Showing papers on "Lyapunov function published in 1969"

New results in linear system stability

Abstract: This paper considers connections between bounded-input, bounded-output stability and asymptotic stability in the sense of Lyapunov for linear time-varying systems. By modifying slightly the definition of bounded-input, bounded-output stability, an equivalence between the two types of stability is found for systems which are uniformly completely controllable and observable. The various matrices describing the system need not be bounded. Other results relate to the characterization of uniform complete controllability and the derivation of Lyapunov functions for linear time-varying systems.

On the stability of linear stochastic systems

Abstract: Necessary and sufficient conditions for stability with probability 1 are developed for the class of linear stochastic systems. A simple technique for constructing quadratic stochastic Lyapunov functions is presented which entails the solution to an n \times n linear matrix equation.

Stability, Instability, Invertibility and Causality

Abstract: 1. Introduction. A number of interesting stability criteria for feedback systems have recently appeared in the control theory literature. The procedures used in proving these criteria can roughly be divided into three classes; the first based on Popov-like methods, the second using Lyapunov theory with Lyapunov functions derived from spectral factorizations or Riccati-type algebraic matrix equations, and the third treating the stability problem from a functional analysis point of view. Each of these methods has relative merits, e.g., the Lyapunov methods seem to be the only ones which allow us to obtain an estimate of the domain of attraction in the case of nonglobal stability. However, the method based on functional analysis appears to be the more satisfactory one, in view of the essential simplicity, of the intuitive nature of the results (loop gain less than one, passivity conditions), and of the fact that it unifies the various criteria (as, e.g., the circle criterion and the Popov criterion). It therefore deserves more investigation and exposition than it has thus far been given. A peculiarity of this method, as presently employed, is that most of the analysis and estimates have to be made on extended spaces which, although derived from normed spaces, are themselves not normed. This entails in general rather cumbersome mathematical manipulations. One however suspects the

On the stability of randomly sampled systems

Abstract: Randomly sampled linear systems with linear or non-linear feedback loops are studied by a stochastic Lyapunov function method. The input in this paper is assumed zero; driven systems will be treated in a later paper. Improved criteria for stability (with prebability one, on s th moment s > 1 , or in mean-square) are given when the sequence of holding times are independent. The method is relatively straightforward to apply, especially in comparison with the direct methods, and allows the study with nonlinear feedback or nonstationary holding times. A randomly sampled Lur'e problem is studied. Numerical results, describing some interesting phenomena, such as, jitter stabilized systems are presented.

Higher order derivatives of Liapunov functions

Abstract: The problem considered is that of inferring asymptotic stability by examining higher order derivatives of a Liapunov function when examination of only the first derivative does not permit such an inference. It is shown that a seemingly plausible approach using only \dot{V} and \ddot{V} is vacuous. It is also shown that it is possible to infer asymptotic stability by examining \dot{V} and \ddot{V} , and perhaps \ddot{V} .

Routh-Hurwitz conditions and Lyapunov methods for the transient-stability problem

Abstract: In this paper, Lyapunov functions are generated to determine the regions of asymptotic stability of power systems under transient disturbances. With suitable assumptions, the swing equations of the synchronous machines connected to a power system are second- or third-order nonlinear autonomous differential equations. The Lyapunov functions V, employed to determine the domains of asymptotic stability of these nonlinear differential equations, are simple quadratic forms, whose coefficients are chosen so that the Routh-Hurwitz criteria are satisfied for the corresponding linear differential equations. For a synchronous machine swinging against an infinite bus, three typical Lyapunov functions are generated, taking transient saliency and positive-sequence damping into account. The application of Aizerman's method to the same problem leads to a fourth Lyapunov function. The domains of stability given by these Lyapunov functions are compared with the actual stability region obtained by numerical integration. Consideration of one time constant of the prime-mover governor leads to a third-order differential equation. A Lyapunov function is generated, and the stability surface obtained is verified by computing the swing curves numerically for various initial conditions. Finally, it is explained how some of these techniques could be extended for generating Lyapunov functions for multimachine systems.

Lyapunov stability and Lyapunov functions of infinite dimensional systems

Abstract: Sufficient conditions are found for the existence of positive definite functions of state which are nonincreasing in time along any trajectory of an autonomous system. The class of systems considered is quite general, and no restriction is made concerning the dimension of the state space or separability of effects of state and input of the subsystems. If certain other relations between the norm of interest on the state space and the positive definite functions are established, Lyapunov or in some cases asymptotic stability in the large can be established. The sufficiency part of the Kalman-Yacubovich lemma as applied to the same problem, is extended to include infinite dimensional systems. That is, it is shown that if the Popov criterion is satisfied, then a Lyapunov function of the Lur'e type exists, even in the infinite dimensional case.

On an inequality of Lyapunov

Abstract: According to Fink and St. Mary [2] the proof of (2) given in [1] is incorrect, and therefore the inequality is, as yet, undecided. For n = 2 and P2 0, (2) is known to be correct and also the best possible. This case was first proved by Lyapunov and is generally referred to as Lyapunov's theorem. A recent proof of this result may be found in Hochstadt [3]. In [2] a similar technique is used to prove (2) for n= 2, where P2 iS merely integrable on [a, b]. In fact, one can extract from [2] the inequality

On the construction of lyapunov functions

Abstract: the same These results are used to obtain new instability results for the Hill equation which extend the classical results of Lyapunov and Haupt Finally, we show that a Lyapunov function for w' = 2Bw can be used to algorithmically obtain a Lyapunov function for y' = By along with certain verifiable conditions from which stability properties of y' = By (and hence of x' = Ax) may be obtained No such algorithm may be found in the present literature 2 Preliminary results The matrix A can be written as a sum of piecewise

Piecewise-quadratic and piecewise-linear Lyapunov functions for discontinuous systems†

Abstract: This paper develops necessary and sufficient conditions for the existence of Lyapunov functions of tho Lur'e type (‘ piecewise-quadratic ’) for relay-control systems and gives explicit formulae- for ‘optimum ’functions of this form for estimating regions of asymptotic stability in special cases, comparing these with results of numerical optimization. Certain similarities and advantages of ‘piecewise-linear ’Lyapunov functions are discussed. Among their benefits is algebraic simplification which allows one to compute ranges of time varying and non-linear parameters which preserve asymptotic stability in a given region in the state space; an example of this type is given.

Numerical application of Szego's method for constructing Liapunov functions

Abstract: A numerical algorithm for the construction of Liapunov functions is described. This algorithm is based upon the method of Szego and is applicable to a particular class of systems. The algorithm is described for second-order systems although extension to systems of higher order would appear to be possible.

Canonical form for the matrices of linear discrete-time systems

Abstract: Some applications of canonical matrices to linear continuous-time systems are reviewed. A canonical form is proposed for real nonderogatory convergent matrices, such as the A matrices which occur in the description of linear discrete-time dynamical systems by vector-matrix difference equations of the form xk+1 = Axk + Buk. The new canonical form is applied to the generation of particular and general solutions of the matrix equation ATLA − L = −K, which occurs in the application of Lyapunov theory to the analysis and design of such systems.

Stochastic stability in nonlinear control systems

Abstract: The stability of nonlinear control systems with stochastic coefficients is studied by applying the Lyapunov theory. A Lyapunov function V(x) is first assumed, similar to the deterministic case. Consider next the natural stochastic analog of \dot{V}(x) as A V(x) , where A is equal to the differential generator. The procedure of establishing stability conditions is illustrated by two second-order examples and four third-order examples.

Synthesis of asymptotically stable linear time-invariant closed-loop systems incorporating multivariable 3-term controllers

Abstract: An inverse Lyapunov technique is presented for the design of asymptotically stable linear time-invariant closed-loop systems incorporating multivariable 3-term controllers.

Path integrals and Lyapunov functionals

Abstract: A method for generating Lyapunov functionals for time-delay systems by means of path integrals in state space is given. The method is derived by making use of a new description of such systems in terms of convolution equations involving distributions with compact support. The important properties of these equations are discussed and it is shown that a suitable state space can be defined. Path integrals in this state space are defined and conditions for path independence are derived. With the aid of some results dealing with the spectral factorization of entire functions of exponential order, it is shown that these path integrals can be used to define Lyapunov functionals for time-delay systems. The method given represents an extension to infinite-dimensional systems of a technique developed by Brockett for systems described by ordinary differential equations. While the present approach differs fundamentally from that used for finite-dimensional systems, the results given here are similar to, and in the special case of finite-dimensional systems reduce to, the results given by Brockett. Hence the method given can be successfully applied even without a deep understanding of either distributions or distributional convolution equations. This is illustrated by a number of examples which show the application of the results to stability analysis as well as to a class of quadratic minimiization problems.

Generation of Liapunov functions for time-varying nonlinear systems

Abstract: A procedure for the generation of Liapunov functions for time-varying nonlinear systems is described. A composite differential one-form is derived from the basic system equations. A line integral of the one-form, with t held constant, produces the candidate for a Liapunov function. An example is included.

Absolute stability of a class of nonlinear sampled-data systems

Abstract: The present investigation studies the problem of absolute stability of a class of nonlinear sampled-data control systems with or without integrators in the loop. An absolute-stability criterion has been obtained by the second method of Lyapunov. The same stability criterion has been derived previously by the authors via the Popov approach. The criterion is shown to be a sufficient condition for the existence of a certain type of Lyapunov function which assures global-asymptotic stability of the class of systems under investigation. In contrast to previous results, the criterion does not place any restriction on the number of integrators in the loop. A systematic step-by-step method for applying the inequality is given, and an example illustrating the application of this frequency-domain inequality and a comparison with previous results are presented. The method is found to be versatile and more effective, and, in general, a better stability boundary can be obtained.

A simpler mean-square stability criterion for a class of linear stochastic systems

Abstract: A simpler criterion for establishing the mean-square stability of a class of n th order linear systems with randomly, time varying parameters is presented. In order to find the necessary and sufficient condition for the boundness of second-order moments, use is made of a theory of Lyapunov's second method for stochastic systems.

Stability of the damped Mathieu equation

Abstract: A method developed to generate Lyapunov functionals for distributed-parameter systems is applied to the damped Mathieu equation, the resulting Lyapunov functions giving rise to an improvement in the known stability boundaries.

Stability of certain systems of second order differential equations

Abstract: Conditions are obtained which are sufficient for a system of ordinary second order differential equations to have an autonomous quadratic Lyapunov function. From these are derived conditions which ensure that every pair of solutions converges as t»+∞

Stability of columns subject to impulsive loading

Abstract: THE STABILITY CRITERION DEVELOPED FOR SIMPLY SUPPORTED COLUMNS DETERMINE A BOUND ON THE TRANSVERSE MOTION OF THE COLUMN RELATIVE TO THE AMPLITUDE AND DURATION OF A PRESCRIBED FORCE PULSE AND A SET OF INITIAL PERTURBATIONS. THE CONDITIONS OF STABILITY ARE DEDUCED FROM THE CHARACTERISTICS OF A CLASS OF LYAPUNOV FUNCTIONS CONSTRUCTED FOR THIS PROBLEM (LYAPUNOV'S DIRECT METHOD). THE STABILITY OF SIMPLY SUPPORTED COLUMNS IS PRESENTED FOR THREE SPECIFIC EXAMPLE LOADINGS: A CONSTANT THRUST OF INFINITE DURATION, A VARIABLE THRUST OF INFINITE DURATION, AND A TRANSIENT THRUST OF PARABOLIC SHAPE. THE QUALITY OF THE BOUNDS IS ASCERTAINED THROUGH COMPARISON WITH THE TRAJECTORY OF THE DIFFERENTIAL EQUATION OF MOTION, OBTAINED BY NUMERICAL INTEGRATION./ASCE/

Acquisition conditions for phase-lock loops†

Abstract: The acquisition behaviour of the frequency of an input signal by means of phase-lock loops is investigated. Sufficient conditions for acquisition are obtained by means of.Lyapunov's direct method. A systematic procedure is developed for the computation of Lyapunov functions and acquisition regions. It is illustrated by two examples. Finally the power and the limitations of this approach are discussed.

Investigation of the stability of control systems with double (frequency and width) pulse modulation by the direct lyapunov method

Abstract: : Discussion is based on a typical control system comprised of a continuous linear section (CLS) and combined pulse modulator (CPM). The modulator converts continuous output signal into a sequence of pulses of constant amplitude and modulated in sign, frequency, and duration. The modulation law is expressed. The development consists of the following: application of the direct Lyapunov method to the study of stability of a system with double pulse modulation; the Lyapunov function and its first difference; conditions of complete stability of a static system; and conditions of complete stability of a set of equilibrium positions of astatic systems. The method presented is applicable not only to systems with double pulse modulation but to all pulse systems which may be regarded as a special case of such a system.

Lyapunov function construction for a class of discrete time-varying systems

Abstract: The construction of Lyapunov functions for discrete systems which may be arranged in the form of a linear, time-varying subsystem with feedback memoryless nonlinearities is considered. The results constitute a generalization of Popov's stability theory.