Theoretical control theory. I: Foundations, synthesis of linear control systems (Q1902291)

When considering the contents of this first volume, we find a good selection of the most important topics of modern systems and control theory system representation and analysis, stability, synthesis in the frequency domain, controllability and reachability, state feedback for linear single-input-single-output systems together with a treatment of the necessary mathematical prerequisites such as fundamentals from linear algebra, transition matrices for discrete and continuous systems. Laplace- and \(z\)-transform. When reading further, one discovers that the author indeed achieves his goal i.e. providing the scientific basis for important fields of control theory. Further, one finds topics (modellig, numerics) scarcely discussed in such detail in books like this one. It starts with an easily understandable treatment of the difference and relations between system, system model and mathematical description of the system model followed by a careful treatment of the various ways of deriving the latter (linearization, bilinearization, sampling etc.) together with a discussion of relations between the most common types of models such as descriptor systems, state space or frequency domain models. All this is illustrated by well-chosen examples allowing engineers to access easily to the theory and -- at the same time -- to allow a deeper understanding of e.g. applied mathematicians for the demands of applications. This excellent way of presentation is found throughout the book and supplemented by remarks concerning numerical problems and methods to overcome them e.g. in connection with the necessary determination of eigenvalues of a matrix. The numerous detailed examples are chosen not only for the sake of treatability but to make a method easy to understand and to demonstrate -- if appropriate -- its limitations. Thus, examples together with exercises (with solutions) allow for a good understanding of the engineering implications -- though the author often ends up with the mathematical sequence definition -- theorem -- proof, in order to obtain precise statements. To sum up: This book is a very interesting introduction to modern control theory (which is to be complemented by vol. 2 dealing with MIMO-systems, state reconstruction, optimization and nonlinear systems) from which not only engineers but also mathematicians will benefit.