Showing papers on "Lyapunov function published in 1974"

Model reference adaptive control with an augmented error signal

Abstract: It is shown how globally stable model reference adaptive control systems may be designed when one has access to only the plant's input and output signals Controllers for single input-single output, nonlinear, nonautonomous plants are developed based on Lyapunov's direct method and the Meyer-Kalman-Yacubovich lemma Derivatives of the plant output are not required, but are replaced by filtered derivative signals An augmented error signal replaces the error normally used, which is defined as the difference between the model and plant outputs However, global stability is assured in the sense that the normally used error signal approaches zero asymptotically

Stable Adaptive Schemes for System Identification and Control - Part II

Abstract: The extension of some of the results contained in Part I [1] to the case when only the plant outputs rather than all its state variables are accessible for measurement is discussed. A unified approach to the synthesis of an adaptive observer is presented whereby the plant state and parameters are simultaneously estimated. Uniform asymptotic stability of the scheme is proved using Lyapunov's direct method. The information provided by the adaptive observer is used to synthesize an adaptive controller for the plant. While all the principal results obtained here are for the case of a single-input single-output plant, extensions to special classes of multivariable systems is indicated.

An identification procedure for discrete multivariable systems

Abstract: A new procedure is presented for the identification of discrete linear multivariable systems. Using Lyapunov's direct method the asymptotic stability of the overall system is established when the inputs are sufficiently general. It is shown that the procedure may also be extended for the identification of a certain class of nonlinear systems.

A partial stability approach to the problem of transient power system stability

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.

On the stability of interconnected systems

Abstract: This paper investigates the stability of systems which can be regarded as composed of interconnected subsystems. A sufficient condition for inputs-output L p stability, in terms of the L p gains of the subsystems and their interconnections is derived. For the case of L 2 stability, it is compared with other criteria for asymptotic stability, obtained by Lyapunov techniques, and shown to give better results for a certain class of systems.

Lyapunov-Popov stability analysis of synchronous machine with flux decay and voltage regulator

Abstract: The transient stability of ft synchronous machine connected to an infinite bus ia studied, including the flux decay effect, and voltage regulator action, through Lyapunov-Popov approach. This problem belongs to the general class of systems having multi-argument non-linearities. A systematic procedure to construct a Lyapunov function is evolved by combining a criterion of Dosoer and Wn (1909) and Kalman's construction procedure (1663). The Lyapunov function is then made use of to estimate the critical fault clearing time. The result is compared with that for the case where only flux decay is considered but. voltage regulator action is neglected and the case where both flux decay and voltage regulator action are neglected. The effect of variation of the main parameters of the voltage regulator is also studied and useful conclusions drawn. The Lyapunov function derived in this paper is considered to be superior to those currently available in the literature for this problem.

New Lyapunov functions for power systems based on minimal realizations

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.

Application of generalized Zubov's method to power system stability†

Abstract: The generalized form of Zubov's partial differential equation (Szego 1962) is written using the dynamical equations of a power system. Then a Lyapunov function which satisfies the partial differential equation is obtained by using a transformation of state variables. The V function obtained is used to describe the region of stability. The application of the method to the power system stability problem is illustrated by considering a synchronous generator connected to an infinite bus, Three examples, using three different models for the synchronous machine, are given.

Conservation laws in classical mechanics for quasi-coordinates

Abstract: Noether's theorem and Noether's inverse theorem for mechanical systems with gauge-variant Lagrangians under symmetric infinitesimal transformations and whose motion is described by quasi-coordinates are established. The existence of first integrals depends on the existence of solutions of the system of partial differential equations — the so-called Killing equations. Non-holonomic mechanical systems are analysed separately and their special properties are pointed out. By use of this theory, the transformation which corresponds to Ko Valevskaya first integral in rigid-body dynamics is found. Also, the nature of the energy integral in non-holonomic mechanics is shown and a few new first integrals for non-conservative problems are obtained. Finally, these integrals are used in constructing Lyapunov's function and in the stability analyses of nonautonomous systems. The theory is based on the concept of a mechanical system, but the results obtained can be applied to all problems in mathematical physics admitting a Lagrangian function.

Stable realization of fixed-lag smoothing equations for continuous-time signals

Abstract: Novel smoother structures are introduced for the optimal fixed-lag smoothing of continuous-time signals in noise. The smoothers have the very desirable property that they are simultaneously realizable, stable in the sense of Lyapunov, and optimal. This is in contrast to those proposed to date which are either optimal and realizable but unstable, realizable and stable but suboptimal, or optimal and stable but unrealizable.

A generalised approach to the stability analysis of PWM feedback control systems

Abstract: A unified approach to stability analysis of feedback control systems with pulse-width modulators is discussed. The proposed procedure is developed on the basis of the discrete analog of Lyapunov's method, with no limiting hypothesis on the structure of the controlled plant and of the modulation law. The analysis is exemplified particularly for lead-type PWM control systems. Significant plant classes are investigated, and the critical values of parameters in a closed form are determined which assure asymptotic stability of the steady-state solution, whatever the reference input may be.

Comments on "On Lyanupov functions for power systems with transfer conductances"

Abstract: This note shows that the generalized Popov criterion cannot be used to construct a Lur'e-type Lyapunov function for a multimachine power system model if transfer conductances are included. The two-machine case of the above paper is an exception.

Lyapunov inequalities and bounds on solutions of certain second order equations

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.

Stability of stochastic functional differential equations

Abstract: A system of functional differential equations with random retardation, ẋ(t) = f(t, xt), is studied, where xt(θ) = x(t + θ), η(t, ω) ≤ θ ≤ 0, − r ≤ η(t, ω) ≤ 0, and η(t, ω) is a stochastic process defined on some probability space (Ω, μ, P). Some comparison theorems are stated and proved in details under suitable assumptions on f(t, xt). Sufficient conditions for stability in the mean for the trivial solution then follow. The usefulness of the sufficient conditions is illustrated by an example with two different Lyapunov functions.

A stability criterion for nonautonomous difference equations with application to the design of a digital 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.

Improved signal synthesis techniques for model reference adaptive control systems

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.

A Lyapunov Theorem for Angular Cones

Abstract: A theorem of Lyapunov which has several equivalent formulations characterizes the matrices whose eigenvalues lie in the right half-plane. By considering one of the formulations from a new point of view, a characterization of the matrices whose eigenvalues lie in an arbitrary open positive convex cone is obtained. The positive cone of the c haracterization generalizes the right half-plane

Generation of Lyapunov functions—a new approach

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.

A new first integral corresponding to Lyapunov's function for a pendulum of variable length

Abstract: Using Noether's theorem and the generalized Killing equations [1], new first integrals of the differential equation of motion for a class of non-conservative mechanical systems with one degree of freedom, a special case of which is a simple pendulum of variable length, are obtained. These integrals are identified as Lyapunov's functions for non-autonomous systems. The stability conditions are established.

Instability of Nonlinear Infinite-Dimensional Feedback Systems Using Lyapunov Functionals

Abstract: Sufficient conditions are given for the $L_2$ instability of a broad class of nonlinear, time-varying feedback systems. The system under consideration is assumed to be decomposed into two subsystems: one passive and nonlinear, not necessarily memoryless, and the other unstable and linear, not necessarily finite-dimensional. The main results essentially state that the feedback system is unstable if the linear subsystem is strictly passive and bounded on a proper subset of $L_2$. The results apply both to instability in the input-output sense and to instability of unforced systems.The principal conceptual tools of the analysis are a Lyapunov function and a state, both of which are defined on the linear subsystem in a manner not depending on the dimensionality of the system. As an application of these results, an instability counterpart to the circle criterion is presented which applies to a class of systems more general than those of previous results. The conditions of this counterpart imply, in addition to...

Set stability for difference equations

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).

Observer feedback for uncertain systems

Abstract: In this study feedback design procedures are developed for linear multivariable systems with uncertain parameters and inaccessible states. An observer is used in the feedback loop. The basic design parameters are the observer gain matrix and the state-estimate feed-back matrix. Two models of uncertainty are considered. One is a "deterministic" model where the parameters are assumed to vary within specified bounds. The other is a "random" model where the parameters are assumed to have white noise random variations. The prime design objective is to reduce as much as possible an integral-quadratic performance measure. In the random case the expected value of the integral-quadratic form is used for minimization while in the deterministic case an upper bound on the integral-quadratic is used. Gradients are used to compute the "optimal" design parameters. The required gradients are computed from the solution of Lyapunov equations.

On Ranges of Lyapunov Transformations II

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

Lyapunov Functions Satisfying $\ddot V > 0$

Abstract: By considering the second time derivative of suitable Lyapunov functions we obtain new results in Lyapunov theory for ordinary differential equations.

On Dirichlet's problem for quasi-linear elliptic equations

Abstract: This paper concerns the existence and uniqueness as well as the approximations of the solutions to the Dirichlet problem for the second-order quasi-linear elliptic equation of n variables in a simply connected domain with Lyapunov boundary. Sufficient conditions on the co-efficients of the equation are established for the existence proofs of the solution so that Vekua's function theoretic method for treating Dirichlet's problems for equations in two variables can be employed for the case n ≥ 3 variables. Based on Warschawski's work, an iterative scheme is also presented for constructing an approximating solution in the three-dimensional case. It is shown that the approximants converge to the actual solution geometrically.

Improvement of continuous-gradient identification techniques by Lyapunov's second method

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.

Computation of regions of transient stability of multimachine power systems

Abstract: A major difficulty in applying Lyapunov theory to the problem of specifying transient stability regions of n -machine power systems is computational complexity, which increases markedly with n . This note outlines a method, requiring only a nominal amount of computation, to determine such regions.