Input to State Stability: Basic Concepts and Results
The analysis and design of nonlinear feedback systems has recently undergone an exceptionally rich period of progress and maturation, fueled, to a great extent, by (1) the discovery of certain basic conceptual notions, and (2) the identification of classe
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1 Introduction The analysis and design of nonlinear feedback systems has recently undergone an exceptionally rich period of progress and maturation, fueled, to a great extent, by (1) the discovery of certain basic conceptual notions, and (2) the identification of classes of systems for which systematic decomposition approaches can result in effective and easily computable control laws. These two aspects are complementary, since the latter approaches are, typically, based upon the inductive verification of the validity of the former system properties under compositions (in the terminology used in [62], the “activation” of theoretical concepts leads to “constructive” control). This expository presentation addresses the first of these aspects, and in particular the precise formulation of questions of robustness with respect to disturbances, formulated in the paradigm of input to state stability. We provide an intuitive and informal presentation of the main concepts. More precise statements, especially about older results, are given in the cited papers, as well as in several previous surveys such as [103, 105] (of which the present paper represents an update), but we provide a little more detail about relatively recent work. Regarding applications and extensions of the basic framework, we give some pointers to the literature, but we do not focus on feedback design and specific engineering problems; for the latter we refer the reader to textbooks such as [27, 43, 44, 58, 60, 66, 96].
2 ISS as a Notion of Stability of Nonlinear I/O Systems Our subject is the study of stability-type questions for input/output (“i/o”) systems. We later define more precisely what we mean by “system,” but, in an intuitive sense, we have in mind the situation represented in Fig. 1, where the “system” may well represent a component (“module” or “subsystem”) of a more complex, larger, system. In typical applications of control theory,
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E.D. Sontag
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Fig. 1. I/O system
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Fig. 2. Plant and controller
our “system” may in turn represent a plant/controller combination (Fig. 2), where the input u = (u1 , u2 ) incorporates actuator and measurement noises respectively, as well as disturbances or tracking signals, and where y = (y1 , y2 ) might consist respectively of some measure of performance (distance to a set of desired states, tracking error, etc.) and quantities directly available to a controller. The goals of our work include: • • •
Helping develop a “toolkit” of concepts for studying systems via decompositions The quantification of system response to external signals The unification of state-space and input/output stability theories
2.1 Desirable Properties We wish to formalize the idea of “stability” of the mapping u(·) → y(·). Intuitively, we look for a concept that encompasses the properties that inputs that are bounded, “eventually small,” “integrally small,” or convergent, produce outputs with the respective property: ⎧ ⎫ ⎧ ⎫ bounded
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