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35
A Characterization of Integral Input to State Stability
, 1998
"... 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 L 2 gains. It plays a central role in recursive design, coprime factorizations ..."
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Cited by 46 (13 self)
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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 L 2 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 "H 2 " theory. It allows to quantify sensitivity even in the presence of certain forms of nonlinear resonance. We obtain here several necessary and sufficient 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 stabi...
Stability and stabilization: discontinuities and the effect of disturbances, in “Nonlinear analysis, differential equations and control
, 1998
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Input to state stability: Basic concepts and results
 Nonlinear and Optimal Control Theory
, 2006
"... 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 decompositi ..."
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Cited by 46 (6 self)
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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
The ISS philosophy as a unifying framework for stabilitylike behavior
 in Nonlinear Control in the Year 2000
, 2000
"... Abstract. The input to state stability (ISS) paradigm is motivated as a generalization of classical linear systems concepts under coordinate changes. A summary is provided of the main theoretical results concerning ISS and related notions of input/output stability and detectability. A bibliography i ..."
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Cited by 44 (6 self)
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Abstract. The input to state stability (ISS) paradigm is motivated as a generalization of classical linear systems concepts under coordinate changes. A summary is provided of the main theoretical results concerning ISS and related notions of input/output stability and detectability. A bibliography is also included, listing extensions, applications, and other current work. 1
A Lyapunov approach to incremental stability
"... . This paper deals with several notions of incremental stability. In other words, we focus on stability of trajectories with respect to one another, rather than with respect some attractor or equilibrium point. The aim is to present a framework for understanding such questions fully compatible with ..."
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Cited by 42 (2 self)
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. This paper deals with several notions of incremental stability. In other words, we focus on stability of trajectories with respect to one another, rather than with respect some attractor or equilibrium point. The aim is to present a framework for understanding such questions fully compatible with the wellknown InputtoState Stability approach. 1 Introduction InputtoState stability (### for short) has proven a valid instrument in order to study questions of robust stability for nitedimensional nonlinear systems. One reason for that is the possibility of dealing at the same time with a body of theory which nicely extends the classic Lyapunov approach to nonautonomous systems, while still allowing for inputoutput descriptions of the system behavior [15]. In this way tools such as smallgain theorems and Lyapunov dissipation inequalities [1, 13] have come together in a unied framework which bridges the gap between the statespace and inputoutput approaches. Stability propertie...
A smooth Lyapunov function from a classKL estimate involving two positive semidefinite functions
 ESAIM, Control Optim. Calc. Var
, 2000
"... Abstract. We consider differential inclusions where a positive semidefinite function of the solutions satisfies a classKL estimate in terms of time and a second positive semidefinite function of the initial condition. We show that a smooth converse Lyapunov function, i.e., one whose derivative alon ..."
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Cited by 41 (11 self)
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Abstract. We consider differential inclusions where a positive semidefinite function of the solutions satisfies a classKL estimate in terms of time and a second positive semidefinite function of the initial condition. We show that a smooth converse Lyapunov function, i.e., one whose derivative along solutions can be used to establish the classKL estimate, exists if and only if the classKL estimate is robust, i.e., it holds for a larger, perturbed differential inclusion. It remains an open question whether all classKL estimates are robust. One sufficient condition for robustness is that the original differential inclusion is locally Lipschitz. Another sufficient condition is that the two positive semidefinite functions agree and a backward completability condition holds. These special cases unify and generalize many results on converse Lyapunov theorems for differential equations and differential inclusions that have appeared in the literature. AMS Subject Classification. 34A60, 34D20, 34B25.
OutputInput Stability and MinimumPhase Nonlinear Systems
 IEEE Trans. Automat. Control
, 2002
"... This paper introduces and studies the notion of outputinput stability, which represents a variant of the minimumphase property for general smooth nonlinear control systems. The denition of outputinput stability does not rely on a particular choice of coordinates in which the system takes a norma ..."
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Cited by 20 (15 self)
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This paper introduces and studies the notion of outputinput stability, which represents a variant of the minimumphase property for general smooth nonlinear control systems. The denition of outputinput stability does not rely on a particular choice of coordinates in which the system takes a normal form or on the computation of zero dynamics. In the spirit of the \inputtostate stability" philosophy, it requires the state and the input of the system to be bounded by a suitable function of the output and derivatives of the output, modulo a decaying term depending on initial conditions. The class of outputinput stable systems thus dened includes all ane systems in global normal form whose internal dynamics are inputtostate stable and also all leftinvertible linear systems whose transmission zeros have negative real parts. As an application, we explain how the new concept enables one to develop a natural extension to nonlinear systems of a basic result from linear adaptive control.
Further constructions of strict Lyapunov functions for timevarying systems
 in Proceedings of the American Control Conference
, 2005
"... We provide explicit closed form expressions for strict Lyapunov functions for timevarying discrete time systems. Our Lyapunov functions are expressed in terms of known nonstrict Lyapunov functions for the dynamics and finite sums of persistency of excitation parameters. This provides a discrete tim ..."
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Cited by 13 (3 self)
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We provide explicit closed form expressions for strict Lyapunov functions for timevarying discrete time systems. Our Lyapunov functions are expressed in terms of known nonstrict Lyapunov functions for the dynamics and finite sums of persistency of excitation parameters. This provides a discrete time analog of our previous continuous time Lyapunov function constructions. We also construct explicit strict Lyapunov functions for systems satisfying nonstrict discrete time analogs of the conditions from Matrosov’s Theorem. We use our methods to build strict Lyapunov functions for timevarying hybrid systems that contain mixtures of continuous and discrete time evolutions. Key Words: Strict Lyapunov functions, discrete and hybrid timevarying systems. 1
A smallgain theorem with applications to input/output systems, incremental stability, detectability, and interconnections
 Journal of the Franklin Institute
, 2002
"... Abstract A general ISStype smallgain result is presented. It specializes to a smallgain theorem for ISS operators, and it also recovers the classical statement for ISS systems in statespace form. In addition, we highlight applications to incrementally stable systems, detectable systems, and to i ..."
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Cited by 11 (3 self)
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Abstract A general ISStype smallgain result is presented. It specializes to a smallgain theorem for ISS operators, and it also recovers the classical statement for ISS systems in statespace form. In addition, we highlight applications to incrementally stable systems, detectable systems, and to interconnections of stable systems.
Intrinsic Robustness of Global Asymptotic Stability
 SYSTEMS & CONTROL LETTERS
, 1999
"... Equivalence is shown for discrete time systems between global asymptotic stability and the so called integral Input to State Stability. The latter is a notion of robust stability with respect to exogenous disturbances which informally translates into the statement "no matter what is the initial cond ..."
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Cited by 10 (1 self)
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Equivalence is shown for discrete time systems between global asymptotic stability and the so called integral Input to State Stability. The latter is a notion of robust stability with respect to exogenous disturbances which informally translates into the statement "no matter what is the initial condition, if the energy of the inputs is small, then the state must eventually be small".