The book A Linear Systems Primer is a streamlined presentation of the book Linear Systems [Boston: Birkhäuser (2006; Zbl 1189.93001)] of the same authors. It provides an introduction to system theory with an emphasis on control theory. The primer has many references to the old book of the authors since some of the proofs and results have been omitted in the new version in order to provide the reader with a clear understanding of the fundamental concepts of systems and control theory. This is the reason that primer focuses more on course use of the material with emphasis on presentation. In the following a general overview of the book’s content is given.
Chapter one deals with ordinary differential equations and initial-value problems that is essential in the study of continuous-time finite-dimensional systems. It starts with a discussion about mathematical models and classification of models. Then the initial value problem is presented and results are provided that ensure the existence and uniqueness of solutions of this problem. The problem of linearization of non-linear differential equations is addressed. The solutions of systems of first-order ordinary differential equations is finally given.
The second chapter addresses the state-space description and the input-output description of systems. Both continuous and discrete time systems are studied and emphasis is given on linear systems.
The third chapter studies the response of continuous- and discrete-time linear systems. An emphasis is given on characterizing all solutions using bases (of the solution vector spaces) and on determining such solutions. Equivalent representation of systems and sampled data systems are also presented.
The fourth chapter addresses the Lyapunov stability and the input-output stability of continuous- and discrete-time systems.
The notions of controllability, reachability, observability and constructibility are introduced and then studied in details in chapter five.
The sixth chapter presents important special forms for the state-space description of time invariant systems. More particular, a special form is presented that separates the controllable (observable) from the uncontrollable (unobservable) part of the system.
Based on this form additional tests for controllability (observability) are presented. The controller and observer state-space forms that are useful in the study of state-space realizations, state feedback and state estimators are also presented. Relations between internal and external descriptions of a system are presented in chapter seven. New characteristics such as, transmission poles and zeros, invariant poles and zeros, decoupling zeros, are introduced and its connections are established. Polynomial matrix and matrix fraction descriptions of a system are also introduced in this chapter.
The problem of state-space realizations of input-output descriptions is defined and the existence of such realizations is addressed in chapter eight. It is shown that a state-space realization is minimal iff it is controllable and observable and all the minimal realizations of a given transfer function are related by similarity transformations. The order of a minimal realization is determined and several realization algorithms are presented.
The problem of pole or eigenvalue assignment of a system by using state feedback is considered in chapter nine. The problem of state estimation is discussed and methods for the determination of full-order and reduced-order estimators (observers) are given in the sequel. A brief introduction to the Linear Quadratic Regulator (LQR) problem and the Linear Quadratic Gaussian (LQG) estimation problem is given. Finally, state feedback static controllers and state dynamic observers are combined to form dynamic output feedback controllers.
Chapter ten starts by considering interconnected systems (with PMD or MFD), connected in parallel, series and feedback configurations and their properties i.e. controllability, observability and internal stability. Then a parametrization of all stabilizing feedback controllers of a given plant (either nonproper or proper) is given. Finally two degrees of freedom controllers are studied.
An appendix is given at the end of the book with preliminary results mainly from linear algebra and matrix theory.
The book is accompanied by many illustrative examples that help the reader to understand the fundamental concepts presented in the theory as well as with many unsolved exercises at the end of each chapter that can be used for homework. At the end of each chapter the reader can find a summary with the most important results presented in the chapter, and a very useful discussion with historical information for the main concepts presented at the chapter accompanied with the relevant references.
Comparing the two books of the same authors, the primer can be easily used as a textbook for senior undergraduate of a typical one-semester course introduction to linear systems primarily for first-year graduate and senior undergraduate students in engineering, but also in mathematics, physics and the rest of the sciences. Linear Systems (2006; Zbl 1189.93001) from the other side can be used for Ph.D. students and researchers in the area of systems and control theory as a complete reference guide to linear systems, like the known ones of T. Kailath [Linear systems. Englewood Cliffs, N.J.: Prentice-Hall (1980; Zbl 0454.93001)], W. A. Wolovich [Linear multivariable systems. New York-Heidelberg-Berlin: Springer-Verlag (1974; Zbl 0291.93002)], H. H. Rosenbrock [State-space and multivariable theory. London etc.: Thomas Nelson & Sons (1970; Zbl 0246.93010)], W. J. Rugh [Linear system theory. 2nd ed. Upper Saddle River, NJ: Prentice Hall (1996; Zbl 0892.93002)], E. D. Sontag [Mathematical control theory. Deterministic finite dimensional systems. 2nd ed. New York, NY: Springer (1998; Zbl 0945.93001)] etc.