The resulting gain is the current gain of the amplifier with a shortcircuit load. Algebra banach algebra a algebra is a vector space with a distributive multiplication operator. The first example uses a baby version of the small gain theorem. Here p s and k s are transfer functions and are assumed to be realrational, proper, and stable. Some robust control problems for a class of nonlinear cascaded systems in the presence of state and input driven unmeasured dynamics are analyzed. Sontag2 department of mathematics rutgers university piscataway, nj 088548019, usa abstract. 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. We next show how a nite gain property can be encoded in this form and provide a generalized small gain theorem for parametric assumeguarantee contracts. Abstract this paper presents nonlinear small gain theorems for both continuous time systems and discrete time systems in a general setting of inputtooutput stability. 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. Multichannel smallgain theorems for large scale networked. This paper presents nonlinear small gain theorems for inputtostate stability and inputtooutput stability properties of largescale timedelay systems consisting of multiple subsystems based on. This paper gives three versions of the small gain theorem with restrictions for. A generalized small gain theorem for outputlagrangeinputtooutput stable systems in network interconnections has been obtained in 19.
This paper presents a smallgain theorem for networks composed of a countably infinite number of finitedimensional subsystems. One of the simplest versions applies to the unit feedback interconnection on. Nonlinear smallgain theorems for largescale timedelay. Advanced vibration control of atomic force microscope scanner. Smallgain theorem, gain assignment and applications nyu. The use of the small gain theorem for control design dates back to the 1960s see 35, 36, and 20, for example. Lectures in feedback design for multivariable systems. This is for a good reason, as relaxing the constraint would make the conclusion invalid. The small gain theorem was proved by george zames in 1966. Ab this paper presents nonlinear smallgain theorems for inputtostate stability and inputtooutput stability properties of largescale timedelay systems consisting of multiple subsystems based on razumikhintype techniques. Find an upper bound on kek2 provided that mand start from rest and kw1k2 r. A gain matrix is used to encode the mutual dependencies of the systems in the network. A nonlinear small gain theorem for the analysis of control.
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. 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. Inputtostate stability iss is a stability notion widely used to study stability of nonlinear control systems with external inputs. Furthermore, for the same class of control systems we derive smallgain theorems for asymptotic gain, uniform global stability and weak inputtostate stability properties. A nonlinear smallgain theorem for largescale time delay systems. One of the simplest versions applies to the unit feedback interconnection on figure 3. Advanced textbooks in control and signal processing. However, the basic interface provided by atps validitysatisfiability checking of formulas has changed little over the years.
The observation provides useful information for designing and analyzing nonlinear control systems based on the iiss smallgain theorem. Most small gain theorems in the past literature were stated for interconnected systems of which the subsystems are either inputtostate stable. Citeseerx document details isaac councill, lee giles, pradeep teregowda. Related results and applications of the nonlinear small gain theorem in nonlinear. Stability of nonlinear interconnected systems smallgain condition. 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. Lyapunov smallgain theorems for not necessarily iss. This paper presents a new approach to eventtriggered control for nonlinear uncertain systems by using the notion of inputtostate stability iss and the nonlinear small gain theorem. In nonlinear systems, the formalism of inputoutput stability is an important tool in studying the stability of interconnected systems since the gain of a system. We consider interconnections of n nonlinear subsystems in the inputtostate stability iss framework. Inputtooutput stability, control systems, smallgain theorem. Small gain theorem the simplest form of spectral theory. Citeseerx citation query smallgain theorem for iss.
A smallgain control method for nonlinear cascaded systems. In section iii the small gain theorem, the main theoretical apparatus in the paper, is employed, and a framework is proposed to broaden its use for the speci. A smallgain approach to robust eventtriggered control of. The smallgain theorem, the passivity theorem and their. The result can be viewed as an extension of the small gain theorem with. The small gain theorem states that a sufficient condition for the closedloop system of figure 1 to be stable is that the norm l of the loop gain is less than 1.
We demonstrate that the small gain analysis framework is very naturally and generally applicable in the context of hybrid systems, and thus has. Revisiting the iiss smallgain theorem through transient. But the substitution theorem cannot use for solving the theorem which has more than two sources which are neither connected in series nor. A smallgain approach to stability analysis of hybrid systems dragan ne. However, the story for statespace treatment of the nonlinear systems is different, since the small gain theorem can only guarantee the system to. Borrowing techniques in modern nonlinear control, especially sontag s notion of inputtostate stability iss. This paper gives three versions of the small gain theorem with restrictions for uncertain timevarying nonlinear systems.
The purpose of this paper is to bring the small gain analysis method to the attention of the hybrid systems community. In mathematics, the classification of the finite simple groups is a theorem stating that every finite simple group is either cyclic, or alternating, or it belongs to a broad infinite class called the groups of lie type, or else it is one of twentysix or twentyseven exceptions, called sporadic. Theconstructiveproofs arebasedon a combinationof astepbystepprocedure,based onh arguments, together with a small gain theorem, recently proved for nonlinear systems by jiang. The concept of the theorem is based on the substitution of one element from another element. 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. The smallgain theorem gives a sufficient condition for finitegain stability of the feedback connection.
The motivation for the current study is to derive stability results for multivariable that is, multiinput multioutput, or mimo positive lure inclusionsnonlinear positive control systemswhich, as already stated, arise in. Feedback connections of two iiss systems are robustly stable with respect to disturbance if an extended small. A smallgain theorem for a wide class of feedback systems. Is smallgain theorem a sufficient and necessary condition for. Small gain theorems for general networks of heterogeneous infinite. We provide a generalized version of the nonlinear small gain theorem for the case of more than two coupled inputtostate stable iss systems. A common feature of stability analysis for complex interconnected. The results build on some generalizations of recent results on the class of monotone aggregation functions and almost inversion of general small gain operators.
The smallgain theorem, the passivity theorem and their equivalence the time interval over which the system is dehed, denoted by s, is normally m, m or t, m. We prove a smallgain theorem for interconnections of n nonlinear heterogeneous inputtostate stable control systems of a general. Stability and the scaled small gain theorem caltech authors. Our asymptotic small gain theorem is, then, another tool for guiding nonlinear control design. Anderson department of electrical engineering university of newcastle, new south wales, australia abstract. A small gain theorem for parametric assumeguarantee. Building on the smallgain theorem, we develop selftriggered control yielding stable closedloop systems.
We introduce a concept of inputtooutput practical stability iops which is a natural generalization of inputtostate stability proposed by sontag. 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. Applications to partial state feedback stabilization problems with. The smallgain 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. Pdf a lyapunovbased smallgain theorem for infinite. Earnshaws theorem simion 2019 supplemental documentation. Stability analysis of hybrid systems via smallgain theorems. Two small gain theorems and a passivity theorem yielding stability results for general feedback systems are shown to be equivalent. 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. We give a stability criterion for a general situation and show how it can be applied in special cases.
A stepwise constructive control methodology is proposed on the basis of the nonlinear smallgain theorem. First, however, we present the discretetime bounded real lemma. Earnshaws theorem earnshaws theorem states that a charged particle cannot be held statically in a stable equilibrium by electrostatic forces alone. 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 gain regulation. Problem 1 small gain theorem t arizona state university. The small gain theorem in this section we construct quadratic lyapunov functions to prove sufficiency in the small gain theorem for the interconnection of a dynamic system and a normbounded memoryless timevarying nonlinearity. A small gain control method for nonlinear cascaded systems with dynamic uncertainties abstract. For each subsystem an iss lyapunov function is given that treats the other subsystems as independent inputs. Application of small gain type theorems in logistics of autonomous processes s. Chapter 3 small gain theorem and integral quadratic constraints.
Lyapunovrazumikhin and lyapunovkrasovskii theorems for. The ultimate objective of control is to achieve required performance specifications such as signal tracking, disturbance suppression and rapid response. Robust synchronization in networks of cyclic feedback systems. The smallgain condition, expressed in terms of the spectral radius of a gain operator collecting all the information about the internal lyapunov gains, can be numerically computed for a large class of systems in an efficient way.
Communications in information and systems c 2006 international press vol. The small gain theorem gives a sufficient condition for finite gain stability of the feedback connection. Stability conditions for infinite networks of nonlinear. This theorem gives intuition on the behaviour of the circuit.
The smallgain theorems developed therein pro vide a construction of an iss lyapunov function if the instabilities of subsystems are of the same type, that is if either. Despite its apparent simplicity at first sight, zamess smallgain theorem plays a crucial role in the development of linear robust control theory. Bounding l2 gain system error generated by approximations. On the robustness of homogeneous systems and a homogeneous. 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. Sanfelice and rafael fierro abstractthis paper investigates stability of nonlinear control systems under intermittent information.
The pole polynomial of a system with transfermatrix gs is the. If kgk small gain theorem in a state space framework for time varying systems. The small gain theorem, the passivity 7jheorem and their equivalencet by b. Lyapunovrazumikhin and lyapunovkrasovskii theorems for interconnected iss timedelay systems sergey dashkovskiy and lars naujok abstractwe consider an arbitrary number of interconnected nonlinear systems with timedelays and investigate them in view of inputtostate stability iss. In general, the main idea behind an iss smallgain theory is to consider a largescale system possibly an in. It can be seen as a generalization of the nyquist criterion to nonlinear timevarying mimo systems systems with. Please report errors in award information by writing to. The above expressions can be substituted into the asymptotic gain model equation to find the overall gain g. The result generalizes all existing results in the literature and exploits notions of weighted, uniform, and nonuniform inputtooutput stability properties. More precisely, we show that a continuous homogeneous system, which is asymptotically stable. Automated theorem provers atps are a key component that many software verification and program analysis tools rely on. Local small gain theorems for performance analysis dxdt fx,w 1,w 2 z hx d. Optimal control encyclopedia of life support systems.
A small gain theorem for parametric assumeguarantee contracts. Robust synchronization in networks of cyclic feedback systems abdullah omar hamadeh, guybart stan, jorge gonc. Citeseerx small gain theorems on inputtooutput stability. Introduction since 1960 small gain type theorems have proved to be. Application of small gain type theorems in logistics of. A lyapunovbased small gain theorem for interconnected switched systems, systems and control letters, v. In contrast to the current work, the less general small gain theorem from dashkovskiy and pavlichkov 2017 actually addresses uniform asymptotic stability for strong solutions of treated as c 1curves in. Smallgain theorem, gain assignment and applications. A smallgain control method for nonlinear cascaded systems with dynamic uncertainties abstract. A banach algebra, x, is a banach space with an associated mapping. 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.
The smallgain condition, expressed in terms of the spectral radius of a gain operator collecting all the information about the internal lyapunov gains, can be numerically computed for a large class of systems in an efficient. Under a small gain assumption on the monotone operator induced by the gain matrix, a locally. The initial small gain theorem involved finite linear gains from a norm of the input to a. Roughly speaking, a control system is iss if it is globally asymptotically stable in the absence of external inputs and if its trajectories are bounded by a function of the size of the input for all sufficiently large times. A small gain approach to robust eventtriggered control of nonlinear systems abstract. A small gain approach to stability analysis of hybrid systems dragan ne. We show that our technique is applicable for different formulations of iss property summation, maximum, semimaximum and discuss tightness of achieved smallgain theorems. 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. One of them is that both internal and external stability properties are discussed in a single framework, while only inputoutput. Read 2 answers by scientists to the question asked by huiyu jin on may 25, 2016.
For wellposedness we shall assume that p or k or both is strictly proper equal to zero at s 0. A lyapunovbased smallgain theorem for infinite networks. A smallgain approach to stability analysis of hybrid systems. T1 smallgain theorem, gain assignment and applications. A lyapunov formulation of the nonlinear smallgain theorem. Smallgain theory for stability and control of dynamical. Borrowing techniques in modern nonlinear control, especially sontags notion of inputtostate stability iss. Section iv presents the main theoretical contribution. This chapter presents a version of the small gain theorem for l2 gains, and then proceeds to introduce the general technique of working with quadratic.
We demonstrate that the smallgain analysis framework is very naturally and generally applicable in the context of hybrid systems, and thus has. This condition is a nonlinear version of the classical small gain condition see, for instance, 2. The memoryless version of the small gain theorem stated in one of the. Sufficient conditions to guarantee the condition 8 are given in io. N2 we introduce a concept of inputtooutput practical stability iops which is a natural generalization of inputtostate stability proposed by sontag. Optimal control huibert kwakernaak encyclopedia of life support systems eolss network and control theory. Smallgain theorem for iss systems and applications.
Chapter 3 small gain theorem and integral quadratic. A stepwise constructive control methodology is proposed on the basis of the nonlinear small gain theorem. We note that while the basic idea of the small gain stability analysis for hybrid systems was announced. This paper further develops a method, originally introduced by angeli and the second author, for proving global attractivity of steady states in.
309 1152 1025 145 1460 492 705 413 338 624 586 1028 1090 484 1062 662 672 730 304 811 1301 86 475 456 613 201 606 799 787 219 1134 888 831 752 721 398 264 619 1215 550