1082 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 45, NO. 6, JUNE 2000 A Characterization of Integral Input-to-State Stability David Angeli, Eduardo D. Sontag, and Yuan Wang Abstract—The notion of input-to-state stability (ISS) is now recognized as a central concept in nonlinear systems analysis. It provides a nonlinear generalization of finite gains with respect to supremum norms and also of finite gains. It plays a central role in recursive design, coprime factorizations, controllers for nonminimum phase systems, and many other areas. In this paper, a newer notion, that of integral input-to-state stability (iISS), is studied. The notion of iISS generalizes the concept of finite gain when using an integral norm on inputs but supremum norms of states, in that sense generalizing the linear “ ” theory. It allows one to quantify sensitivity even in the presence of certain forms of nonlinear resonance. We obtain here several necessary and suf- ficient characterizations of the iISS property, expressed in terms of dissipation inequalities and other alternative and nontrivial characterizations. These characterizations serve to show that integral input-to-state stability is a most natural concept, one that might eventually play a role at least comparable to, if not even more important than, ISS. Index Terms—Dissipation inequalities, finite gain, input-to-state stability, nonlinear systems, tracking. I. INTRODUCTION O NE OF the main issues in control design concerns the study of closed-loop sensitivity to disturbances, and, more generally, of the dependence of state trajectories on actuator and measurement errors, magnitudes of tracking signals, and the like. In linear systems theory, classical frequency-domain measures of performance such as root loci and gain-phase char- acteristics have led to the modern theories of “ ” control and its variants. During the last ten years or so, the notion of input-to-state stability (ISS) was formulated (in [23]), and quickly became a foundational concept upon which much of modern nonlinear feedback analysis and design rest. As an illustration, let us point out Kokotovic’s recent survey paper [10], intended as a sum- mary of current work and future directions on nonlinear design, in which the notion of ISS plays a central unifying concept. Sev- eral current textbooks and monographs, including [12]–[14] and [21], make use of the ISS notion and results, sometimes in an essential manner. Applications of input-to-state stability are now widespread. Besides the many applications to recursive design in the above- Manuscript received July 6, 1998; revised September 15, 1999. Recom- mended by Associate Editor, L.-S. Wang. This work was supported in part by the U.S. Air Force under Grant F49620-98-1-0242 and in part by the National Science Foundation under Grant DMS-9457826. D. Angeli is with the Dipartimento Sistemi e Informatica, University of Flo- rence, Firenze 50139 Italy (e-mail: angeli@dsi.unifi.it). E. D. Sontag is with the Department of Mathematics, Rutgers—The State University, New Brunswick, NJ 08903 USA (e-mail: sontag@hilbert.rut- gers.edu). Y. Wang is with the Department of Mathematics, Florida Atlantic University, Boca Raton, FL 33431 USA (e-mail: ywang@math.fau.edu). Publisher Item Identifier S 0018-9286(00)05275-2. mentioned books, let us merely cite a few additional references: singular perturbation analysis [3], powerful global small-gain theorems [11], foundations of tracking design [19], supervi- sory/switching adaptive control [6], observers [8], almost-dis- turbance decoupling for non-minimum-phase systems [9], and feedback stabilization with bounded controllers [29]. Moreover, this concept has many equivalent versions, which indicates that it is mathematically natural: there are characterizations in terms of dissipation, robustness margins, and classical Lyapunov-like functions; see, e.g., [25] and [26]. As remarked in [24], input-to-state stability is a nonlinear generalization both of finite gain with respect to supremum norms and of finite gain (“nonlinear ”); this property takes account of initial states in a manner fully compatible with classical Lyapunov stability and replaces finite linear gains, which represent far too strong a requirement for general nonlinear operators, with “nonlinear gains.” A system that is ISS exhibits low overshoot and low total energy response when excited by uniformly bounded or en- ergy-bounded signals, respectively. These are highly desirable qualitative characteritics. However, it is sometimes the case that feedback design does not render ISS behavior, or that only a weaker property than ISS is verified in a step in recursive de- sign. One such weaker, but still very meaningful, property was given the name of integral input-to-state stability (iISS) in a recent paper [24]. This property reflects the qualitative property of small overshoot when disturbances have finite energy and provides a qualitative analog of “finite norm” for linear systems. This is a property with obvious physical significance and relevance. The paper [24] showed that iISS is, in general, strictly weaker than ISS, and provided a very conservative Lyapunov-type sufficient condition. This paper provides sev- eral foundational results, showing that the iISS property is a most natural one to be expected for well-behaved nonlinear systems, being equivalent to the combination of well-known dissipation and detectability properties, and admitting elegant Lyapunov-theoretic characterizations. We are confident that once the results in this paper become more widely known, iISS will play a role at least as prominent as the one that ISS currently has. In fact, the notion of iISS, and the results in this paper, which were previously announced in electronic preprint form, have already played a role in several recent control works. For example, the iISS property appears in the latest approaches to supervisory design in adaptive control. In [7], Hespanha and Morse—citing preprints of this work—studied the closed-loop system obtained when a high-level supervisor directs the switching among a family of candidate controllers for an uncertain plant. Their convergence analysis was based on the 0018–9286/00$10.00 © 2000 IEEE