Thyagarajan T. - Linear and Non-Linear System Theory

T. Thyagarajan and D. Kalpana

First edition published 2021

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ISBN: 978-0-367-34014-8 (hbk)

ISBN: 978-0-429-32345-4 (ebk)

Typeset in Palatino

by codeMantra

To my adorable grandson, R. Aarav

T. Thyagarajan

To my beloved husband, Mr. C. Kamal, ME

D. Kalpana

This textbook is intended for undergraduate and postgraduate students of electrical, chemical, electronics and instrumentation, and instrumentation and control branches of engineering disciplines pursuing advanced control system and linear and non-linear system theory as a subject. It will also be useful to research scholars pursuing research in similar disciplines. This book focuses on the basics of state space approach, analysis of non-linear systems, optimal control and optimal estimation.

discusses the review of statistical tools, state estimation in linear time-invariant systems using Kalman filter, continuous-time and discrete-time Kalman filter and introduction to Extended Kalman Filter (EKF).

Errors might have crept in, despite utmost care to avoid them. We shall be grateful to the readers if any errors are brought to our attention, along with suggestions for improvements to the book. We solicit your ever-granted support for the success of this professional endeavour.

At the outset, we express our gratitude to Almighty for his blessings. We are indebted to our Vice Chancellor, Prof. M. K Surappa, and to our Registrar, Prof. L. Karunamoorthy, of Anna University for their ever-granted patronage and encouragement. Our special thanks to our colleagues from the Department of Instrumentation Engineering for their valuable suggestions. We are also indebted to the teaching fraternity and student community for their support.

Our family members (wife, Mrs. Meera Thyagarjan, and daughter, Mrs. T. Priyanka, of the first author and mother, Mrs. D. Sarasu, son, K. Rahul Ramesh, and daughter, K. Vakshika, of the coauthor), who missed our quality time during holidays, weekends, vacations, and late evenings, while we were preparing the manuscript, deserve our wholehearted approbation.

Dr. Gagandeep Singh and his team at Taylor & Francis deserve our encomiums for their cooperation in bringing out this book in record time.

T. Thyagarajan
D. Kalpana

T. Thyagarajan is a Professor in the Department of Instrumentation Engineering at MIT Campus, Anna University. He earned a PhD in intelligent control at Anna University. With a National Science Council Fellowship, he pursued postdoctoral research at NTU, Taiwan. He is an erudite professor and a socially conscious researcher and has been engaged in teaching, research and administration for 33years. He has published over 105 research papers, authored/coauthored 3 textbooks, guided 14 PhD scholars, coordinated R&D funding worth Rs 15.5 crores through various funding agencies. He made technical visits to the United States, Europe, Southeast Asia and the Middle East. He received 13 awards, including DTE award for guiding the best BE project, Sisir Kumar Award for publishing the best research paper, 2 best paper awards in IEEE International Conferences, Best Teacher award and Best Researcher award. He is a Senior Member of IEEE. His current teaching and research interests include process modelling and control, auto-tuning, applied soft computing and healthcare instrumentation.

D. Kalpana is an Assistant Professor of instrumentation engineering at MIT Campus, Anna University. She earned a BE (EIE), an ME (instrumentation engineering), and a PhD at Anna University. She received a Young Scientist Fellowship from Tamilnadu State Council for Science and Technology and pursued postdoctoral research work at NIT, Warangal. She has been engaged in teaching and research for 14years. She has received the Dr A.P.J. Abdul Kalam Award for her contribution to teaching. She has published 25 papers in peer-reviewed journals and conference proceedings. She is a member of ISA. Her teaching and research interests include linear and non-linear system theory, auto-tuning, modelling and control of industrial processes and applied soft computing.

Determinants find their usefulness in solving linear systems. The mathematical expressions of linear systems can be simplified by representing them in the form of determinants.

A determinant contains elements that are arranged along horizontal and vertical lines.

Rows: The elements of a determinant that are arranged along horizontal lines form the rows.

Columns: The elements of a determinant that are arranged along vertical lines form the columns.

It consists of four elements, namely, a11,a12,a21 and a22 which are arranged in two rows and two columns and is expressed as follows:

The above determinant consists of two rows and two columns.

It consists of nine elements, namely, a11, a12, a13, a21, a22, a23, a31, a32 and a33 which are arranged in three rows and three columns and is expressed as follows:

The above determinant consists of three rows and three columns.

The minor Mij of an element aij of a determinant is the determinant obtained by deleting the ith row and jth column which intersect the element aij.

For example, if =|a11a12a13a21a22a23a31a32a33|

Then, the minor of a22 is given by M22=|a11a13a31a33|=[(a11a33)(a13a31)].

Similarly, the minor of a33 is given by M33=|a11a12a21a22|=[(a11a22)(a12a21)]