Showing papers on "Lyapunov equation published in 1981"

Quadratic-type Lyapunov functions for singularly perturbed systems

Abstract: Asymptotic and exponential stability of nonlinear singularly perturbed systems are investigated via Lyapunov stability techniques. A quadratic-type Lyapunov function for a singularly perturbed system is obtained as a weighted sum of quadratic-type Lyapunov functions of two lower order systems. Estimates of domain of attraction, of upper bound on perturbation parameter, and of degree of exponential stability are obtained. The method is illustrated by studying the stability of a synchronous generator connected to an infinite bus.

Explicit solutions of the discrete-time Lyapunov matrix equation and Kalman-Yakubovich equations

Abstract: A general solution for the nonsquare nonsymmetric Lyapunov matrix equation in a canonical form is presented. The solution is shown to be a Toeplitz matrix which may be calculated using the backwards Levinson algorithm This solution is then applied to the Kalman-Yakubovich equations to derive a method for generating strictly positive-real functions via the positive-real lemma. This latter result has an application in system identification.

Lagrange-Charpit method and stability problem of power systems

Abstract: The paper applies the Lagrange-Charpit method to construct a Lyapunov function for stability studies of power systems. The Lyapunov function is given for the single-machine system taking into account saliency and the effect of variable damping. The stability boundary obtained is compared with those obtained by conventional Lyapunov functions and the true stability boundary. It is shown that the application of the Lagrange-Charpit method results in considerable improvement of stability-boundary estimates over those which have been currently available concerning this problem. Another model including the effects of constant damping and the velocity governor with one time constant is also studied. In this 3rd-order system, cross-sections of the stability surface for various planes are given, showing the superiority of the proposed Lyapunov function.

Quasi-solutions, vector Lyapunov functions, and monotone method

Abstract: The notion of quasi-solution which was introduced earlier has been developed in this paper. Further, the notion of coupled quasi-solutions and coupled maximal and minimal solutions has been introduced. It is shown that the idea of quasi-solutions leads to isolated subsystems, and has obtained error estimates between solutions and quasi-solutions. Also, monotone iterative techniques have been developed to obtain coupled maximal and minimal quasi-solutions.

A note on the Lyapunov matrix equation

Abstract: In [5] bound for the determinant of the solution to the Lyapunov matrix equation was reported. This note gives an another bound for this value.

Robust Lyapunov stability results and adaptive systems

Abstract: Lyapunov stability results are summarized which are robust in character. These results are applied to examine the behaviour of adaptive identification and control algorithms applied to non-stationary plants where certain persistency of excitation conditions are satisfied.

A proof of a discrete stability test via the Lyapunov theorem

Abstract: In this note a discrete stability test is proved by using the Lyapunov theorem. The proof is simple and elegant, and is similar to the celebrated proof of the Routh-Hurwitz criterion by Parks [1].

Construction of power system transient stability equivalents using the Lyapunov function

Abstract: Owing to increasing size and complexity in power systems, the study of the stability equivalent is receiving a great deal of attention. This paper describes a systematic method for the recognition of coherent machines by means of the Lyapunov function which is used for transient stability analysis. A group of generators, whose partial Lyapunov function has a small value compared with that for the whole system is aggregated into one equivalent. generator. This method does not require a long time simulation of the entire system. The parameter of the simplified system are also determined by using the Lyapunov function. The method in applied to 10-machine and 50-machine sample systems and the results are shown.

Efficient solution of Lyapunov equation for matrix autoregressive models and its application to the inverse Levinson problem

Abstract: A novel efficient method for solving the discrete Lyapunov equation is presented, for the case where a matrix autoregressive model is assumed. This leads to an efficient procedure for solving the inverse Levinson problem, namely - constructing ladder realizations for given AR models (rather than for given covariance sequences). The method is based on a recently described method of inverting matrices that are sums of block-Toeplitz and block-Hankel matrices. The procedure is then shown to yield a stability test for the given autoregressive model.

On a class of Lyapunov functionals of the Boltzmann equation and their generalizations

Abstract: A family of Lyapunov functionals L α (generalized entropy functionals) of Boltzmann's collision equation is used to judge the sensitivity of the tendency towards the equilibrium against relevant classes of persistent perturbations of the Boltzmann equation. The embedding of the Boltzmann equation into a hierarchy of BBGKY-type is considered from the stability point of view by means of generalized functionals μ L α . The derivation of the Boltzmann equation from the hierarchy by a factorization assumption is restated as optimization problem.

Efficient solution of the inverse Levinson problem in the multichannel case via the Lyapunov equation

Abstract: A new efficient method for solving the discrete Lyapunov equation when the system matrix is block companion is presented; hence for solving the equivalent inverse Levinson problem in the multichannel case (constructing ladder realizations for a given matrix autoregressive (AR) model). The condition for the existence of a unique solution is specified. In addition, nice generalizations of some results of the scalar case are obtained.

Control of Synchronous Orbit Using Lyapunov Stability Theory

Abstract: A state-dependent control law is proposed using the Lyapunov stability theorem and its usefulness for closed-loop control of a synchronous orbit is investigated. It is shown that a sequential scheme using a modified control law provides a more practical means of orbit control.

On the construction of generalized lyapunov's function

Abstract: In this paper, we try to obtain a better understanding of geometrical meaning of Lyapunov's second method from the topological viewpoint and shall give an affirmative con- clusion of the existence and concrete construction of the family of surfaces,i.e. the generalized Lyapunov's function. Since the topological character of integral curves of differential equation is a regular curve, the existence problem of the family of closed surfaces, or the family of closed curves in plane, can be solved completely by means of the topological viewpoint, if the undisturbed move- ment is asymptotically stable in a generalized way.

Non-linear perturbations of dynamical systems with bounded inputs

Abstract: The non-linear variations-of-constants formula is used to derive state estimates when a non-linear system is subject to bounded inputs. Both input-output and Lyapunov methods are examined

Statistical irreversibility and instability of motion according to Lyapunov

Abstract: The stability of the motion of a system of many particles according to Lyapunov is discussed. It is shown that the set of initial displacements of the particles decomposes into sets which result in stable, and consequently mixed, motions. The mixing rate is characterized by the average statistical Lyapunov index, for which an expression is obtained in terms of the interaction potential and a function modulating the initial displacements of the particles. An integral equation is obtained for the latter.

Further corrections to "Comments on ́On the Lyapunov matrix equatioń"

Abstract: The purpose of this correspondence is to point out that the proof of Theorem 3 in the above paper is invalid.

Lyapunov stability of scalar charged solitons

Abstract: The direct Lyapunov method is used to investigate the stability of charged solitons of pulson type described by a relativistic complex scalar field in a model of general form. It is shown that the stability can only be conditional. Some necessary and sufficient conditions for stability of stationary solitons for a fixed charge are formulated. Examples of models with power-law and logarithmic nonlinearities are considered.

Asymptotic stability of a class of stochastic interconnected systems

Abstract: The stability of a class of stochastic interconnected systems is examined in the Lyapunov sense. Subsystem Lyapunov functions are used to construct Lyapunov functions for the composite system. The results presented are sufficient conditions for asymptotic stability with probability one.