Input to State Stability and Allied System Properties

The problems of design and analysis of the nonlinear control systems are at the center of attention of the scientific community for several decades, but it was only during the last two decades that an appreciable progress was marked in the design and application of various methods of the theory of nonlinear control. On the one hand, this success is due to the development of the computer engineering enabling fast and thorough study of the nonlinear processes, and to the increased demands on the control performance that can be satisfied by considering more complicated nonlinear models of the control plants and more complicated control algorithms. On the other hand, this is due to the development of an adequate theoretical apparatus for design and analysis of the nonlinear systems. The 1990s were noted for the advent and development of a whole series of nonlinear system methods that were christened “constructive” in [1]. The concept of stability of the dynamic systems from input to the state vector that is considered in what follows may be sorted with such “discoveries” [2, 3]. It is distinguished for that it enables one to formalize the conditions for stability of the nonlinear system under input perturbations or, stated differently, to formalize the notion of system “robustness” to external perturbations. The present review discusses the main results related with this concept and the allied notions of stability. The basic property of robust stability is introduced in Section 2. Sections 3 and 4 introduce various adjacent notions, and their use is considered in Section 5. The auxiliary material and all abbreviations used in the review are compiled in Section 6, and the results left outside the present review are compiled in Section 7.