The object of this book is primarily to introduce the reader to the field of applied mathematics, which deals with the description of the mathematical structure of systems in general, through the mathematical structure of a certain class of mathematical models of systems, known as the class of "linear" systems.

Through such mathematical models of systems, we examine the ways and possibilities of controlling the stability and, more generally, the dynamic behavior of systems in general, without specializing our study to a specific system. The control of the stability and temporal behavior of a given system is achieved through its interconnection, in "feedback" topology with another system which constitutes the "stabilizing controller" or "stabilizing compensator".

The new system that arises from this interconnection is said to constitute a system of "automatic control". The determination of the mathematical model of a "stabilizing controller" is one of the main problems of Mathematical Control Theory, and its solution is one of the main objects of study in this book.

As mentioned, a basic precondition for the study in this book is that the systems under examination can be described by a specific class of mathematical models: the class of "linear systems". Linear systems (or better, linear models of systems) constitute a mathematical idealization of the behavior of physical systems based on the assumption of "linearity".

A consequence of the assumption of linearity is a specific mathematical calculus which leads the researcher mathematician-engineer to draw conclusions about the mathematical structure, operation, and stability of "linear systems". These conclusions, in turn, dictate methodologies for calculating the stabilizing controllers of physical systems, the (automatic) control of the behavior of which concerns us.

Of course, the assumption of linearity in Nature and in the systems encountered in it generally does not hold. Linearity, if it holds, does so under constraints. Unfortunately, or fortunately, Nature is "nonlinear". The differential equations and the relationships that describe the temporal behavior of physical phenomena and systems are either unknown or, at best, "nonlinear".

Approaches to such nonlinear phenomena and processes through linear differential equations or relationships are the best we can do to address such problems and reach practical solutions through a mathematical calculus that arises from these approaches.

This book consists of two volumes. Both volumes deal with linear systems. In the first volume, the one you are holding, the mathematical structure of systems is examined through linear mathematical models that connect the input to the output of the system. This investigation: "input-output" is known as classical control theory, and its mathematical foundations were established, developed, and evolved mainly from 1850 to 1950.

In the second volume, the mathematical model of the system additionally introduces the very significant concept of the "state" of the system for its internal structure. These mathematical models are known as state-space models and the methodology that describes them is known as modern control theory.