Wirth december 20, 2007 abstract we provide a generalized version of the nonlinear small gain theorem for the case of more than two coupled inputtostate stable iss systems. N2 we introduce a concept of inputtooutput practical stability iops which is a natural generalization of inputtostate stability proposed by sontag. Here, it is also shown how this result may be used in a large variety of speci. The smallgain theorem, the passivity theorem and their. Small gain theorem with restrictions 119 the following small gain theorem with restrictions for time invariant nonlinear systems was established in appendix b of 6.
This condition is a nonlinear version of the classical small gain condition see, for instance, 2. In 2, the authors presented an isstype small gain theorem in terms of operators that recovers the. From the small gain theorem, there is deduced the passivity theorem, which is actually a special case of a tleorem predioting stability in case certain conicity results are satisfied. A smallgain theorem for a wide class of feedback systems. The small gain theorem was proved by george zames in 1966.
Revisiting the iiss smallgain theorem through transient plus iss smallgain regulation hiroshi ito, randy a. In our case, however, the class of uncertainties includes a phase constraint. This is for a good reason, as relaxing the constraint would make the conclusion invalid. Pdf implications of the small gain theorem in the design. The iiss is a more general robustness property than iss, 14, and the theorem in 6, 9 includes the iss smallgain theorem as a special case. A nonlinear small gain theorem for the analysis of control. T1 smallgain theorem for iss systems and applications. The result generalizes all existing results in the literature and exploits notions of weighted, uniform, and nonuniform inputtooutput stability properties. More recently, smallgain tools have been used extensively to study feedback interconnections of nonlinear statespace. Inputoutput triggered control via the small gain theorem.
The modeling of matrix uncertainty by means of a fictitious feedback loop linear fractional transformation is directly analogous to the small gain h parameter uncertainty model of 32. It can be shown that the cyclic small gain theorem can recover both the conventional small gain results and the recently developed small gain results for dynamical networks. Umodel, lms, convergence speed, learning rate, small gain theorem. It specializes to a small gain theorem for iss operators, and it also recovers the classical statement for iss systems in statespace form. A cyclic small gain theorem for hybrid dynamic networks with the impulses of the subsystems triggered asynchronously is developed in liu, jiang, and hill 2012b. Sontag2 department of mathematics rutgers university piscataway, nj 088548019, usa abstract. Citeseerx small gain theorems on inputtooutput stability. Assume that subsystem 6 is iss with restrictions x1. Small gain theorems on inputtooutput stability english.
The base of robustness analysis for such dynamic uncertainty is the small. Stability analysis of hybrid systems via smallgain theorems 423 the state dynamics describing the evolution of these variables with respect to time are composed of continuous evolution and discrete events. More recently, small gain tools have been used extensively to study feedback interconnections of nonlinear statespace. Assuming that each subsystem is exponentially inputtostate stable. It can be shown that the cyclicsmallgain theorem can recover both the conventional smallgain results and the recently developed smallgain results for dynamical networks. The smallgain theorem gives a sufficient condition for finitegain stability of the feedback connection. Under the term inputoutput triggering, we refer to selftriggering based on the values of the plants external input and output in the last feedback transmission. Applications to partial state feedback stabilization.
The purpose of this paper is to bring the small gain analysis method to the attention of the hybrid systems community. The memoryless version of the small gain theorem stated in one of the. Indeed, this feature is illustrated by many examples in the paper. This nonlinear iss small gain theorem differs from classical small gain theorems and the nonlinear small gain theorem of 7 and 26 in several aspects. A vector small gain theorem 311 functions for the case of iss stability. The small gain theorem then becomes a statement that stability follows if yh, yh, c 1. Engineering applications and computational algorithms guelph, on, 2003, dynamics of continuous, discrete and impulsive systems series b. A new small gain theorem is presented for general nonlinear control systems and can be viewed as unification of previously developed nonlinear small gain theorems for systems described by ordinary differential equations, retarded functional differential equations and hybrid models. Motivated by a paradigm shift towards a hyperconnected world, we develop a computationally tractable small gain theorem for a network of infinitely many systems, termed as infinite networks. Lyapunov formulation of the largescale, iss cyclicsmall. For this result the interconnection gains are described in a nonlinear gain matrix, and the small gain condition requires bounds on the image of this gain matrix. A cyclicsmallgain theorem for hybrid dynamic networks with the impulses of the subsystems triggered asynchronously is developed in liu, jiang, and hill 2012b. Pdf implications of the small gain theorem in the design of.
In particular, retail is considered as a feedback phenomenon, and some fundamental limits on learning from data are identified. Local small gain theorems for performance analysis dxdt fx,w 1,w 2 z hx d. In 2, the authors presented an isstype small gain theorem in terms of operators that recovers the case of state space form, with applications. Introduction the smallgain theorem is a classical tool for analyzing inputoutput stability of feedback systems. This paper presents a lyapunov formulation of the iss cyclic small gain criterion for discretetime dynamical networks. Making use of this necessary condition for the stability, this paper gives a new interpretation to the iiss small gain theorem as transient plus iss small.
This paper presents a small gain theorem for networks composed of a countably infinite number of finitedimensional subsystems. Smallgain theory for stability and control of dynamical. Small gain theorems for large scale systems and construction. Freeman, and antoine chaillet abstract recently, the smallgain theorem for inputtostate stable iss systems has been extended to the class of integral inputtostate stable iiss systems. The result can be viewed as an extension of the small gain theorem with restrictions for timeinvariant nonlinear systems or the small gain theorem without restrictions for timevarying nonlinear systems. Small gain theorem the simplest form of spectral theory. The smallgain theorem states that, the feedback interconnection between two stable linear systems is stable if the loopgain between the two systems is strictly less than one. An iss small gain theorem for general networks springerlink. The paper also discusses about the robustness of adaptation algorithm in the presence of noise and suggests optimal choices for faster convergence speeds.
Sanfelice and rafael fierro abstractthis paper investigates stability of nonlinear control systems under intermittent information. Smallgain theorem, gain assignment and applications. In addition, we highlight applications to incrementally. The first one states that the general interconnection of two iops systems is again an iops system if an appropriate composition of the gain functions is smaller than the identity function. Abstract this paper presents nonlinear small gain theorems for both continuous time systems and discrete time systems in a general setting of inputtooutput stability. Citeseerx document details isaac councill, lee giles, pradeep teregowda. Introduction the small gain theorem is a classical tool for analyzing inputoutput stability of feedback systems. The proposed smallgain theorem addresses exponential inputtostate stability with respect to closed sets, which enables us to. A nonlinear small gain theorem is presented that provides a formalism for analyzing the behavior of certain control systems that contain or utilize saturation. We introduce a concept of inputtooutput practical stability iops which is a natural generalization of inputtostate stability proposed by sontag. Smallgain theorem for iss systems and applications nyu. Inputoutput triggered control via the small gain theorem and. One of them is that both internal and external stability properties are discussed in a single framework, while only inputoutput stability is addressed in previous small gain theorems. In 5, a small gain theorem was provided in terms of isslyapunov functions.
A smallgain approach to stability analysis of hybrid systems. Another feature, which is not frequently recognised, is that vector lyapunov functions can handle largescale systems more easily than single lyapunov functions. Sufficient conditions to guarantee the condition 8 are given in io. An iss small gain theorem for general networks sergey dashkovskiy bj. Chapter 3 small gain theorem and integral quadratic constraints. Smallgain theorem for iss systems and applications. The result can be viewed as an extension of the small gain theorem with restrictions for timeinvariant nonlinear systems or the small gain theorem. Iss small gain theorem for general networks 95 in 29 teel gives a small gain theorem for systems with outputs satisfying an issrelated stability property and having saturated interconnections. The proposed small gain theorem addresses exponential inputtostate stability with respect to closed sets, which enables us to analyze diverse stability problems in a unified manner. Interpreting the iiss smallgain theorem as transient plus. The small gain theorem gives a sufficient condition for finite gain stability of the feedback connection. The smallgain theorem gives a sufficient condition for finite gain stability of the feedback connection. Chapter 3 small gain theorem and integral quadratic. Revisiting the iiss smallgain theorem through transient.
Abstract this report explores the challenges and opportunities inherent in developing a retail laboratory. The proposed smallgain theorem addresses exponential inputtostate stability with respect to closed sets, which enables us to analyze diverse stability. A timedelay version of the cyclicsmallgain theorem can be found in tiwari, wang. Pdf smallgain theorem for stability, cooperative control. We provide a generalized version of the nonlinear small gain theorem for the case of more than two coupled inputtostate stable systems. Find an upper bound on kek2 provided that mand start from rest and kw1k2 r. Most small gain theorems in the past literature were stated for interconnected systems of which the subsystems are either inputto. In general, the main idea behind an iss smallgain theory is to consider a largescale system possibly an in. The small gain condition, expressed in terms of the spectral radius of a gain operator collecting all the information about the internal lyapunov gains, can be. Advanced vibration control of atomic force microscope scanner.
This paper presents a lyapunov formulation of the iss cyclicsmallgain criterion for discretetime dynamical networks. A vector smallgain theorem for general nonlinear control. A timedelay version of the cyclic small gain theorem can be found in tiwari, wang, and jiang 2012. A smallgain theorem, which can be applied to a wide class of systems that includes systems satisfying the weak semigroup property, is presented in the present work. This paper further develops a method, originally introduced by angeli and the second author, for proving global attractivity of steady states in. Stability analysis of hybrid systems via smallgain theorems. Motivated by a paradigm shift towards a hyperconnected world, we develop a computationally tractable smallgain theorem for a network of infinitely many systems, termed as infinite networks. Pdf a lyapunovbased smallgain theorem for infinite networks. The triggering event, that is to be precluded in our approach, is the violation of the small gain condition. In the recent work karafyllis and tsinias, 2004, a small gain theorem was proved for the weightediss property for time varying systems followed by an interesting application of smooth output. It can be seen as a generalization of the nyquist criterion to nonlinear timevarying mimo systems systems with multiple inputs and multiple outputs.
The theorem is used to show that an iterative procedure can be derived for controlling systems in a general nonlinear, feedforward form. Fm11 1150 proceedings of the 33rd conierenm on deddon and control lake buena vista, fl december 1994 smallgain theorem, gain assignment and applications z. Problem 1 small gain theorem t arizona state university. Pdf a lyapunovbased smallgain theorem for infinite. Robust stabilization with positive real uncertainty.
We have shown that, providing the switching signals neither switch too frequently nor activate noniss subsystems for too long, a smallgain theorem can be used to conclude global asymptotic stability gas of the interconnected system. Building on the smallgain theorem, we develop selftriggered control yielding stable closedloop systems. This paper gives three versions of the small gain theorem with restrictions for uncertain timevarying nonlinear systems. The small gain theorem states that, the feedback interconnection between two stable linear systems is stable if the loop gain between the two systems is strictly less than one. The observation provides useful information for designing and analyzing nonlinear control systems based on the iiss small. A small gain theorem, which can be applied to a wide class of systems that includes systems satisfying the weak semigroup property, is presented in the present work.
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