Showing papers on "Lyapunov function published in 1971"
TL;DR: In this paper, the problem of automatically constructing a quadratic Lyapunov function V = x'Ax for a high order non-linear system given by x@? = f(x), f(0) = 0, where f is a continuous function of x which guarantees uniqueness of solutions of the system is dealt with.
188 citations
TL;DR: In this article, it was shown that a state space realization of an input-output stable dynamical system is globally asymptotically stable in the sense of Lyapunov if it is uniformly observable and if every state is reachable.
Abstract: This paper discusses the relationship between properties of input-output descriptions and state space models for dynamical systems. It is shown that a state space realization of an input-output stable dynamical system is globally asymptotically stable in the sense of Lyapunov if it is uniformly observable and if every state is reachable. This result is proved in the context of abstract dynamical systems and leads to the equivalence of input-output stability and asymptotic stability for uniformly controllable and uniformly observable linear finite-dimensional systems. The generation of Lyapunov functions is subsequently considered, and variational techniques for the construction of Lyapunov functions are presented. Passivity and related energy concepts are particularly exploited in this context. These results yield the Lyapunov functions used in the proofs of the circle criterion and the Popov criterion as particular cases. The generality of the approach, however, makes these ideas applicable to much more ...
115 citations
TL;DR: A deterministic analysis of limit-cycle oscillations, which occur in fixed-point implementations of recursive digital filters due to roundoff and truncation quantization after multiplication, is performed in this paper.
Abstract: A deterministic analysis of limit-cycle oscillations, which occur in fixed-point implementations of recursive digital filters due to roundoff and truncation quantization after multiplication, is performed. Amplitude bounds, based upon a correlated (nonstochastic) signal approach using Lyapunov's direct method, as well as a general matrix formulation for zero-input limit cycles, are derived and tested for the two-pole filter. The limit cycles are represented on a successive-value phase-plane-type diagram from which certain symmetry properties are derived. The results are extended to include limit cycles under input-signal conditions.
107 citations
TL;DR: In this paper, it is shown that the classical direct methods, which are based on energy considerations, can be derived and generalized by means of Lyapunov's second method.
Abstract: This paper deals with recent advances in developing direct methods for studying the transient stability problem of single-machine and multimachine power systems. The paper starts out with the construction of the mathematical model that is usually employed in the analyis of power system transient stability. Computer simulation methods are then briefly discussed, and it is indicated why accurate direct methods for transient stability investigations would be most welcome. It is shown that the classical direct methods, which are based on energy considerations, can be derived and generalized by means of Lyapunov's second method. The main purpose of the paper is to give an exposition of the interesting results that have been obtained by applying Lyapunov's second method to the transient stability problem of single-machine and multimachine power systems. In the final portion of the paper some areas for further research are discussed.
103 citations
TL;DR: In this paper, the steady state optimal linear regulator with state and control dependent noise was analyzed in a manner similar to that developed by Wonham [1], and conditions under which an optimal control exists no matter how large the noise is.
Abstract: The steady state optimal linear regulator with state and control dependent noise is analyzed in a manner similar to that developed by Wonham [1]. By state dependent noise we mean Gaussian white noise with coefficient linear in the state variable, and similarly for control dependent noise. Using a Lyapunov criterion for the existence of stationary probability distributions due to Zakai, it is possible to treat equations leading to diffusion processes with degenerate differential generators. It is found that if the noise is sufficiently small, then an optimal control exists. Further analysis, again using Lyapunov methods, yields conditions under which an optimal control exists no matter how large the noise is.
66 citations
TL;DR: The direct method of Lyapunov applied to the problem of transient power system stability is developed to permit the practical study of n-machine systems and several ideas for a further generalization and optimization of the method and a possibility for the design and control of power systems are proposed.
Abstract: The direct method of Lyapunov applied to the problem of transient power system stability is developed to permit the practical study of n-machine systems. A simple physical interpretation and a proof are given for two Lyapunov functions. A new and better Lyapunov function is defined, and several possibilities for its evaluation are studied. Through an analysis of the singular points of the system the causes of the conservativeness of the results are established, and a further improvement on the Lyapunov function is made. A general structure for a computer program is proposed, and results of an example with a 10-machine system solved by the conventional and the new methods are presented; the computing time by the Lyapunov method was about 40-percent shorter. Finally, several ideas for a further generalization and optimization of the method and a possibility for the design and control of power systems are proposed.
45 citations
01 Aug 1971
TL;DR: In this article, the transient stability of a synchronous machine swinging against an infinite busbar has been investigated using a Lyapunov function construction method, where the transient saliency and the variable-field flux linkage are taken into account, but changes in prime-mover input have been neglected.
Abstract: In this paper, Zubov's method of construction of Lyapunov functions is applied to the transient-stability problem of a synchronous machine swinging against an infinite busbar. In the machine model, the transient saliency and the variable-field flux linkage are taken into account, but changes in prime-mover input have been neglected. Cross-sections of the stability surface for various principal planes are shown and compared with the actual stability surfaces as obtained by numerical integration. It is shown that the application of Zubov's method results in considerable improvement of stability-boundary estimates over those given by the ‘quadratic-plus-an-integral-of-the-nonlinearity’ type of Lyapunov functions which have previously been used for transient stability studies. Zubov's method generally requires machine computations. A section of the paper is, therefore, devoted to providing guidelines for digital-computer implementation of this method.
35 citations
01 Mar 1971
TL;DR: In this paper, a new situation has been considered where vector Lyapunov functions play a further useful role and a new type of stability, namely, strict partial stability has been defined.
Abstract: It has been proved that the use of a vector Lyapunov function is more advantageous in certain situations rather than a scalar function. Moreover, each function needs to satisfy less rigid requirements. In this paper a new situation has been considered where vector Lyapunov functions play a further useful role. For this purpose, a new type of stability, namely, strict partial stability has been defined. The principal tool employed is the second method of Lyapunov and a comparision theorem of a more general type.
29 citations
TL;DR: In this paper, the authors considered non-linear sampled-data control systems consisting of a continuous linear part and a nonlinear pulse modulator (PM), where the PM modulates a sequence of rectangular pulses by sign, frequency and duration, as a function of a linear combination of discrete values of the system coordinates.
28 citations
01 Jan 1971
TL;DR: In this paper, it is shown that the classical direct methods, which are based on energy considerations, can be derived and generalized by means of Lyapunov's second method.
Abstract: Absfruct-This paper deals with recent advances in developing direct methods for studying the transient stability problem of singlemachine and multimachine power systems. The paper starts out with the construction of the mathematical model that is usually employed in the analyis of power system transient stability. Computer simulation methods are then briefly discussed, and it is indicated why accurate direct methods for transient stability investigations would be most welcome. It is shown that the classical direct methods, which are based on energy considerations, can be derived and generalized by means of Lyapunov’s second method. The main purpose of the paper is to give an exposition of the interesting results that have been obtained by applying Lyapunov’s second method to the transient stability problem of single-machine and multimachine power systems. In the final portion of the paper some areas for further research are discussed. I. INTRODUCTION T HE PROBLEM of transient stability of power systems becomes increasingly important as t,he size of the interconnected areas becomes very large. Indeed, the tendency of a system to lose synchronism and disintegrate, and the resulting possibility of oscillations in the power transfer between interconnected areas is much more prevalent for large systems than it is for relatively small isolated groups. Optimum control and stability investigations are presently used to a large extent in the analysis and design of power systems. A comp1et.e survey of the application of optimum control to power systems wa.s presented at the 1968 Joint Automa.tic Control Conference [l]. It is the aim of this paper to give an exposition of recent results and mbthods concerning the transient stability problem of power systems. The writing of t.his paper is motivated on one hand by a desire t.0 expose this problem to the cont,rol audience at large, and on the other hand by the timeliness of illustrating an area of application for sophisticated stability analysis techniques to a practica.1 real-world problem. This is done in the hope that such an exposition could. lead to a contribution in bridging the well-publicized gap between t.heory a.nd practice. r\’ot,e that, even in the optimization of t,he steadystat.e operation of power systems [I], stability crit,eria are important, since they const,itute some of the opthizat,ion constraints. In recent yem a large amount of papers on stability has been published in t,he automatic control literature. The applicat,ion of these ideas and methods to actual
21 citations
01 Jan 1971
TL;DR: In this paper, it was shown that the problem of topological stability is not a trivial problem in the sense of [1] and does not permit an algebraic solution, even if cases of codimensionality infinity (and even codimensionsality 103) are neglected.
Abstract: Further on, it is proved that the problem of stability, and therefore the problem of topological classification as well, is not a trivial problem in the sense of [1]: i.e., it does not permit an algebraic solution, even if cases of codimensionality infinity (and even codimensionality 103) are neglected.
TL;DR: In this paper, it was shown that non-adiabatic perturbations can be handled as easily as adiabatic ones; in fact there is no need to differentiate between the two and Lyapunov's method provides an elegant way to unify them.
TL;DR: In this paper, it was shown that the oscillatory solution is Lyapunov stable under small perturbations in the coefficients of the equation and that whenever the coefficients are quasi-periodic and analytic, the almost periodic oscillation is in the same class.
Abstract: This paper studies forced almost periodic oscillations in a nonlinear system of two Volterra integral equations. It improves certain results in an earlier paper on the same topic in two ways. First it is shown that the oscillatory solution is Lyapunov stable under small perturbations in the coefficients of the equation. Secondly, it is shown that whenever the coefficients are quasi-periodic and analytic, the almost periodic oscillation is in the same class.
01 Nov 1971
TL;DR: In this article, the authors present a more accurate approach to the transient stability problem of multimachine power systems by Lyapunov's direct method, which is more rigorous than those already taken into account in that the machines' inertia coefficients are no longer considered as constant but varying with angular speed.
Abstract: The paper is a contribution to the transient-stability study of multimachine power systems by Lyapunov's direct method. The mathematical model used is more rigorous than those already taken into account in that the machines' inertia coefficients are no longer considered as constant but varying with angular speed. This constitutes a more accurate approach to the transient-stability problem. A 9-machine realistic power system illustrates the proposed method, and permits comparisons and conclusions to be made.
TL;DR: An improved stability criterion for the damped Mathieu equation using periodic Lyapunov functions was derived in this paper, where the stability criterion was based on the stability of the periodic Lipschitz constant.
Abstract: An improved stability criterion is derived for the damped Mathieu equation using periodic Lyapunov functions
01 Oct 1971
TL;DR: In this article, the main aim of the report is to make transparent most of the interesting ideas involved in the design of adaptive systems using the direct method of Lyapunov, by considering primarily systems described by first-order differential equations.
Abstract: : The main aim of the report is to make transparent most of the interesting ideas involved in the design of adaptive systems using the direct method of Lyapunov, by considering primarily systems described by first-order differential equations. A study is made of three adaptive designs for such systems. (Author)
01 Jul 1971
TL;DR: Some global results are given in terms of arbitrary sets which can be employed as tools in dealing with various problems of stability and boundedness.
Abstract: 1. The proofs of many results in the theory of stability and boundedness basically depend on dividing the vicinity of some kind of invariant set (or other convenient set) into suitable subsets and then trying either to prove that solutions cannot leave such sets or to estimate the escape time. This observation makes it possible to give some global results in terms of arbitrary sets which can be employed as tools in dealing with various problems of stability and boundedness. In applications, these tools enlarge the class of useful Lyapunov like functions and also offer more flexibility.
TL;DR: In this article, an analogous 2n -dimensional Lyapunov matrix equation was derived for second-order n-dimensional systems, which is shown to be more readily solvable than the equivalent 2n-dimensional LyAPunov equation.
Abstract: Equations analogous to the Lyapunov matrix equation are derived for second-order n -dimensional systems. These are shown to be more readily solvable than the equivalent 2n -dimensional Lyapunov matrix equation.
TL;DR: In this article, a procedure for the design of stable adaptive model reference control systems is described, which combines the Lyapunov method with the use of sensitivity coefficients, and the resulting system is shown to be asymptotically stable in the sense of "eventual stability" as used by La Salle and Rath (1963).
Abstract: A procedure is described for the design of stable adaptive model reference control systems. This procedure combines the Lyapunov method with the use of sensitivity coefficients. A basic algorithm is first obtained and the resulting system is shown to be asymptotically stable in the sense of ‘eventual stability’ as used by La Salle and Rath (1963). Modifications of this algorithm are then proposed which are easier to implement. One of these uses sensitivity coefficients and is studied in some detail.
TL;DR: In this article, a non-linear discrete equivalence of a class of pulse-frequency modulation feedback systems is developed. But the analysis of feedback systems with pulsefrequency modulation has not been investigated using the continuous-time domain approach.
Abstract: In the past few years analysis of feedback systems with pulse-frequency modulation has been investigated using the continuous-time domain approach. This paper develops a non-linear discrete equivalence of a class of pulse-frequency modulation feedback systems. Using the non-linear discrete equivalence, stability of feedback systems with pulse-frequency modulation is analysed through the second method of Lyapunov.
TL;DR: In this paper, it is shown that if the mistuning is not too big, specifically if |mistuning| < (half-power IF bandwidth), then solutions which are bounded away from zero amplitude approach the natural equilibrium point.
Abstract: It is known that frequency feedback demodulators can show instability in their response to step changes (mistuning) in input frequency. This work reports on some mathematical analyses of this phenomenon as described by differential equations arising from simple IF and feedback filters in the demodulator. These equations are studied for local and global stability by geometric or phase-plane analysis, by means of Lyapunov functions, and by the topological Poincare-Bendixson methods. A typical result is for the case of no feedback filter and one-pole baseband analog of the IF filter, and states in physical terms that if the mistuning is not too big, specifically if |mistuning| < (half-power IF bandwidth)(1 + feedback gain) then solutions which are bounded away from zero amplitude approach the natural equilibrium point. Examples are given in which a sufficiently large mistuning makes the equilibrium point unstable.
TL;DR: In this article, the Lyapunov matrix equation is shown to be of importance in generating the optimal gains for distributed-parameter feedback control systems with controllers based on the values of the state variables at discrete points.
Abstract: The Lyapunov matrix equation is shown to be of importance in generating the optimal gains for distributed-parameter feedback control systems with controllers based on the values of the state variables at discrete points. It is shown that the Lyapunov matrix equation can be considerably reduced for such systems. In a typical example, the number of unknowns is reduced from 1225 to 49.
TL;DR: In this article, a Krasovskii-type Lyapunov function is applied to a system represented by 2nd-order nonlinear state equations, and the resulting computational advantage can then be exploited in determining regions of asymptotic stability.
Abstract: A Krasovskii-type Lyapunov function is applied to a system represented by 2nd-order nonlinear state equations. The resulting computational advantage can then be exploited in determining regions of asymptotic stability.
TL;DR: Using the second method of Lyapunov, a new procedure for performing sensitivity calculations in stability analysis has been developed that is simple, and lends itself readily to programming on a digital computer.
Abstract: Using the second method of Lyapunov, a new procedure for performing sensitivity calculations in stability analysis has been developed. The procedure is simple, and lends itself readily to programming on a digital computer.
TL;DR: In this paper, the Lyapunov operator corresponding to a stable system is described in terms of its mapping of the set of positive definite matrices, and its image is shown to have geometrical descriptions which simplify the analysis.
Abstract: In this paper, the Lyapunov operator corresponding to a stable system is described in terms of its mapping of the set of positive definite matrices. This set and its image are shown to have geometrical descriptions which simplify the analysis. This geometrical approach also yields a method for finding the ‘best’ Lyapunov function of the quadratic type for a system with a variable parameter
TL;DR: In this article, the stability of a distributed nuclear core system with nonlinear temperature feedbacks is analyzed, and an extension of Lyapunov's theory is employed to obtain stability criteria for this coupled system.
Abstract: In the study of reactor system stability, the past work is mainly concentrated in two areas: the study of reactor core stability using either the point model or distributed model, and the study of plant dynamics using a lumped parameter model with time delay. The former study assumed no interaction between the core dynamics and the rest of the components in the system whereas the latter neglected the distributed effect, thus some salient features in the system may be lost. The work presented in this paper analyzes the stability of a distributed reactor system which consists of the core dynamics with nonlinear temperature feedbacks, the external primary heat exchanger, and the coolant loop interconnecting these two major components. The coupled set of nonlinear partial differential equations is treated directly without recourse to the usual lumped parameter approximation. An extension of Lyapunov's theory, which has been successfully used in the study of distributed core dynamics, is employed to obtain stability criteria for this coupled system.
TL;DR: In this article, a new approach involving the use of two signum functions together with a suitably chosen Lyapunov function is employed to investigate the boundedness property of solutions of two special cases of (1.3).
Abstract: In this paper a new approach involving the use of two signum functions together with a suitably chosen Lyapunov function is employed to investigate the boundedness property of solutions of two special cases of(1.3). This approach makes for considerable reduction in the conditions imposed on f, g in an earlier paper[1].