Showing papers on "Lyapunov function published in 1975"
TL;DR: In this paper, a Lyapunov function for a nonlinear system has been constructed for each subsystem and then the weighted sum u of the WOVE + w2v2 +... + w&z& (Wi > 0) is used to reduce the effort involved.
139 citations
TL;DR: In this article, the authors studied the infinite-time quadratic cost control problem for a general class of linear autonomous hereditary differential systems using an approach which clarified the system-theoretic relationship between stabilizability, stability and existence of a solution of an associated operator equation of Riccati type.
Abstract: This paper studies the infinite-time quadratic cost control problem for a general class of linear autonomous hereditary differential systems. It uses an approach which clarifies the system-theoretic relationship between stabilizability, stability and existence of a solution of an associated operator equation of Riccati type. For this purpose the stability problem is studied and an operator equation of the Lyapunov type is derived. In both cases we obtain equations which characterize the kernels of the Lyapunov and the Riccati equations.
136 citations
TL;DR: In this article, it is shown that an accurate determination of these points is not necessary and using the analogy of 1-mechine-infinite bus example, (2n-1-1) u.p.'s can be approximately obtained.
Abstract: One of the major difficulties in using the Lyapunov method for on-line transient stability is the determination of the critical value of the V-function which describes the stability boundary. In one method the V critical is taken to be the minimum value of the V-function evaluated at (2n-1-1) unstable equilibrium points (u.e.p.). In this paper it is shown that an accurate determination of these points is not necessary and using the analogy of 1-mechine-infinite bus example, (2n-1-1) u.e.p.'s can be approximately obtained. Two multimachine systems (4 and 9- machine) are used as examples. The error in the stability boundary by using approximate method is shown to be acceptable. The use of Newton-Raphson method to calculate the post-fault stable equilibrium point is suggested. Also the type of mathematical model which is sufficient to represent both uniformly and non-uniformly damped multimachine systems is discussed.
96 citations
TL;DR: In this article, a new approach to the stability analysis of large scale power systems using vector Lyapunov functions is introduced, by decomposing the composite power system into a number of low-order subsystems, and using this function together with the stability properties of individual subsystems and the nature of interactions between subsystems criteria for the stability of the overall system are derived.
43 citations
TL;DR: In this article, the influence of initial data on dynamic behaviour and stability properties of non-stationary systems is analyzed and new stability properties are discovered and defined as well as corresponding stability domains.
Abstract: The paper presents a refined analysis of the influence of initial data on dynamic behaviour and stability properties of non-stationary systems. As a result, new stability properties are discovered and defined as well as the corresponding stability domains. Furthermore, general necessary conditions are established for asymptotic stability of the unperturbed motion of these systems, the instantaneous asymptotic stability domain of which can be either time-invariant or time-varying and then possibly asymptotically contractive. It is shown that the classical Lyapunov stability conditions cannot be applied to the stability test as soon as the system instantaneous domain of asymptotic stability is asymptotically contractive. In order to investigate asymptotic stability of the motion in such cases novel criteria are established. Under the criteria the Eulerian derivative of a system Lyapunov function may be non-positive only and still guarantee asymptotic stability of the unperturbed motion The results ...
41 citations
TL;DR: In this paper, the authors explore the problem of maximizing the size of a region of asymptotic stability (RAS) over a class of Lyapunov functions (LF) in order to estimate the domain of attraction of a continuous or discontinuous (relay) system.
Abstract: This paper explores the problem of maximizing the size of a region of asymptotic stability (RAS) over a class of Lyapunov functions (LF) in order to estimate the domain of attraction of a continuous or discontinuous (relay) system. Analytic and numerical evidence shows that a subclass of LF may exist with multiple-tangency for the RAS problem, and can cause some serious convergence problems. Several systems are considered with two classes of LF for continuous and one class for relay systems.
35 citations
TL;DR: The use of Lyapunov functions in the design of a fixed controller for model reference systems results in a bang-bang control as discussed by the authors, where the motion in the neighborhood of the switching surface is analyzed and conditions are found that ensure stable trajectories.
Abstract: The use of Lyapunov functions in the design of a fixed controller for model reference systems results in a bang-bang control. The motion in the neighborhood of the switching surface is analyzed and conditions are found that ensure stable trajectories. It is shown that the Lyapunov design verifies these conditions. The last part of the trajectories appears to be a sliding motion, forcing the plant to track the model. The results are illustrated by a simulation example.
28 citations
TL;DR: In this paper, the authors deal with the construction of Lyapunov functions for the finite dimensional linear system ẋ = Ax when the entries of the generating matrix A satisfy various conditions requiring dominance of its diagonal elements and nonnegativity of its off-diagonal elements.
23 citations
TL;DR: In this paper, it was shown that eventually stable sets are asymptotically invariant, which enables us to regard Lyapunov stability as a special case of eventual stabilities.
TL;DR: In this article, a constructive and simple proof is given of a generalization of a result of Lyapunov on the convexity of the range of vector measure, which is used in this paper.
Abstract: A constructive and simple proof is given of a generalization of a result of Lyapunov on the convexity of the range of vector measure.
TL;DR: In this article, generalization of the method of Lympunov functions is outlined, whereby the convergence of numerical methods of optimization may be proved, and theorems on sufficient conditions on convergence of continuous and discrete schemes are stated and proved.
Abstract: GENERALIZATIONS of the method of Lympunov functions are outlined, whereby the convergence of numerical methods of optimization may be proved. Theorems on sufficient conditions for the convergence of continuous and discrete schemes are stated and proved. The application of the theorems is illustrated by examples.
TL;DR: In this article, the Glansdorff-Prigogine stability criterion is examined in the light of Lyapunov's theory of stability, and it is found that its application to the study of stability of far from equilibrium is sound, although in general the criterion is expected to leave a large region of undertermined stability when used to delimit the stability domain in the manifold of steady states.
TL;DR: In this paper, the problem of determining the positive definiteness of an arbitrary multivariable polynomial is transformed into an equivalent problem which is solvable by rational operations.
Abstract: The problem of determining the positive definiteness of an arbitrary multivariable polynomial is transformed into an equivalent problem which is solvable by rational operations. Nontrivial examples which illustrate different aspects of the algorithm are given in connection with stability determination via Lyapunov's method.
TL;DR: For the continuous real valued functions, p, m and g, with p(x)≥0, and m(x), g(x, m, g, and v>0 being reals, the differential equations y″(x)+p(x)|y(x)),μsgn y(x)= =m(x|y(g(x))|vsgn yǫ(gǫ) y(g)ǫ)) as mentioned in this paper.
Abstract: For the continuous real valued functions, p, m and g, with p(x)≥0, and m(x)≥0, and μ>0, v>0 being reals, the differential equations y″(x)+p(x)|y(x)|μsgn y(x)= =m(x)|y(g(x))|vsgn y(g(x)) is considered. Lyapunov type integral inequalities are established which yield implicit lower bounds on the distance between consecutive zeros of a nontrivial solution of the above equation, and several others. The same is done for a problem involving the distance from a zero of a solution y to the next greater zero of its derivative y′. Special conditions are placed on the corresponding initial functions. They allow for application of results to oscillatory solutions of the given equation, and also to non-trivial solutions having a zero initial function. When p(x) ≡ 0. the results take on a special form; and when in addition m(x)>0, g(x)
TL;DR: In this article, a gradient parameter estimation technique is presented for multivariable discrete linear systems and a class of nonlinear systems, and the second method of Lyapunov is used to prove the asymptotic stability of the identification procedure.
Abstract: A gradient parameter-estimation technique is presented for multivariable discrete linear systems and a class of nonlinear systems. The second method of Lyapunov is used to prove the asymptotic stability of the identification procedure.
TL;DR: In this paper, a modified approach for the design of Lyapunov model reference adaptive systems is presented, which introduces the use of a function of the parameter misalignment, which can be generated with minor demands for additional hardware.
Abstract: A modified approach is presented for the design of Lyapunov model reference adaptive systems. The approach introduces the use of a function of the parameter misalignment, which can be generated with minor demands for additional hardware. Simulation confirms the acceleration in convergence of the system error compared to previous design procedures.
TL;DR: In this paper, an inertia parameter is introduced which simplifies and clarifies investigation of the problem for all possible rigid bodies, and the results of Rumiantsev's Lyapunov analysis are used to provide part of the stable zone.
Abstract: The classical stability problem of the spinning rigid body with a fixed point under the influence of uniform gravity, is examined by several approaches. First, an inertia parameter is introduced which simplifies and clarifies investigation of the problem for all possible rigid bodies. Instability zones are then found using the Floquet theory. The results of Rumiantsev's Lyapunov analysis are used to provide part of the stable zone. Finally, the remaining regionR, for which a stability (in the large) decision is not guaranteed either by the Floquet theoryor the Lyapunov analysis, is discussed.
TL;DR: In this article, a number of results on perturbations of second order differential equations of the form $x'' + f(x )h( {x'} )x' + g( x ) = 0$ are presented.
Abstract: In this paper we present a number of results on perturbations of second order differential equations of the form $x'' + f( x )h( {x'} )x' + g( x ) = 0$. This is accomplished by constructing a variety of Lyapunov functions. We then show how these Lyapunov functions can be converted to Lyapunov functionals for the delay equation $x'' + f( x )h( {x'} )x' + g( {x( {t - \tau ( t )} )} ) = 0$, thereby obtaining boundedness results. Some of the work is generalized to higher order systems. We also present some continuation results for higher order delay equations. Several examples are given.
TL;DR: In this paper, the authors extend the mathematical theory for total stability into a form which can be used directly in applications to obtain explicit expressions for maximum magnitudes of the initial condition and the disturbances in the equations of motion.
Abstract: This article discusses the total stability, or stability under constantly acting disturbances, of a system of nonlinear ordinary differential equations Total stability differs significantly from Lyapunov stability in that the former allows for a disturbance in the equations of motion, aswell as a disturbance in the initial condition The purpose of this study is to extend the mathematical theory for total stability into a form which can be used directly in applications To do this, a specific Lyapunov function is constructed Then, using this Lyapunov function in a new total stability theorem we obtain explicit expressions for maximum magnitudes of the initial condition and the disturbances in the equations of motion These maximum magnitudes are expressed in terms of the prescribed bound of the motion from the equilibrium, and in terms of the parameters of the physical system which the differential equations describe
01 Jan 1975
TL;DR: This paper presents a stable implicit adaptation algorithm for model reference control using Lyapunov's second method and does not depend on perfect model following between the system and the reference model.
Abstract: This paper presents a stable implicit adaptation algorithm for model reference control. The constraints for stability are found using Lyapunov's second method and do not depend on perfect model following between the system and the reference model. Methods are proposed for satisfying these constraints without estimating the parameters on which the constraints depend.
TL;DR: In this paper, the authors derived instability criteria for feedback systems with a time-invariant linear element in the forward branch and a timevarying nonlinear feedback amplifier.
Abstract: Instability criteria are derived for feedback systems containing a time-invariant linear element in the forward branch, and a time-varying nonlinear feedback amplifier. The transfer function of the linear element is not restricted to rational functions. The results are obtained using Lyapunov techniques in function space. They are considerably stronger than those in early papers on the subject, which are confirmed as special cases of the present ones.
TL;DR: A procedure, based on classical Lyapunov techniques, is presented for computing a lower bound of sustained oscillation amplitudes in unstable nonlinear feedback loops.
01 Dec 1975
TL;DR: In this article, a simple technique for obtaining improved regions of transient stability of power systems is presented, which requires a simple transformation of the system equation before Popov's; stability condition is employed for generating the Lyapunov function.
Abstract: The paper presents a simple technique for obtaining improved regions of transient stability of power systems. This requires a simple transformation of the system equation before Popov's; stability condition is employed for generating the Lyapunov function. Results are presented for computation of the regions of both absolute and exponential stability.
TL;DR: In this paper, the Lyapunov functions for linear multivariable time-invariant systems are developed by use of construction techniques which are based on Lyapinov's second method.
01 Jan 1975
TL;DR: In this article, a second order approximation is derived from a linearized error characteristic equation for Lyapunov designed model-reference adaptive systems and used to estimate the maximum error between the model and plant states, and the time to reach this peak following a plant perturbation.
Abstract: A second order approximation is derived from a linearized error characteristic equation for Lyapunov designed model-reference adaptive systems and is used to estimate the maximum error between the model and plant states, and the time to reach this peak following a plant perturbation. The results are applicable in the analysis of plants containing magnitude-dependent nonlinearities.
TL;DR: A technique is presented for multivariable model reference adaptive control systems that are synthesised according to the second method of Lyapunov by the use of a function of the parameter misalignment.
Abstract: A technique is presented for multivariable model reference adaptive control systems that are synthesised according to the second method of Lyapunov. Improved error convergence is obtained by the use of a function of the parameter misalignment.