Showing papers on "Lyapunov equation published in 1973"

Asymptotic stability and instability of large-scale systems

Abstract: The purpose of this paper is to develop new methods for constructing vector Lyapunov functions and broaden the application of Lyapunov's theory to stability analysis of large-scale dynamic systems. The application, so far limited by the assumption that the large-scale systems are composed of exponentially stable subsystems, is extended via the general concept of comparison functions to systems which can be decomposed into asymptotically stable subsystems. Asymptotic stability of the composite system is tested by a simple algebraic criterion. By redefining interconnection functions among the subsystems according to interconnection matrices, the same mathematical machinery can be used to determine connective asymptotic stability of large-scale systems under arbitrary structural perturbations. With minor technical adjustments, the theory is broadened to include considerations of unstable subsystems as well as instability of composite systems.

The circle criterion and quadratic Lyapunov functions for stability analysis

Abstract: In this note the general equivalence is pointed out between the stability results obtained by means of the circle criterion and those derived from an optimum quadratic Lyapunov function.

Mean square stability criteria for stochastic feedback systems

Abstract: In this paper several stability criteria are obtained for a class of stochastic feedback systems containing a multiplicative white noise element ; continuous time as well as discrete time systems are dealt with. Mean square stability properties are derived by means of Lyapunov theory ; the Lyapunov functions are generated by means of the Lyapunov equation, the path integral technique, and the Kalman-Yacoboviteh lemma. Results similar to the Routh-Hurtwitz conditions and the circle criteria for deterministic feedback systems are discussed. Different interpretations of the white noise element (Ito, Stratonovitch, etc.) are considered.

On Lyapunov functions for power systems with transfer conductances

Abstract: The problem of a two-machine power system for stability studies is formulated, taking into consideration the transfer conductances in the transmission network. Lyapunov function is then obtained using a generalization of Popov's criterion due to Anderson.

Lyapunov methods and equations of parabolic type

Abstract: In this paper we describe the use of Lyapunov's second method to study the stability and instability of equilibrium solutions of nonlinear parabolic equations. First~ we shall define Lyapunov functionals in a manner which is appropriate for evolution equations on a Banach space (and which is different from the usual definitions for ordinary differential equations). In the second section these Lyapunov functionals are used to obtain results on the stability of solutions of evolution equations. The third section is a study of certain classes of nonlinear parabolic equations. For these equations Lyapunov methods provide a rigorous justification of certain "energy principles." In particular, we shall show that the equilibrium solutions which minimize a certain functional are stable and that many other equilibrium solutions are unstable. Some results on the use of Lyapunov methods for nonlinear partial differential equations have already been obtained by Zubov (9) and Chafee and Infante (3). Lyapunov methods have been used for nonlinear parabolic equations in chemical engineering: see Aris (2) and the references therein.

Lyapunov functions and autonomous differential equations in a Banach space

Abstract: where A is a continuous function from E into E. In particular, sufficient conditions are established to ensure that (1) has a unique critical point which is globally asymptotically stable and, with additional conditions on A, an iterative method is developed which converges to this critical point. The main purpose of this paper is to point out that many of the techniques used in the theory of monotone operators can be applied in a more general situation (see, for example, Hartman [8]). Instead of using the norm on E to study the solutions to (1), we assume the existence of a function V from E× E into [0, or) which has the following basic properties:

Lyapunov Theory and Perturbations of Differential Equations

Abstract: In this paper we discuss qualitative properties of the solution of systems of ordinary differential equations and perturbations of such systems in the event a Lyapunov function is known whose derivative along solutions of the system satisfies a strong negative definite condition. Boundedness and stability of sets are discussed along with the observation that a Lyapunov function with a strongly negative definite derivative must be positive definite and radially unbounded. These results are used to discuss certain types of perturbations of systems of differential equations. Several examples are given to illustrate the main results.

A synthesis of Lyapunov functions for non-linear time-varying control systems

Abstract: The formulation of a class of Lyapunov functions for time-varying nonlinear control systems which occur in the stability analysis of control systems is considered. A new approach is presented, which is an extension of a generation technique for time-invariant non-linear systems. This approach, which uses by analogy the classical theory of Hamilton, permits stability and transient information to be obtained from system equations with time-varying damping as well as time-varying gain. A second and third-order example is used to illustrate the application of this new result.

On the stability of symmetric nonlinear output-coupled multivariable systems

Abstract: A Lyapunov function together with a modified Meyer-Kalman-Yakubovitch lemma are used to develop absolute stability conditions for autonomous symmetric nonlinear output-coupled multivariable systems. The nonlinear characteristics are restricted in both sector and slope. As an example, a coupled-core nuclear reactor with typical parameters is considered to illustrate the concepts developed in this correspondence.

Comments on "Lyapunov functions for quadratic differential equations with applications to adaptive control"

Abstract: The adaptation laws existing in known model reference systems are shown to be unfavorably determined by the underlying basic structure. Simulation results are also given, which demonstrate that the adaptivity itself can be limited, if not missing, when the chosen structure implies stability domain restrictions.

Stochastic Lyapunov stability of the phase-locked loop†

Abstract: In this paper stability bounds for the stochastic phase-locked loop are studied, using the second method of Lyapunov. The fundamental configurations for the phase-locked loop and the related equations describing loop operation are given. The principles necessary to establish the idea of a stochastic Lyapunov function are presented, with special emphasis on the Ito differential form. An examination of the stochastic stability of a second-order phase-locked loop is carried out qualitatively. The stability analysis technique employed in this paper yields probabilistic data for cycle-slipping in the phase-locked loop, and thus represents a significant departure from the traditional approach based on Fokker—Planck techniques. The concepts presented, however, are applicable to a much wider variety of control system stochastic stability problems.