prepared the cover page of this
lecture note.




Daniel Abebe




©
All rights reserved!!!
Debre Berhan University
Department of Electrical and Computer Engineering
Debre Berhan, Ethiopia.
June 10, 2022


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, Contents

Dedication i

Acknowledgment ii

Acronym’s (Abbreviation) vi

1 State Space Representation of LTI Systems 1
1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Mathematical Formulation of Control System Problems . . . . . . . . 1
1.3 Mathematical Modeling and State Space representation of Dynamic
Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.3.1 State Space Representation of Series RLC Circuit . . . . . . . 3
1.3.2 State Space Representation of Two degree of Freedom Quarter
Car Suspension Model . . . . . . . . . . . . . . . . . . . . . . 4
1.3.3 State Space Representation of DC Motor . . . . . . . . . . . . 6
1.3.4 State Space Representation of Inverted Pendulum on a Cart
(IPC) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.3.5 State Space Representation of Quadruple Tank System . . . . 11
1.4 State Space Representation From Transfer Function . . . . . . . . . . 13
1.5 Transfer Function From State Space Representation . . . . . . . . . . 17
1.6 Review on Eigenvalues and Eigenvectors . . . . . . . . . . . . . . . . 18
1.7 Solution of State Space Equations . . . . . . . . . . . . . . . . . . . . 18
1.8 Similarity Transformation . . . . . . . . . . . . . . . . . . . . . . . . 20
1.9 Phase Plane and Stability Analysis of LTI Systems . . . . . . . . . . 22
1.9.1 Stability Analysis of LTI Systems . . . . . . . . . . . . . . . . 22
1.9.2 Phase Plane Analysis of LTI systems . . . . . . . . . . . . . . 24
1.10 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
1.11 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

2 Observability and Controllability of LTI Systems 47
2.1 Controllability of LTI Systems . . . . . . . . . . . . . . . . . . . . . . 47
2.2 Observability of LTI Systems . . . . . . . . . . . . . . . . . . . . . . 48
2.3 Modality Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
2.4 Kalman Decomposition and Weaker Conditions for LTI Systems . . . 51
2.5 Controllability and Observability in Sense of Transfer Function . . . . 53
2.6 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
2.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59




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