Showing papers on "Lyapunov equation published in 1974"
TL;DR: In this article, it is shown that a number of ambiguous points arising in transient stability analysis of power systems can be clarified if partial or output stability concepts are used instead of stato stability, which solves the question of the number of state variables and the validity of using positive semi-definite Lyapunov functions.
Abstract: In this paper it is shown that a number of ambiguous points arising in transient stability analysis of power systems can be clarified if partial or output stability concepts are used instead of stato stability. This solves the question of the number of state variables and the validity of using positive semi-definite Lyapunov functions. It is further discussed how a very general Lur'e-type Lyapunov function can be derived for the power system stability problem, which unifies and generalizes numerous Lyapunov functions available in the technical literature.
55 citations
TL;DR: In this paper, a fundamental inequality which is satisfied by the extremal eigenvalues of the matrices Q and P, provided A is a stability matrix, is established, which is extremely useful in determination of suboptimal controllers for the minimum time problem.
Abstract: Given the Lyapunov matrix equation A'Q + QA = -P a fundamental inequality which is satisfied by the extremal eigenvalues of the matrices Q and P , provided A is a stability matrix, is established. This result, besides being interesting from a theoretical standpoint, is extremely useful in the determination of suboptimal controllers for the minimum time problem [1].
42 citations
TL;DR: In this article, the state models of an n-machine power system for stability studies are obtained in the minimal state space based on the concept of the degree of a rational function matrix, and Lyapunov functions are then constructed for these models in a systematic manner using Anderson's theorem for multi-nonlinear systems.
Abstract: In this paper the state models of an n-machine power system for stability studies are obtained in the ‘ minimal state space ’ based on the concept of the degree of a rational function matrix. Lyapunov functions are then constructed for these models in a systematic manner using Anderson's theorem for multi-non-linear systems. These Lyapunov functions are different from those currently obtainable in the literature for power systems.
36 citations
TL;DR: In this paper, the identically zero function is a solution, i.e., f(t, 0) = 0 for all time t. We denote this special solution simply by 0.
25 citations
TL;DR: In this article, the authors considered the problem of continuous real valued functions on a non-trivial interval J of reals under the conditions (H 1 and H 2) and (H 2).
Abstract: In this paper we consider the equation (1.1) (r(t)y′(t))′+p(t)f(y(t)) = 0 under the conditions ((H 0): the real valued functions r, r′ and p are continuous on a non-trivial interval J of reals, and r(t)>0 for t∈J; and (H1):f:R→R is continuously differentiable and odd with f'(y)>0 for all real y. We also consider the equation (1.2) y″(t)+m(t)y′(t)+n(t)f(y(t)) = 0 under the conditions (H 1) and (H 2): the real valued functions m and n are continuous on a non-trivial interval J of reals.
16 citations
TL;DR: In this article, the authors derived a criterion for the asymptotic stability of a system governed by a non-autonomous difference equation of second order, which is pertinent to the design of a simple digital frequency-shift-keying (FSK) oscillator.
Abstract: We derive, by use of the Lyapunov theory for difference equations, a criterion for the asymptotic stability of a system governed by a nonautonomous difference equation of second order. The results are pertinent to the design of a simple digital frequency-shift-keying (FSK) oscillator. A feature of our analysis that is of major importance is the technique used in our application of the Lyapunov theory. It is shown that the technique yields the best stability criterion that can be obtained by the use of any positive definite quadratic form as a Lyapunov function. Thus we show how one might overcome one of the major obstacles to the practical application of the Lyapunov theory, i.e., the problem of choosing an appropriate (in some meaningful sense) Lyapunov function.
11 citations
TL;DR: In this article, improved signal synthesis techniques are developed for nonlinear, time-varying systems having unspecified plant parameters lying within known bounds, which satisfy a Lyapunov stability theorem which guarantees that they approach zero asymptotically.
Abstract: Improved signal synthesis techniques are developed for nonlinear, time-varying systems having unspecified plant parameters lying within known bounds. Output system errors satisfy a Lyapunov stability theorem which guarantees that they approach zero asymptotically. Improved results are derived first by using a quadratic-form Lyapunov function of system errors and then by using expanded Lyapunov functions involving derivatives of system errors to yield smoother input adaptive signals for a broad class of systems. An example is provided to illustrate the indicated design improvements.
9 citations
TL;DR: In this article, a modified method of differential moments for the generation of Lyapunov functions is presented, which is an improvement over the existing method that it fixes the maximum number of moment equations to be considered for a given system, to obtain one or more V-functions for the study of stability of non-linear autonomous systems.
Abstract: A modified method of differential moments for the generation of Lyapunov functions is presented. The modified formulation given in this paper is an improvement over the existing method that it fixes the maximum number of moment equations to be considered for o given system, to obtain one or more V-functions for the study of stability of non-linear autonomous systems. The flexibility in obtaining one or more V-functions is illustrated by examples of second- and third-order systems.
7 citations
TL;DR: In this article, several definitions of stability for a set relative to a difference equation are considered, and stability theorems are established by employing the idea of a positive semi-definite Lyapunov function which has a strongly negative definite difference.
Abstract: Several definitions of stability for a set relative to a difference equation are considered, and stability theorems are established. These theorems provide sufficient conditions for stability and are obtained by employing the idea of a positive semi-definite Lyapunov function which has a strongly negative definite difference. These results represent the discrete-time analogues of the recent work of Grimmer and Haddock (1973).
7 citations
TL;DR: In this article, the Stein-Pfeffer Theorem characterizes those where B is similar to A and H is positive definite, and several extensions of this theorem are proved.
Abstract: The Lyapunov transformation corresponding to the matrix is a linear transformation on the space of hermitian matrices of the form Given a positive stable , the Stein-Pfeffer Theorem characterizes those where B is similar to A and H is positive definite. Here several extensions of this theorem are proved
TL;DR: In this article, the authors studied the solvability of the Hermitian solution of the equation and obtained various criteria for the normality of A in terms of Hermitians.
Abstract: We study the solvability of the equation $AX + XA^ * = XA + A^ * X = I$ In particular, we obtain various criteria for the normality of A in terms of Hermitian solutions of the equation which satisfy additional conditions
TL;DR: In this article, the second time derivative of suitable Lyapunov functions is considered for ordinary differential equations and new results are obtained for the case of ODEs with respect to the second-order derivative.
Abstract: By considering the second time derivative of suitable Lyapunov functions we obtain new results in Lyapunov theory for ordinary differential equations.
TL;DR: In this paper, a modification of continuous-gradient techniques is presented for the identification of linear and a class of nonlinear systems, and the second method of Lyapunov is used to prove the asymptotic stability of the parameter difference between the system and a model.
Abstract: A modification of continuous-gradient techniques is presented for the identification of linear and a class of nonlinear systems. The second method of Lyapunov is used to prove the asymptotic stability of the parameter difference between the system and a model.
01 Apr 1974
TL;DR: In this paper, a method for the application of Lyapunov's second method to the synthesis of relay-control systems for plants that have undamped responses was presented for attitude control of a flexible space vehicle.
Abstract: A method is presented for the application of Lyapunov's second method to the synthesis of relay-control systems for plants that have undamped responses. A technique is presented for overcoming the problems associated with the solution of the Lyapunov matrix equation when the plant eigenvalues have zero real part. The method is applied to the attitude control of a flexible space vehicle.
TL;DR: In this paper, the relation between the first and second methods of Lyapunov is considered and a useful nth order generating technique for LyAPunov functions is developed which applies to linear and nonlinear systems.
Abstract: This paper considers the relation which exists between the first and second methods of Lyapunov. As a result of this study a useful nth order generating technique for Lyapunov functions is developed which applies to linear and nonlinear systems. The function obtained has a negative definite time derivative so that regions of asymptotic stability or instability can be determined. The function is derived from the equations of the system and the curl relations. The final result requires the calculation of single and inverse coefficients using recursive formulae. The basic method for generating Lyapunov functions makes use of the first method of Lyapunov. It is then extended to systems where the first method fails, to variations which have proved useful, and to more general and, hence, more complex forms. Two theorems are proved and examples are given in each section to illustrate the method presented.
TL;DR: In this paper, a generalized Lyapunov function for power systems is obtained based on a previous work of the author and it is shown that it includes the different results in the literature and gives larger regions of asymptotic stability.
Abstract: A generalized Lyapunov function for power systems is obtained based on a previous work of the author. It is shown that it includes the different results in the literature and gives larger regions of asymptotic stability.
TL;DR: In this paper, the stability of a quasi-linear system is investigated by assuming stability properties of associated linear systems, and the task of finding suitable Lyapunov functions may be avoided in certain cases.
Abstract: The stability of a quasi-linear system is investigated by assuming stability properties of associated linear systems. By this method, the task of finding suitable Lyapunov functions may be avoided in certain cases.
TL;DR: In this paper, the stability, boundedness and boundedness in the Lagrange sense of solutions of difference equations with discrete variables were investigated and the existence of Lyapunov functions for linear systems was shown.
Abstract: In the viewpoint of engineering, difference equations are often used to analyse sampled -data systems in which the stability and boundedness problems are considered to be very important The purpose of this paper is to give some results on the stability, boundedness and boundedness in the Lagrange sense of solutions of difference equations with discrete variables. A result shows the existence of Lyapunov functions for linear systems which will be often used to discuss the stability and boundedness problems for perturbed systems.
01 Jan 1974
TL;DR: The problems of model reduction, filter-observer and feedback control design in linear systems when treated from the parameter optimization point of view are shown to be mathematically equivalent under a wide variety of conditions.
Abstract: The problems of model reduction, filter-observer and feedback control design in linear systems when treated from the parameter optimization point of view are shown to be mathematically equivalent under a wide variety of conditions. The common basis is an optimization problem of a scalar criterion with a Lyapunov equation side constraint, called the canonical problem. Aside from the conceptual advantage of showing that a wide variety of design concepts and techniques are mathematically equivalent, the canonical formulation provides a compact set of necessary conditions for the solution from which known and new results can be easily derived. Finally it also permits development of one general purpose program to handle many different objectives, criteria, and types of disturbances.
01 Dec 1974
TL;DR: In this article, a method for genarating Lyapunov functions for time-invariant systems starting from a simple initiating function has been presented, which is simple, effective and offers a systematic procedure for constructing a large number of LyAPunov function for a given problem.
Abstract: A method for genarating Lyapunov functions for time-invariant systems starting from a simple initiating function has been presented in the paper. The method is simple, effective and offers a systematic procedure for constructing a large number of Lyapunov functions for a given problem.
TL;DR: In this article, sufficient conditions for the stability and asymptotic stability of the null solution of a neutral functional differential equation were established by employing the idea of a Lyapunov function which has a strongly negative definite derivative on an annulus.
Abstract: Sufficient conditions for the stability and asymptotic stability of the null solution of a neutral functional differential equation are established. These stability results are obtained by employing the idea of a Lyapunov function which has a strongly negative definite derivative on an annulus, and they represent an extension of the recent work of Haddock (1971).
TL;DR: For a large class of functional differential equations of the neutral type, sufficient conditions for set stability and uniform set stability were developed in this article, which arc obtained via Lyapunov function arguments and are extension of the recent work of Heinen and Dimino (1973).
Abstract: For a large class of functional differential equations of the neutral type, sufficient conditions for set stability and uniform set stability are developed. The results arc obtained via Lyapunov function arguments and are extension of the recent work of Heinen and Dimino (1973).