Linear and Non-Linear System Theory, by T. Thyagarajan and D. Kalpana, is a timely and substantive contribution to the literature for system theory. Linear and Non-Linear System Theory is a textbook that is definitely designed and written with the student in mind. And any sophisticated reader who attempts to use this book as a reference book may have a mixed reaction regarding the format. This also includes the lecture concepts and style. This book is full of educational philosophy and a wealth of both linear and non-linear systems and their control concepts. The basic prerequisite of understanding linear and nonlinear system theories is strong mathematical background.
The book has numerous features that make it unique and special. The book covers a strong comprehensive treatment of matrix fraction descriptions, the theories of rational and polynomial matrices. It covers state feedback compensator design and state observer theory thoroughly. Generally, it also covers control fundamentals to anchor, deepen, reinforce, and to emphasize the value of the observability, controllability, and realization concepts that are well introduced in different chapters. The authors have given good summaries of mathematical concepts to the systems concepts. They achieved this through intricate interplay of different mechanisms of presentation. The earlier popular texts on the theory of systems like Desoer (1970) and Chen (1970), Linear Systems gives an extensively on optimal control and state estimation using Kalman filter based on different conditions. Thyagarajan and Kalpana also base their arguments on structural considerations of systems and their fundamental functions in vital engineering problem cases.
After a good but brief introductory chapter, the authors go immediately into considerations of external and internal system descriptions and the concept of system realization. The exploration of these concepts is definitely at the graduate level. They assume a knowledge of elementary transform, matrix, signal, and system theory as captured in Mffiillem and Cooper (1974), Oppenheim et al. (2001), and Willsky and Young (1997). This basic knowledge is sufficient for study, as more than half of text, since all key systems concepts are clearly introduced and fully developed and multivariable cases are treated.
The other part of the book is better placed to challenge a graduate student who is devoted to advanced cases. Here the algebraic and geometric interpretations are more prevalent and mathematical concepts are even clearer.
The presentation of key topics such as observability and controllability is distributed throughout the book. Treatment of these topics consists of a carefully staged sequence of encounters with each concept. Each encounter is deeper and more thought provoking with well-illustrated examples and explanations. Practical encounters and several ideas surrounding each concept are presented, exemplified, thoroughly discussed, and the concept is mathematically formalized correctly are good for all system engineering students. The authors have taken a great amount of care to establish the numerous references. This provides a good historical background and perspective how the concepts have evolved into second decade of the 21st century. These references also give special insight from the early investigators and authors. Linear and Non-Linear Systems contains numerous and illustrative cases that are good supplements used as references with other texts. The authors are careful not to let material accumulate very much before engaging and testing the user’s, student’s, or researcher’s understanding of the just explained concept.
Chapter 1 contains a brief review of the assumed math background and general description about determinants and matrices. It then looks into the limitations of transfer function approach and introduces the state model for electrical, mechanical and other physical systems.
In Chapter 2, the concepts focus on state space approach, its merits and skills needed to represent the time-invariant systems. State space form is vital for analyzing the systems. While Chapter 3 looks at state variable feedback, designs, and asymptotic observer-controller compensator design. And features to gauge if the system is stabilizable, controllable, observable, and detectable.
Chapter 4 has the different types of nonlinearity existing in a given system. It also discusses the method used to analyze the non-linear system based on phase plane analysis and linearizing the non-linear systems using describing function approach. In Chapter 5, entitled “Stability of Non-Linear Systems” presents the concepts of stability and techniques used to assess the stability of nonlinear system. Methods discussed here include describing function, Lyapunov stability, Popov’s stability and circle criterion. Chapter 6 is the pinnacle of bifurcation theory, types of bifurcation, introduction to chaos theory: Lorentz equations, stability analysis of the Lorentz equations and chaos in chemical systems.
In Chapter 7, the authors deal with comparison of optimal control with classical control theory, basic definitions, performance measures, design of linear quadratic regulator, linear quadratic tracking and linear quadratic Gaussian systems. Finally, Chapter 8 reviews of statistical tools, state estimation in linear time-invariant systems using Kalman filter, continuous-time and discrete-time Kalman filter. It also gives basic introduction to Extended Kalman Filter.
Readers interested in both systems presented here are advised to resist the inevitable temptation of not using other references. The time invested by authors exploring the context and putting it together is indeed valuable. Knowledgeable readers will be comfortable decoupling one chapter to another.
The style used in this book makes it an excellent self-study text guide and self-explanatory. Successful use of this textbook in both senior undergraduate and graduate courses will require substantial assimilation of the authors’ philosophy. Substantial reorganization of traditional linear and non-linear systems course lectures and summaries are worth doing by any professors who will use this textbook in their classes.
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Krakow, Poland
References
- Desoer, C. A. (1970), Notes for a Second Course on Linear Systems, New York: Van Nostrand Reinhold.
- Chen, C. T. (1970), Introduction to Linear System Theory, New York: Holt, Rinehard and Winston.
- Mffiillem, C.D. and Cooper, G.R. (1974), Continuous and Discrete Signal and System Analysis, New York: Holt, Rinehart and Winston.
- Oppenheim, A. V., Buck, J. R., and Schafer, R. W., (2001), Discrete-Time Signal Processing, Vol. 2, Upper Saddle River, NJ: Prentice Hall.
- Willsky A. S. and Young, I. T. (1997), Signals and Systems, Prentice-Hall International.