Preface

Phenomenal progress in nonlinear systems theory has been made during the last decades. It has been reflected in two aspects. On the one hand, internal and external global stability notions have been studied intensely for uncertain nonlinear systems. On the other hand, the applications of these advanced stability results to control engineering systems have led to numerous novel methodologies for the design of nonlinear feedback controllers. It is fair to say that input-to-state stability (ISS), a notion invented by E.D. Sontag in the late 1980s, plays an influential role in the work of many researchers including the authors of this book. ISS has bridged the gap which previously existed between the input–output and the state-space methods, two popular approaches within the control systems community. Roughly speaking, the importance of ISS for the study of nonlinear systems is reflected by the intriguing fact that it captures two main stability notions: Lyapunov stability (i.e., the behavior of the zero-input response with respect to nonzero initial conditions) and input– output stability (i.e., the behavior of the zero-state response with respect to nonzero external inputs). Nonlinear systems are encountered frequently in almost all branches of science and engineering. In fact, in engineering, physics, economics, and biology, nonlinearity is the rule, and linear systems are rare (which almost exclusively exist only in our computer programs). Despite the importance of nonlinear system theory, graduate students or researchers in mathematics, engineering, physics, economics, and biology often have difficulties in taking advantage of recent advances in mathematical systems and control theories. There are several excellent textbooks that provide nice introductions to nonlinear systems theory, but many recent stability results are scattered in the vast literature. Motivated by this observation, we set our hands to write this monograph about a year and half ago. The specific objectives of this book are described in the following. The first aim of the book is to provide the basic knowledge needed for a graduate student in order to be able to understand the current research in nonlinear stability theory and nonlinear control theory. A relatively high level of mathematical background is assumed: the reader is required of having basic knowledge in differential equations, calculus, and real analysis. Measure theory is not needed (although