Showing papers on "Lyapunov function published in 1968"
TL;DR: In this paper, the problem of designing an attitude regulator for an arbitrary rigid body was considered, and the design criterion was that the system should have only one equilibrium point, and this point should be asymptotically stable for arbitrary initial conditions.
Abstract: The problem considered in this paper is that of designing an attitude regulator for an arbitrary rigid body. The design criterion is that, in the presence of no disturbance torques on the body, the system should have only one equilibrium point, and this point should be asymptotically stable for arbitrary initial conditions. By use of Lyapunov techniques and an appropriate choice of state variables, it is shown that this criterion can be satisfied by employing a linear feedback law with constant coefficients.
119 citations
78 citations
51 citations
TL;DR: In this article, the transient stability of an a.c. generator is analyzed by utilizing Lyapunov's direct method, which includes the effects of flux docay and simplified representations of speed governer and voltage regulator.
Abstract: The transient stability of an a.c. generator is analysed by utilizing Lyapunov's direct method. Lyapunov functions are developed, which include the effects of flux docay and simplified representations of speed governer and voltage regulator. It is shown that Lyapunov's method can be utilized to estimate reasonably accurately the critical fault clearing time with considerably fewer computations and in much shorter time than by conventional techniques. Furthermore, it is shown how a Lyapunov function can indicate the relative effects of various factors on the stability property of a generator. The Lyapunov functions have been developed essentially by trials based on physical considerations. Various existing methods of constructing the Lyapunov functions were also considered but were found to be of no help in the present study.
40 citations
TL;DR: In this paper, the existence of a finite invariant measure for a Feller process on a a-compact metric (state) space has been studied and necessary and sufficient conditions have been formulated.
Abstract: In the study of dynamical systems perturbed by noise, it is important to know whether the stochastic process of interest has a stationary distribution. Four necessary and sufficient conditions are formulated for the existence of a finite invariant measure for a Feller process on a a-compact metric (state) space. These conditions link together stability notions from several fields. The first uses a Lyapunov function reminiscent of Lagrange stability in differential equations; the second depends on Prokhorov's condition for sequential compactness of measures; the third is a recurrence condition on the ergodic averages of the transition operator; and the fourth is analogous to a condition of Ulam and Oxtoby for the nonstochastic case.
38 citations
01 Jun 1968
TL;DR: The ‘M.I.T. rule’ and two Lyapunov synthesis methods of design are critically reviewed and the input modification scheme of Grayson and Parks' method for the determination of parameter-adjusting feedback loops is reviewed.
Abstract: The ‘M.I.T. rule’ and two Lyapunov synthesis methods of design are critically reviewed. The latter are the input modification scheme of Grayson and Parks' method for the determination of parameter-adjusting feedback loops. For large inputs such methods are unsatisfactory: this is demonstrated by root locus plots for the step response of a second-order system with adaptive gain.A Lyapunov synthesis procedure is proposed which is a composite solution. The feedback loops of Parks are employed but an input modification signal is superimposed. This can be conveniently chosen from existing signals to add a fourth order negative definite term to V, thus promoting a greater degree of stability for large inputs. Results from an analogue computer study are included. The method is general in application.
29 citations
01 Sep 1968
TL;DR: In this paper, a Lyapunov function for transient power-system stability was determined for a synchronous machine connected to an infinite bus, which includes pole saliency and damping torques.
Abstract: A Lyapunov function is determined for the problem of transient power-system stability; only the stability of a synchronous machine connected to an infinite bus is considered. It is shown that the obtained Lyapunov function includes governor action, pole saliency and damping torques, and is hence an extension of earlier results.
27 citations
25 Jan 1968
TL;DR: In this paper, the authors present an account of general problems of dynamics and stability theory of a body with liquid connected with the systematic application of analytic mechanics methods and, especially, the second method of Lyapunov.
Abstract: : The book consists of two independent parts The first part contains an account of general problems of dynamics and stability theory of a body with liquid connected with the systematic application of analytic mechanics methods and, especially, the second method of Lyapunov In this case the body and the liquid in its cavity are considered as one mechanical system, and nonlinear equations of motion are investigated The second part is dedicated to the theory of small oscillations of liquid and of the body with liquid In contrast to the first part here only linear problems are discussed The central problem investigated in the second part appears to be a traditional problem of small oscillations - study of the structure of a spectrum
25 citations
TL;DR: A theoretical basis for identifying stationary or slowly varying parameters by means of a gradient algorithm and it is shown that the Lyapunov-optimum gain matrix is also an error-correction gain matrix.
21 citations
TL;DR: In this article, the authors provided sufficient conditions on the absolute value of the excitation applied to the column problem in order to insure mean-square global stability, which is applicable for any general continuous random process.
Abstract: This paper deals with Lyapunov-type analysis of the dynamic stability of a linear elastic column subjected to an axial stochastic load. Within the past decade, the interest in stability of stochastic systems of differen tial equations has rapidly increased, as indicated by the large number of papers on the subject. The intent of this paper is to provide sufficient conditions on the absolute value of the excitation applied to the column problem in order to insure mean-square global stability. Although this problem has been investigated by several authors by considering the concept of almost sure stability, the bounds provided are related to only the mean value of the absolute value of the excitation, or, at best, the standard deviation of a Gaussian process with zero mean. In the present paper, we are able to relate the required bounds, for mean-square global stability, to the mean and variances of the excitation. The bounds are applicable for any general continuous random process. The sufficient bounds are obtained by using a Lyapunov type of approach, introduced by Bertram and Sarachik and extended here for stability in the mean-square sense.
20 citations
TL;DR: The result concerning a significant new class of nonlinearities as well as the extension of the third previously known time-invariant result to cover time-varying systems are original with this paper.
Abstract: For a quite general class of dynamic systems having a single memoryless time-varying nonlinearity in the feedback path some frequency domain stability criteria are developed using Lyapunov's second method. Four classes of nonlinearities are considered, and it is seen that as the behaviour of the nonlinearity is restricted, the stability conditions are relaxed. For the first three classes of non-linearities, the results for time invariant systems are well known, and for two of the classes the result has previously been extended to apply to time-varying situations. The result concerning a significant new class of nonlinearities as well as the extension of the third previously known time-invariant result to cover time-varying systems are original with this paper.
01 Aug 1968
TL;DR: In this article, a generalization of the direct method of Lyapunov is employed to determine the stability of neuristor waveforms, and the validity of the linear approximation for distributed systems is discussed.
Abstract: A generalization of the direct method of Lyapunov is employed to determine the stability of neuristor waveforms. The validity of the linear approximation for distributed systems is discussed. It is shown that a neuristor waveform on a RC ± G transmission line is stable if its derivative represents the minimum eigenvalue solution of the linearized perturbation equation.
TL;DR: It is shown in this paper how the known function may be used to construct new systems, based on the original one, which are also asymptotically stable, and how the parametric perturbation and the synthesis procedures are generalised to include autonomous and non-autonomous non-linear systems.
01 Jan 1968
TL;DR: In this paper, the buckling stability analysis of long cylindrical shells with random imperfections subjected to axial load is treated using two different approaches: Lyapunov method and truncated hierarchy.
Abstract: The buckling stability analysis of long cylindrical shells with random imperfections subjected to axial load is treated using two different approaches. The first study is based on a Lyapunov method which enables one to establish sufficient conditions for buckling stability of a long cylindrical shell with axisymmetric random imperfections. A perturbed system of equations in the neighborhood of the prebuckling solution is investigated. By reducing the problem to a system of integral equations, it is observed that the stability boundary value problem of a long shell is similar to that of a dynamical system with random parametric excitations.
Initial imperfections were assumed to have Gaussian distribution and an exponential cosine correlation function. The critical load was obtained as a function of the root mean square of the imperfections. Results obtained are qualitatively similar to those of Koiter for a periodic imperfection (Ref. 1).
The second part is based on the approximate method of truncated hierarchy. The prebuckling state of equilibrium for asymmetric imperfections is found by a successive substitution technique. A homogeneous variational system of equations is set up in order to examine the existence of bifurcation in the neighborhood of the equilibrium state. These last equations involve random parametric terms. The truncated hierarchy method is applied and characteristic equations are obtained. Various exponential cosine correlation functions associated with asymmetric imperfections are examined numerically. Qualitatively the results obtained are as anticipated.
TL;DR: In this article, a frequency-domain criterion for the stability of systems containing many non-linearities is derived in terms of the positive realness of the product of a diagonal multiplier matrix and the transfer function matrix of the linear part.
Abstract: A frequency-domain criterion for the asymptotic stability-in-the-large of systems containing many non-linearities is derived in terms of the positive realness of the product of a diagonal multiplier matrix and the transfer function matrix of the linear part Several sub-classes of monotonically increasing non-linear functions are considered and it is shown that the elements of the multiplier matrix can be permitted to have complex conjugate poles and zeros whon the non-linearities possess at least a restricted odd asymmetry A Lyapunov function of the quadratic plus multi integral type and a matrix version of the Meyer—Kalman-Yakubovich lemma are used in deriving the results
TL;DR: In this paper, it was shown that Lyapunov functions similar to the Ralston-Parks and Kalman-Bertram forms (which were employed to derive the Routh-Hurwitz conditions through Lynpunov theory) can be formed from the time-varying coefficients of time-changing differential equations for the study of stability.
Abstract: It is shown that Lyapunov functions similar to the Ralston-Parks and Kalman-Bertram forms (which wore employed to derive the Routh-Hurwitz conditions through Lynpunov theory) can be formed from the time-varying coefficients of time-varying differential equations for the study of stability. These Lyapunov functions are used to conclude asymptotic stability of solutions of differential equations whoso time-varying coefficients approach constant values as time tends to infinity.
01 Jun 1968
TL;DR: Liapunov stability theory applied to class of partial differential equations, and generation of contraction groups related to equivalent norms is discussed in this paper, where the authors apply it to a class of PDEs.
Abstract: Liapunov stability theory applied to class of partial differential equations, and generation of contraction groups related to equivalent norms
01 Sep 1968
TL;DR: In this article, Lyapunov's second method gives a sufficient condition for the asymptotic stability of a dynamic steady-state waveform on a nonlinear active transmission line.
Abstract: Lyapunov's second method gives a sufficient condition for the asymptotic stability of a dynamic steady-state waveform on a nonlinear active transmission line. The result is consistent with that of an eigenfunction expansion technique.
TL;DR: In this article, a method of determining a suitable Lyapunov functional guaranteeing asymptotic stability for a class of wave equations is developed. But the method is based on the construction of Lyapinov functions for ordinary differential equations and results identical to those determined from an eigenfunction expansion are obtained.
Abstract: For a class of wave equations, a method of determining a suitable Lyapunov functional guaranteeing asymptotic stability is developed. The method is based on the construction of Lyapunov functions for ordinary differential equations. For a particular example, results identical to those determined from an eigenfunction expansion are obtained.
TL;DR: In this article, a system theory description of positive real matrices is used to generate Liapunov functions for finite dimensional systems with time-varying feedback, which satisfy the conditions of the circle criterion for stability.
Abstract: Finite dimensional systems with time-varying feedback, which satisfy the conditions of the circle criterion for stability, are considered. A recent result giving a system theory description of positive real matrices is used to generate Liapunov functions.
TL;DR: In this article, the stability of the equilibrium position of nonlinear oscillating systems relative to perturbations acting constantly on the system is investigated by using Lyapunov function apparatus.
Abstract: Questions of the stability of the equilibrium position for nonlinear oscillating systems relative to perturbations acting constantly on the system are considered. Theorems are proved governing the stability and instability conditions of the equilibrium point of a nonlinear system of general form. The stability is investigated by using Lyapunov function apparatus.
TL;DR: In this article, a stability condition for non-linear systems with delay and a single nonlinearity was derived via the state space approach, and the criterion can be reverted to a, form identical to that for systems without delay.
Abstract: Popov's frequency method has been extended to non-linear sampled-data systems with a single non-linearity and the stability condition thus derived has been shown to be the necessary and sufficient condition for the existence of a certain type of Lyapunov's function. It has been proved that Popov's frequency stability condition for continuous systems without delay is also applicable to systems with delay. By combining different area inequalities, the present investigation derived via the state space approach, a stability condition for non-linear systems with delay and containing a single non-linearity. The criterion can be reverted to a, form identical to that for systems without delay. It is also shown that the stability criterion derived is also applicable to systems without delay.
TL;DR: In this article, the construction of a quadratic form of Lyapunov function following the method of McGee and MacLellan is presented. But, for certain systems containing a nonlinearity which is a function of several variables, previously established sufficient conditions for stability are substantially weakened.
Abstract: Results are obtained by the construction of a quadratic form of Lyapunov function, following the method of McGee and MacLellan. A comparison with Fallside and Ezeilo shows that, for certain systems containing a nonlinearity which is a function of several variables, previously established sufficient conditions for stability are substantially weakened.
01 Jun 1968
TL;DR: Liapunov stability theory generalized and applied to class of partial differential equations was applied to the class of PDEs in this article, where the authors applied it to the case of partial DDEs.
Abstract: Liapunov stability theory generalized and applied to class of partial differential equations
TL;DR: The invariance of the Lyapunov property for functions under the basic operations of addition, multiplication and, in some cases, division was shown in this article, and the results were compared with those known for positive real functions.
Abstract: The invariance of the Lyapunov property for functions under the basic operations of addition, multiplication and, in some cases, division is shown. The results are compared with those known for positive real functions.
01 Jan 1968
TL;DR: Extending Liapunov second method by simultaneous use of several functions and of higher order derivatives in criteria for motion stability as mentioned in this paper is an extension of the second-order derivative method.
Abstract: Extending Liapunov second method by simultaneous use of several functions and of higher order derivatives in criteria for motion stability
TL;DR: By means of Lyapunov theory, a theorem for the asymptotic stability of a linear system with a single periodic element was proved in this paper, and the theorem was later generalized to the case of linear systems with multiple periodic elements.
Abstract: By means of Lyapunov theory, a theorem is proved for the asymptotic stability of a linear system with a single periodic element.
TL;DR: Diagrams are presented of the allowable input step for stable response, as a function of the initial input equilibrium state, of the global phase-portrait concept of Poincare.
TL;DR: Bondixson's criterion to rule out the possibility of closed trajectories (in a simply connected region) of a second-order nonlinear differential equation is well known (Stern 1965). In as discussed by the authors, it is shown that this criterion can be proved and interpreted through Lyapunov theory.
Abstract: Bondixson's criterion to rule out the possibility of closed trajectories (in a simply connected region) of a second-order non-linear differential equation is well known (Stern 1965). In this note it is shown that this criterion can be proved and interpreted through Lyapunov theory.