This theorem recovers the classical small gain theorem as a special case and its derivation highlights the connection between assumeguarantee reasoning and smallgain results. Here p s and k s are transfer functions and are assumed to be realrational, proper, and stable. We provide a generalized version of the nonlinear small gain theorem for the case of more than two coupled inputtostate stable systems. The small gain condition, expressed in terms of the spectral radius of. Read a small gain theorem for systems with noncausal subsystems, automatica on deepdyve, the largest online rental service for scholarly research with thousands of academic publications available at your fingertips. A small gain theorem for parametric assumeguarantee. Were upgrading the acm dl, and would like your input. Smallgain theorem for iss systems and applications. In proceedings of the 10th international conference on hybrid systems.
A wellknown result in linear control theory is the socalled small gain theorem stating that if given two plants with transfer matrix functions t1 and t2 in h. 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. The purpose of this paper is to bring the smallgain analysis method to the attention of the hybrid systems community. By the application of the liss lyapunovkrasovskii smallgain theorem we derive conditions, which guarantee stability of the. This note introduces a new notion of random dynamical system with inputs and outputs, and sketches a small gain theorem for monotone systems which generalizes a similar theorem known for deterministic systems. The small gain theorem, the passivity theorem and their equivalencet by b. This paper constructs the smallgain theorem upon a general class of sturmliouville systems. This paper studies a smallgain theorem for nonlinear stochastic equations driven by additive white noise in both trajectories and stationary distribution. For instance, as it is shown in 7, the use of the small gain theorem proved in this note. A lyapunovbased smallgain theorem for infinite networks. Building on the smallgain theorem, we develop selftriggered control yielding stable closedloop systems. A banach algebra, x, is a banach space with an associated mapping. A mixed small gain and passivity theorem for an interconnection of linear timeinvariant systems a mixed small gain and passivity theorem for an interconnection of linear timeinvariant systems. The small gain theorem was proved by george zames in 1966.
The result generalizes all existing results in the literature and exploits notions of weighted, uniform, and nonuniform inputtooutput stability properties. Smallgain stability theorems for positive lure inclusions. 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. Wirth, title small gain theorems for large scale systems and. A new smallgain theorem is presented for general nonlinear control systems and can be viewed as unification of previously developed nonlinear smallgain theorems for systems described by ordinary differential equations, retarded functional differential equations and hybrid models. This extends a recent smallgain theorem of angeli and sontag for monotone systems with singletonvalued characteristics. Motivated by the fact that many feedback control laws lead to discontinuous closed loop systems, we pose no continuity assumptions on the system dynamics. Exponential smallgain theorem and fault tolerant safe. A generalization of the nonlinear smallgain theorem for largescale. We present a nonlinear lyapunov function based small gain theorem for analyzing inputtostate stability of discretetime largescale systems. A smallgain theorem for nonlinear stochastic systems with inputs. To construct the theorem, spatiotemporal transferfunction and geometrical isomorphism. Under a small gain assumption on the monotone operator induced. A lyapunov formulation of the nonlinear smallgain theorem.
This note introduces a new notion of random dynamical system with inputs and outputs, and sketches a smallgain theorem for monotone systems which generalizes a similar theorem known for deterministic systems. Stability analysis of hybrid systems via smallgain theorems. Sufficient conditions to guarantee the condition 8 are given in io. V applications of the small gain theorem 1 continuoustime systemslp stability 2 l1 stabilitycircle criterion 3 exponential weightingu stability 4 discretetime systemslp stability 5 slowlyvarying linear systems 6 nonlinear circuit example 7 existence of periodic solutions 8 popov criterion 9 instability notes and references 7. 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. If kgk small gain theorem with restrictions for uncertain timevarying nonlinear systems. This paper gives three versions of the small gain theorem with restrictions for uncertain timevarying nonlinear systems. Smallgain theorem for iss systems and applications centre. As an illustration of the interest of these results, we address the. The purpose of this paper is to develop a weak version of the small gain theorem, in the spirit of the matrosov theorem derived in 2. A generalization is given of the usual smallgain theorem which allows greater flexibility in use of gain bounds for stability analysis of nonlinear feedback. Pdf application of the liss lyapunovkrasovskii smallgain.
Revisiting the iiss smallgain theorem through transient plus. Pdf this paper presents a small gain theorem for networks composed of a countably infinite number of finitedimensional subsystems. We note that while the basic idea of the smallgain stability analysis for hybrid systems was announced. One of the simplest versions applies to the unit feedback interconnection on figure 3. This paper presents a small gain theorem for networks composed of a countably infinite number of finitedimensional subsystems. Sanjit seshias publications a small gain theorem for. We present a nonlinear lyapunov function based smallgain theorem for analyzing inputtostate stability of discretetime largescale systems. We provide a new global smallgain theorem for feedback interconnections of monotone inputoutput systems with multivalued inputstate characteristics.
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. A smallgain theorem for a wide class of feedback systems. Revisiting the iiss smallgain theorem through transient. The proposed smallgain theorem addresses exponential inputtostate stability with respect to closed sets, which enables us to analyze diverse stability problems in a unified manner.
The miller theorem and the frequency response of the. A generalization of the smallgain theorem for nonlinear feedback. A generalized small gain theorem for outputlagrangeinputtooutput stable systems in network interconnections has been obtained in 19. The work on small gain theorems for the nonlinear case began with the work 12, where a nonlinear generalization of the classical smallgain theorem was proposed for both continuous and discretetime feedback systems. We next show how a finite gain property can be encoded in this form and provide a generalized smallgain theorem for parametric assumeguarantee contracts. Algebra banach algebra a algebra is a vector space with a distributive multiplication operator. As an illustration of the interest of these results, we address the problem of global. Pdf a lyapunovbased smallgain theorem for infinite. We introduce a concept of inputtooutput practical stability iops. 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. The small gain theorem gives a sufficient condition for finite gain stability of the feedback connection. We consider interconnections of n nonlinear subsystems in the inputtostate stability iss framework. A smallgain theorem for a wide class of feedback systems is proposed in 7.
Chapter 3 small gain theorem and integral quadratic. The mixed small gain and passivity property is described mathematically using the notion of dissipativity of systems, and finitegain stability of the interconnection is proven via a stability result for dissipative interconnected systems. Pdf the aim of this paper is to give a general quantitative requirement a feedback gain. 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. This paper gives three versions of the small gain theorem with restrictions for. The smallgain theorem gives a sufficient condition for finitegain stability of the feedback connection. We formulate sufficient conditions in terms of weighted onenorms, reminiscent of the smallgain condition, which ensure that the zero. This is the correct answer for the closed loop gain as we shall later see. It includes as a special case the previous nonlinear smallgain theorems with. 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 small gain theorem for timevarying systems vasile drxgan institute of mathematics, str. 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 springerlink. The purpose of this paper is to bring the small gain analysis method to the attention of the hybrid systems community. The requirement to evaluate a gain over the whole signal space is one of the restrictions in the wellknown small gain theorem. An iss small gain theorem for general networks springerlink. A smallgain theorem for monotone systems with multivalued. We also describe another approach, based on constructing a weak nonstrictly decreasing lyapunov function and applying the lasalle invariance principle for hybrid systems from 6. Small gain theorems for networks of heterogeneous systems. The memoryless version of the small gain theorem stated in one of the. Sanfelice and rafael fierro abstractthis paper investigates stability of nonlinear control systems under intermittent information. Finally, small gain theorems for large scale interconnected systems and for systems interconnectedby means of communication channels have recently been developed in 3. It is shown that an intrinsic feedback structure, mapping the noise sequence and the initial weight guess to the a priori estimation errors and the final weight estimate, can be associated with such schemes. A small gain theorem for a wide class of feedback systems is proposed in 7. Revisiting the iiss small gain theorem through transient plus iss small gain regulation hiroshi ito, randy a.
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. The result can be viewed as an extension of the small gain theorem with restrictions fortimeinvariantnonlinear systems or the smallgain theorem without restrictions for timevarying nonlinear systems. It appears that the feedback connection of two sturmliouville subsystems is guaranteed of wellposedness, hurwitz, dissipativity and passivity in l2spaces provided the loop gain is less than 1. 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. Small gain theorem the simplest form of spectral theory. Small gain theorems for general networks of heterogeneous infinite. The work on small gain theorems for the nonlinear case began with the work 12, where a nonlinear generalization of the classical small gain theorem was proposed for both continuous and discretetime feedback systems. The second one shows an example of gain function assignment by feedback. The smallgain theorem, the passivity theorem and their. The mixed small gain and passivity property is described mathematically using the notion of dissipativity of systems, and finite gain stability of the interconnection is proven via a stability result for dissipative interconnected systems. Assuming that each subsystem is exponentially inputtostate stable, we show that if the gain operator, collecting all the information about the internal lyapunov gains, has a spectral radius less than one, the overall infinite network is exponentially inputtostate. Academiei 14 ro70109, bucuresti, romania received and accepted march 1993 abstracta small gain result is proved by liapunov methods for timevarying systems. Communications in information and systems c 2006 international press vol. Aug 23, 2018 stability results are presented for a class of differential and difference inclusions, socalled positive lure inclusions which arise, for example, as the feedback interconnection of a linear positive system with a positive setvalued static nonlinearity.
The purpose of this paper is to develop a weak version of the smallgain theorem, in the spirit of the matrosov theorem derived in 2. For each subsystem an iss lyapunov function is given that treats the other subsystems as independent inputs. The first example uses a baby version of the small gain theorem. The observation provides useful information for designing and analyzing nonlinear. Author links open overlay panel andrii mironchenko. Using the concepts of local gain and strict causality a local form of small gain theorem is proposed, which can be used to analyze input magnitude dependent stability problems of feedback nonlinear systems, such as a volterra system. The motivation for the current study is to derive stability results for multivariable that is, multiinput multioutput, or mimo positive lure. We prove a smallgain theorem for interconnections of n nonlinear heterogeneous inputtostate stable control systems of a general. Pdf in this paper we consider general autonomously controlled production networks. It can be seen as a generalization of the nyquist criterion to nonlinear timevarying mimo systems systems with multiple inputs and multiple outputs. Consider a system with a stable loop transferfunction ls. We prove a smallgain theorem for interconnections of n. A relaxed small gain theorem for interconnected discretetime systems.
Wirth, title small gain theorems for large scale systems and construction, booktitle of iss lyapunov functions, siam journal on control and optimization 2010, in press. A small gain theorem for parametric assumeguarantee contracts. For wellposedness we shall assume that p or k or both is strictly proper equal to zero at s 0. The smallgain condition, expressed in terms of the spectral radius of. Smallgain theorem, gain assignment and applications.
Applications to partial state feedback stabilization. Read a lyapunov formulation of the nonlinear smallgain theorem for interconnected iss systems, automatica on deepdyve, the largest online rental service for scholarly research with thousands of academic publications available at your fingertips. The wellknown smallgain theorem implies that a loopgain of less than unity ensures the stability of dynamical feedback systems. This condition is a nonlinear version of the classical smallgain condition see, for instance, 2. This theorem recovers the classical small gain theorem as a special case and its derivation highlights the connection between assumeguarantee reasoning and small gain results. Bernsteinexplicit construction of quadratic lyapunov functions for the small gain, positivity, circle and popov theorems and their applications. Problem 1 small gain theorem t arizona state university. Such a theorem has been widely used for stability analysis and control design of nonlinear systems in the past several decades ito. In nonlinear systems, the formalism of inputoutput stability is an important tool in studying the. The observation provides useful information for designing and analyzing nonlinear control systems based on the iiss small. Pdf this paper presents a smallgain theorem for networks composed of a countably infinite number of finitedimensional subsystems. We provide a new global smallgain theorem for feedback interconnections of monotone inputoutput systems with multivalued. This paper provides a timedomain feedback analysis of gradientbased adaptive schemes with emphasis on stability and robustness issues. 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.
A gain matrix is used to encode the mutual dependencies of the systems in the network. Finally, smallgain theorems for large scale interconnected systems and for systems interconnectedby means of communication channels have recently been developed in 3. A small gain theorem for timevarying systems sciencedirect. 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. Small gain theorem for distributed feedback control of. Freeman, and antoine chaillet abstract recently, the small gain theorem for inputtostate stable iss systems has been extended to the class of integral inputtostate stable iiss systems. Small gain theorems for large scale systems and construction. Advanced vibration control of atomic force microscope scanner. We can even use the miller theorem to calculate the gain of this circuit for finite opamp gain g. Anderson department of electrical engineering universiv of newcastle, new south wales, australia aeact. We next show how a finite gain property can be encoded in this form and provide a generalized small gain theorem for parametric assumeguarantee contracts. 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 smallgain regulation.
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