Showing papers on "Lyapunov equation published in 1980"

Stability analysis of 2-D systems

Abstract: A polynomial stability criterion for 2-D systems is taken as a starting point for introducing a frequency dependent Lyapunov equation. The Fourier analysis of its matrix solution leads to an infinite dimensional quadratic form which provides a Lyapunov function for the global state of the system. The Fourier coefficients are explicitly obtained as the sum of series involving the system matrices. The convergence of these series constitutes a necessary and sufficient stability condition, which generalizes the analogous condition for 1-D systems.

Connections between finite-gain and asymptotic stability

Abstract: The relationship between input-output and Lyapunov stability properties for nonlinear systems is studied. Well-known definitions for the input-output properties of finite-gain and passivity, even with quite reasonable minimality assumptions on a state-space representation, do not necessarily imply any form of stability for the state. Attention is given to the precise versions of input-output and observability properties which guarantee asymptotic stability. Particular emphasis is given to the possibility of multiple equilibria for the dynamical system.

On stability with large parameter variations: Stemming from the direct method of Lyapunov

Abstract: A new approach with a computer algorithm for determining the "largest" upper bound for parameter variations of a linear constant coefficients system for which the system remains stable is established. This approach then provides the most efficient way of evaluating such "largest" variations of the coefficients of a stable polynomial.

A comparison of Lyapunov and hyperstability approaches to adaptive control of continuous systems

Abstract: This paper contains a brief survey of certain aspects of Lyapunov's stability theory and the hyperstability theory. Conditions which have to be satisfied for the two approaches to be successfully applied to adaptive observers and controllers are examined. When all the signals in the plant are uniformly bounded (as in adaptive observers and some control problems) the two approaches yield the same results. When the plant signals cannot be assumed to be uniformly bounded (as in the general control problem) neither approach works directly and special analysis is needed.

Generating covariance sequences and the calculation of quantization and rounding error variances in digital filters

Abstract: A linear algorithm is given for the generation of covariance sequences for rational digital filters using numerator and denominator coefficients directly. There is no need to solve a Lyapunov equation or to solve for the residues of a spectrum, as in other methods. By appealing to certain results from the theory of inners, we show that the algorithm provides a unique solution, provided only that the filter is stable. Our results may be used to compute error variances due to product rounding and signal quantization, and to generate covariance strings {r_{k}}min{0}\max{K} used in other studies involving second-order properties of digital filters.

The lyapunov equation and the problem of stability for linear bounded discrete-time systems in Hilbert space

Abstract: This paper is devoted to a study of the properties of the equationA*FA−F=−G, where F∈L(ℌ) is unknown, A∈L(ℌ), G∈L(ℌ) is positive andℌ is a Hilbert space. It is shown that necessary and sufficient (in some sense) conditions for the existence of positive definite solutions of this equation are directly connected with the stability of infinite dimensional linear systemxk+1=Ax k . The relationships between stability of such a system and stability of a continuous-time system generated by a strongly continuous semigroup are given also. As an example the case of the delayed system in Rn\(R^n \dot x\left( t \right) = A_0 x\left( t \right) + A_1 x\left( {t - 1} \right)\) is considered.

Application of the Lagrange-Charpit method to analyse the power system's stability

Abstract: A systematic procedure for constructing Lyapunov functions is considered. The linear partial differential equation is written using the system equation and the arbitrary non-negative function φ.The Lyapunov function is obtained by integrating the partial differential equation, using the Lagrango-Charriit method. Then, the arbitrary function φ is handled from the integrable standpoint of the characteristic equation of the partial differential equation. The Lyapunov function obtained is used to describe the region of stability. The superiority of this method is illustrated by applying it to a single-machine system, which includes the damping effect, comparing the results with those obtained by conventional methods.

Nonlinear control of bilinear systems

Abstract: The dual problems of designing state observers and state-feedback laws for bilinear systems are considered. Subsequently, the synthesis of dynamic controllers using state observers is discussed. In each case, global asymptotic stability of the complete system is pursued using a Lyapunov approach. The design is essentially reduced to finding a class of positive-definite matrices satisfying a given set of equality and inequality constraints. To each matrix satisfying these constraints there corresponds a class of controllers which globally stabilise the closed-loop system

Local stability of composite systems--Frequency-domain condition and estimate of the domain of attraction

Abstract: This paper is concerned with such composite systems whose subsystems contain one nonlinearity each and whose interconnections are functions of the scalar outputs of subsystems. A frequency-domain condition which assures local asymptotic stability is given under the assumptions that each nonlinearity satisfies a sector condition, that interconnections are linearly bounded, and that linear parts of subsystems may have unstable poles. In deriving the above result, such Lyapunov functions of subsystems are constructed so that their weighted sum is a Lyapunov function of the overall system. A method to estimate the domain Of attraction based on the above Lyapunov functions is also studied. When the bounds on nonlinearities hold true in the entire space and when the linear parts do not have unstable poles, the present condition turns out to be the same with the L 2 -stability condition which was obtained before by Araki.

A new estimate of the stability regions of large-scale systems

Abstract: A new estimate of the stability region of a large-scale system is presented under the assumption that the system can be decomposed into subsystems and each subsystem has a second-order Lyapunov function. This estimate is potentially larger than those previously obtained by Weissenberger and by Bitsoris and Burgat, especially when the subsystems have Lyapunov functions with strongly-eccentric isoplethic curves. As with the example studied in this paper, the new estimate is considerably larger than that of Weissenberger.

On the behaviour of dynamical systems subject to bounded disturbances

Abstract: For the problem of obtaining bounds on the outputs of a system when the inputs satisfy known constraints, two methods, using impulse responses and Lyapunov functions, ore considered and compared.

Frequency-domain condition for the existence of a Lyapunov functional for the problem of Lurie

Abstract: A sufficient condition is presented for the existence of a Lyapunov functional of the type quadratic form plus integral of the nonlinearity. It is expressed in the frequency domain and takes account of both sector and slope restrictions on the nonlinear element.

Development of the lyapunov functions method in non-linear mechanics

Abstract: We present here some results obtained when developing the method of the Lyapunov function for the Problems leading to the systems which contain a small positive parameter. Stating the Lyapunov-Caplygin problem for the standard systems and the proof of its solvability in case of a simple and perturbed Lyapunov function are given here; we obtained the stability conditions of solutions of the non-linear systems with a non-asymptotically stable generating system. The problem of constructing the Lyapunov functions by means of perturbations is solved in the present paper. An equation with damping of Mate is considered here as an example illustrating the technique of construction of the Lyapunov functions.

Lyapunov-stackelberg estimates for reachable sets

Abstract: A method is presented for estimating reachable sets for nonlinear control systems. The approach combines controllability, abnormality, Lyapunov stability, and Stackelberg optimality. The method is applied to a multiple species fishery model.

Stability of large-scale systems with lagged interconnections

Abstract: The aggregation-decomposition method is used to derive sufficient conditions for the uniform stability, uniform asymptotic stability and exponential stability of the null solution of large-scale systems described by functional differential equations with lags appearing only in the interconnections. The free subsystems are described by ordinary differential equations for which converse theorems involving Lyapunov functions exist and thus enable the sufficient conditions to be expressed in terms of Lyapunov functions rather than the more complicated Lyapunov functionals.

On the absolute stability of continuous-time nonlinear systems

Abstract: Sufficient conditions in frequency domain are derived for the absolute stability of a class of continuous-time nonlinear feedback systems. The results are obtained via Lyapunov's direct method and are presented in a more general setting wherefrom some of the existing stability criteria [1], [2] can readily be obtained.

A Note on the Lyapunov Stability Analysis of a Linear Railway Wheelset

Abstract: SUMMARY In this paper a Lyapunov function is generated for the linearized equations governing a steel wheelset on steel rails. Thus the authors attack the asymmetric problem, and successfully apply Ingwerson's method for constructing Lyapunov functions. The stability criteria applied to this function yield a closed-form expression for the critical forward speed of the wheelset. Thus the authors retain the advantage of Lyapunov's direct method by obtaining an explicit solution for the critical speed. And, although this linearized result has been obtained by other methods, the result is provocative because it suggests that a similar attack on the intractable nonlinear problem (currently being mounted by the authors) may indeed bear fruit.

Construction of a Lyapunov function on a model computer with a large-block mode

Abstract: The article is devoted to the construction of computer software for computing a Lyapunov function satisfying a certain special partial differential equation. The program is written for M-222.