Input-to-state stability. Theory and applications (Q6553913)

This monograph focuses on input-to-state stability (ISS), a specific notion of stability introduced by E. Sontag in the late 1990s. It addresses nonlinear systems control analysis and controller design problems through the manipulation of functional inequalities satisfied along systems trajectories, as pioneered by H. Poincaré and A. M. Lyapunov at the end of the 19th century.\N\NThe book is rather technical, and not really introductory. It is aimed at experts. It is mostly mathematical, and the ``applications'' mentioned in the book's subtitle are academic. The author's choice is to focus from Chapter 1 on controlled ordinary differential equations (ODEs), or ODEs with inputs, relegating ODEs without inputs to Appendix B (which is otherwise a very interesting source of information on various notions of stability).\N\NThe definition of ISS, central to the book, comes at the beginning of Chapter 2 on page 47. In order to understand the definition, the reader is however expected to be already familiar with the zoology of comparison functions (K, L, KL and P functions) whose detailed descriptions are relegated to the end of the book in Appendix A (they are also listed on page xvi in the Abbreviations and Symbols preface, however without further explanations). Remark 2.4 is instrumental to understanding why ISS is a natural extension to controlled ODEs of global asymptotic stability of uncontrolled ODEs. Indeed, ISS holds whenever the flow of the controlled ODE is bounded in norm by the sum of two terms: a first function controls the transient behavior, whereas a second function controls the influence of the input. This second term is absent for uncontrolled ODEs. Ensuring ISS amounts to finding these two comparison functions. Since ISS boils down to the system's trajectories satisfying specific functional inequalities, most of the remainder of the book consists of manipulating these inequalities.\N\NChapter 3 describes the use of small-gain theorems to ensure ISS of networks of interconnected ODEs, a major application of the notion of ISS, practically relevant for e.g. large-scale biological systems. Chapter 4 extends ISS to common nonlinearity classes such as saturation or actuator limitations. For this purpose, the second term (input gain) in the ISS inequality is replaced with an integral term. Chapter 5 shows how ISS inequalities can be manipulated to design robust nonlinear controllers and observers. Chapter 6 deals with ISS for infinite networks, which are systems described by a countable number of interconnected controlled ODEs. This is a first step toward an extension to infinite-dimensional systems described by partial differential equations (PDEs), briefly mentioned in the concluding Chapter 7.\N\NIn summary, this book targets systems control theory experts willing to reinforce their understanding of nonlinear Lyapunov function techniques. It serves as a useful technical reference on ISS, a still active research trend that made its way into nonlinear systems control textbooks.