Showing papers on "Lyapunov function published in 1976"

The stability of nonlinear dissipative systems

Abstract: This short paper presents a technique for generating Lyapunov functions for a broad class of nonlinear systems represented by state equations. The system, for which a Lyapunov function is required, is assumed to have a property called dissipativeness. Roughly speaking, this means that the system absorbs more energy from the external world than it supplies. Different types of dissipativeness can be considered depending on how one chooses to define "power input." Dissipativeness is shown to be characterized by the existence of a computable function which can be interpreted as the "stored energy" of the system. Under certain conditions, this energy function is a Lyapunov function which establishes stability, and in some cases asymptotic stability, of the isolated system.

Stabilizing feedback control for dynamical systems with bounded uncertainty

Abstract: We consider a class of dynamical systems subject to parameter and input uncertainty whose values range in a given compact set. Utilizing the philosophy of "worst case" design, we deduce a feedback control that assures uniform asymptotic (Lyapunov) stability of the origin under all admissible uncertainties.

General aggregation of large-scale systems by vector Lyapunov functions and vector norms

Abstract: A new approach is proposed to an advantageous application of both vector Lyapunov functions and vector norms to aggregation of large-scale systems. As a result, the test of the stability property of a large-scale system is achieved without knowledge of stability properties of its subsystems. Furthermore, new completely relaxed stability conditions are established for non-stationary non-linear dynamic large-scale systems, which reduce the stability test to verification of either the Popov criterion or stability of constant matrices of order reduced to the number equal to, or possibly less than, the number of the subsystems. As by-products of the paper, linear and Aiserman conjectures are proved for classes of systems on arbitrary hierarchical level.

On the uniform asymptotic stability of certain linear nonautonomous differential equations

Abstract: : The ordinary differential equation dx/dt = -P(t)x where P(t) is symmetric positive semi-definite time-varying matrix arises often in mathematical control theory. In this paper the authors consider the stability properties (in the sense of Lyapunov) of the equilibrium state x(t) identically equal to 0. It is a relatively trivial exercise to show that the origin is stable but (uniform) asymptotic stability does not generally hold unless P(t) is positive definite. The semi-definite case arises much more frequently in practice than the definite one and the main effort in this paper is directed towards finding conditions implying uniform asymptotic stability in such a case.

Cellular systems—II. stability of compartmental systems

Abstract: In this work, by employing the concept of vector Lyapunov functions, and the theory of differential inequalities, the stability analysis of compartmental systems is initiated in a systematic and unified way. Furthermore, an attempt is made to formulate and partially resolve the “complexity vs. stability” problem in open compartmental systems. The recent stability results obtained for open chemical systems are applied to open compartmental systems in a natural way. As a byproduct of our analysis, we obtain an estimate for washout functions. Finally, it has been demonstrated that the stability analysis of compartmental systems provides a common conceptual framework for innumerable, apparently unrelated systems in biological, medical, and physical sciences.

Cellular systems—I. stability of chemical systems

Abstract: In this work, we attempt to formulate and partially resolve the “complexity vs. stability” problem in open chemical systems, in the framework of Lyapunov's second method. Sufficient conditions are given for stability of the steady state of a chemical system under structural perturbations caused by interactions among the chemical species in the system. As a byproduct of this analysis, we will show important properties of the self-inhibitory chemical systems, and establish the tolerance of the stability to a broad class of perturbations. Finally, biological open systems are exhibited in order to show the scope of our stability analysis.

On Perturbing Lyapunov Functions

Abstract: A new idea of perturbing Lyapunov functions is presented which permits one to discuss nonuniform properties of solutions of differential equations under weaker assumptions

Stable Identification Schemes

Abstract: Publisher Summary System characterization and system identification play a central role in systems theory and the past two decades have witnessed considerable research activity in these areas. System characterization is concerned with setting up mathematical models for different classes of systems, and the aim of system identification is the determination of the characteristics of a specific model, which approximates the given system in some sense. At the present time, many formulations of the identification problem exist and numerous techniques have also been developed to estimate system parameters. This chapter describes a general identification procedure using a model reference approach developed in recent years. It deals specifically with model reference adaptive systems using Lyapunov's direct method. In this approach, a model of the plant to be identified is assumed to be available, and both model and plant have the same input u(t). The model parameters are continuously adjusted as functions of the error between the plant and model output and other accessible signals in the system. The basic philosophy of the approach is to determine the adaptive laws for adjusting the model parameters so that the stability of the overall system is assured from the outset.

Fast determination of stability regions for online transient power-system studies

Abstract: The paper deals with the rapid determination of the boundary of stability regions which has been the only time consuming step in transient stability studies by Lyapunov's criterion. Three realistic examples, namely a 5-, a 9- and a 15-machine system illustrate and confirm the proposed procedure.

The application of Lyapunov methods to large power systems using decomposition and aggregation techniques

Abstract: This paper deals with the application of Lyapunov methods to the problem of transient stability of large power systems by decomposing and aggregating the complete system model. This approach decreases the difficulty of calculating the unstable equilibrium points of the system by reducing its dimension. A procedure for calculating the critical switching time based on this technique is described. The results of the validity of the method, as applied to two different systems, are reported.

On vector Lyapunov functions for stochastic dynamical systems

Abstract: Vector Lyapunov functions are used in the stability analysis of large-scale stochastic systems described by Ito differential equations (with stochastic disturbances in the subsystems and in the interconnecting structure). Sufficient conditions for asymptotic stability and exponential stability with probability 1 and in probability are established, in all cases the objective is the same: to analyze large-scale systems in terms of their lower order (and simpler) subsystems and in terms of their interconnecting structure. Use of the method presented makes it often possible to circumvent difficulties usually encountered when the Lyapunov method is applied to high-dimensional systems and to systems with complicated interconnecting structure. In order to demonstrate the usefulness of the present approach, a specific example is considered.

Stability conditions and estimates of the stability region of complex systems

Abstract: In this paper the stability of complex systems composed of interconnected subsystems is investigated. The results are derived by studying a comparison system, the dimension of which is equal to the number of subsystems. Special attention is given to the estimation of the stability region of the complex system. The stability analysis of the comparison system is carried out without using a Lyapunov function. This approach permits an extension of the results of Weissenberger (1973) who treated an analogous problem.

Biophysics of communities of living organisms

Abstract: This article reviews theoretical and experimental studies on fundamental problems of ecology. Theoretical study of the dynamics of ecosystems was initiated by the studies of V. Volterra, A. Lotka, A. N. Kolmogorov, and A. A. Lyapunov. These have been primarily models of point systems of the predatorprey type and models of competitive interrelationships between populations. The conclusions of the theory are illustrated by the experimental data of G. F. Gause, Nicholson, et al. It is shown how one might introduce the Lagrangian and Hamiltonian formalism to describe multispecies ecosystems. Considerable space is given to describing closed ecosystems whose development is limited by a biogenic element that constitutes the ecological minimum. It is shown that auto-oscillations can arise in such systems, while the stability of such ecosystems is determined by the diversity of species and some other factors, in particular, the specialization of species. Since real ecosystems consist of a large number of species, it becomes necessary to apply the methods of statistical mechanics to study these systems. Yet one cannot use the Gibbs method, owing to lack of ergodicity. The possibility is discussed of selecting macroparameters for describing ecosystems and constructing a Fokker-Planck equation for the populations.

Extended Lyapunov stability criterion using a nonlinear algebraic relation with application to adaptive control

Abstract: A new nonlinear stability criterion is developed by use of a class of Lyapunov functionals for model-reference adaptive systems (MRAS). Results are compared with traditional results, and a comparative design technique is used to illustrate its function in improving the transient response of an MRAS controller. For a particular system structure and class of input signals, the new stability criterion is shown to include traditional sufficiency stability conditions as a special case. An example is cited to illustrate the use of the nonlinear criterion and its definite advantages in helping improve the adaptive error transient response of a system. Analysis of results is effected by use of a linearization technique on the resulting adaptive equations.

Numerical analysis of the Lyapunov equation with application to interconnected power systems

Abstract: The Lyapunov equation is fundamental to control theory. A number of numerical solution methods are compared, with special emphasis placed on applicability to large scale system matrices of a general sparse structure. An iterative decoupling algorithm is developed to exploit this special form, and a computer program that realizes this method is reported. An introduction to linear error analysis is included, and a number of results are developed and extended to several Lyapunov equation solution techniques. Thesis Supervisor: Nils R. Sandell, Jr. Title: Assistant Professor of Electrical Engineering and Computer Science.

Transient asymptotic stability of power systems as established with Lyapunov functions

Abstract: The use of Lyapunov functions in Lyapunov's Direct Method for analyzing power system transient asymptotic stability is studied. A previous study by other investigators claimed to establish the asymptotic stability of a multimachine system with damping included. In the present paper it is demonstrated that the Lyapunov function used in the earlier study is not sufficient for this purpose. In addition the weakness of Lyapunov-type functions obtained from energy considerations in the study of asymptotic stability is discussed. Then, a method is presented which explains how to construct a satisfactory Lyapunov function for study of asymptotic stability for electric power systems. This method uses linear algebra to convert n second order differential equations to a set of 2n first order equations so that a certain vector product may be used as the Lyapunov function. This Lyapunov function is used to obtain a region of asymptotic stability for such a system. Practical uses of these results in generating system design and expansion are discussed. Finally, a numerical example illustrating the technique is presented.

Process design in a dynamic environment: Part I. A decomposition technique to study the stability of chemical engineering systems

Abstract: The stability of various chemical engineering systems is studied through the use of vector Lyapunov functions. A decomposition technique is employed which reduces the problem to small, independent, free subsystems that can be analyzed through Lyapunov's second method. The stability characteristics of the overall system are then related to the behavior of a linear system which is easily analyzed.

An iterative decoupling solution method for large scale Lyapunov equations

Abstract: A great deal of attention has been given to the numerical solution of the Lyapunov equation. A useful classification of the variety of solution techniques are the groupings of direct, transformation, and iterative methods. The paper summarizes those methods that are at least partly favorable numerically, giving special attention to two criteria: exploitation of a general sparse system matrix structure and efficiency in resolving the governing linear matrix equation for different matrices. An iterative decoupling solution method is proposed as a promising approach for solving large-scale Lyapunov equation when the system matrix exhibits a general sparse structure. A Fortran computer program that realizes the iterative decoupling algorithm is also discussed.

On ranges of Lyapunov transformations IV

Abstract: This result answers the question to what extent does 2?A{PSD{n)) characterize A. The proof in [4] is by induction on n, the order of A, and involves several tedious computations. In Section 2 we give a simpler proof of this result based on a theorem by Schneider [7] which characterizes all linear transformations on the real space #Cn that map PSD(n) on to itself. It is not difficult to see that A and B satisfy (1) or (2) if and only if

On Liapunov Stability of Stiff Non-Linear Multistep Difference Equations,

Abstract: : The concepts of G-stability and (G,mu)-stability recently introduced by Dahlquist are useful for discussing Liapunov stability of solutions to systems of non-linear difference equations, generated by applying linear multistep formulas to monotone, dissipative, arbitrarily stiff systems of non-linear differential equations. In this paper, the theory of G-stability and (G,mu)-stability is reviewed and a construction is proposed which facilitates the finding of a quadratic Liapunov function. By this construction it is proved that, for the four-parameter family of all three-step formulas which are second-order accurate, A-stability is necessary and sufficient for G-stability. Some results on (G,mu)-stability are also obtained by this construction.

Stability of Couette flow between two rotating cylinders

Abstract: We investigate the stability in the large of Couette flow between two cylinders rotating in the same direction. For the case of infinitesimally small perturbations, a sufficient condition for stability of the Couette flow (the Synge condition) was obtained in [1, 2]. In [3] for an investigation of the stability in the nonlinear case, to this condition we must add certain constraints on the initial energy and the angular velocities. In the proposed study, using the second method of Lyapunov, sufficient conditions for stability in the large are obtained, which differ little from the Synge condition. In this case these conditions approach the Synge condition as the distance between the cylinders is decreased.

Stable model reference adaptive systems

Abstract: A method is presented for the design of multivariable model reference adaptive control systems by Lyapunov's second method. The method can be applied to a class of systems in which the misaligned elements of the plant matrices cannot be compensated independently.