Stability Of Large-scale Systems Research Articles

Partial Stability of Large Scale Systems via Comparison Method

In this paper the Comparison Method involving vector Lyapunov functions is applied to study the partial stability properties (i.e. with respect to only a part of the state variables) for large scale systems through the decomposition aggregation approach. Some general theorems are stated, which are based on classic results of the partial stability theory. An application of the results is given for a problem of partial controllability.

On the Decentralized Stabilization of Large Scale Systems with Time Delays

The problem of decentralized stabilization of large-scale interconnected systems with delays using local feedback controllers is investigated. By employing the aggregation technique based on the Lyapunov vector function, sufficient degree of stability for each isolated subsystem are determined in order to garantee the overall system stability. This sufficient condition presents a more general decentralized stability test.

A note on the stability of large scale systems with delays

A stability test is proposed for large scale systems with delays by employing both the aggregation technique based on a Lyapunov function and the strictly quasi-diagonal dominance.

Large-scale dynamic systems: Stability and structure [Book reviews

This monograph presents the state of the art in the theory and applications of large-scale systems via the concept of vector Lyapunov functions in a very readable way and, consequently, should be of immense value to applied mathematical scientists and engineers interested in the analysis and control of complex dynamic systems. The book is written with a distinct purpose of initiating a systematic inquiry into the intricate relationship between complexity, stability, and reliability of large scale dynamic systems. The central new concept of the book is stability under structural perturbations called connecrice srabihy. The concept is of considerable practical interest since in complex processes if is common that the subsystems do not stay together for all time during operation. Furthermore, the knowledge of interconnections among the subsystems is seldom satisfactory due to model uncertainties, imprecise measurements, equipment failures, and the like. A large system is considered connectively stable if it is stable under structural perturbations whereby subsystems are disconnected from each other and again connected together during operation. This type of stability comes often for free when stability is established by vector Lyapunov functions. From the point of liew of Lyapunov theory, connective stability for large scale systems is analogous to absolute stabihty concept for Lure-Popov systems. Both concepts represent strengthenings of the standard global asymptotic stability in order to extract all that is available in the results obtained by using Lyapunov functions. It is, therefore, not surprising that for certain classes of dynamic systems it is possible to derive necessary and sufficient conditions for connective stability. An equally important aspect of the connective stability concept is the fact that stability of this type .can be established for a class of systems by showing stabihty of a single member of that class. a s fa ct may be used as a general recipe for stability analysis of systems with uncertain parameters.

A relaxed dissipativeness test for the stability of large-scale systems

A recently published stability criterion allows checking the stability of a large-scale system whose subsystems are dissipative. (Dissipativeness is a property, akin to, but more general than, passivity.) The criterion requires that a certain test matrix be positive definite. It is now shown that one can sometimes deduce stability when the test matrix is merely nonnegative definite.

The use of simplified models for stability analysis of large-scale systems

Following the general approach called composite system method, the analysis of the stability of large-scale systems is carried out on the basis of a simplified input-output description of each subsystem. Two different cases are considered and investigated. The a priori knowledge is assumed to be for the first case a simplified input-output description of each subsystem, and for the second case a simplified description of the internal structure. The resulting stability criterion is also applicable to subsystems with unstable uncertainties and, therefore, provides an extension of the applicability limits of already known procedures.

Stability of large-scale systems with lagged interconnections

AbstractThe 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.

Stability of Large Scale Systems Via Limiting Equations

When the decomposition-aggregation method is used to analyze the stability properties of large scale systems, it can happen that because of the intrinsic nature of subsystems and/or interconnection structure this analysis becomes very difficult. In these cases the limiting equations approach can be usefully applied. By means of this procedure stability properties of the solutions of limiting equations infer informations about the qualitative behavior of the solutions of the assigned system. In this paper, by adopting this approach, some results concerning the uniform asymptotic stability and the eventual uniform asymptotic stability of large scale systems are given.

Lotka-Volterra-like approach to large-scale systems stability

A new aggregation form of large-scale systems is proposed in this paper. It is of the ‘Lotka-Volterra’ type and appears natural for the generalized Lotka-Volterra large-scale (possibly time-varying) systems, which is shown in the paper. The Lotka-Volterra aggregation form yields a comparison system of the generalized Lotka-Volterra type, which is composed of a linear and a non-linear part. In addition this aggregation form leads to new stability conditions for time-varying large-scale systems, and enables an extension of Weissenberger's results on exponential stability domain estimates to the asymptotic stability domain estimate of time-varying systems. An advantageous feature of the system aggregation and stability analysis developed in the paper is the omission of requirements for (any particular, e.g. exponential) stability of disconnected subsystems.

Asymptotic stability of large-scale systems with application to power systems. Part 1: domain estimation

Stability of nonlinear, stationary, large-scale and continuous-time systems is treated within the general context of the decomposition-aggregation method, and estimates of the asymptotic stability domain are established. The results of the study are intended for transient stability analysis of electrical power systems; they must therefore be the least conservative possible. To meet this general requirement of relaxed stability conditions, the paper discusses stability conditions at the subsystem level as well as estimation of the overall interconnected system.

Asymptotic stability of large-scale systems with application to power systems. Part 2: transient analysis

Based on the general decomposition-aggregation method, a transient stability analysis of power systems is explained. With respect to previous related studies, the proposed improvements are twofold. From a purely theoretical viewpoint, advantage is taken of the relaxation of the stability conditions obtained in the first part of the paper, of the choice of well adapted comparison functions, and of the definition of asymptotic stability domain estimates. From the power systems point of view, use is made of more refined mathematical models than usual; thus, in addition to transfer conductances and electromechanical damping, electromagnetic damping and governor's action are taken into account. A simple numerical example is used for illustrative purposes.

Partial stability of large-scale systems

In this paper a theorem concerning the asymptotic partial stability for large-scale systems is given. By describing high-order systems as collections of lower order interconnected subsystems, the theorem takes the form of conditions on the interconnection structure so that asymptotic partial stability property of isolated subsystems infer the same property of the over-all system. An application is given for a problem of partial controllability in the large.

Stability criteria for large-scale systems

Recent research into large-scale system stability has proceeded via two apparently unrelated approaches. For Lyapunov stability, it is assumed that the system can be broken down into a number of subsystems, and that for each subsystem one can find a Lyapunov function (or something akin to a Lyapunov function). The alternative approach is an input-output approach; stability criteria are derived by assuming that each subsystem has finite gain. The input-output method has also been applied to interconnections of passive and of conic subsystems. This paper attempts to unify many of the previous results, by studying linear interconnections of so-called "dissipative" subsystems. A single matrix condition is given which ensures both input-output stability and Lyapunov stability. The result is then specialized to cover interconnections of some special types of dissipative systems, namely finite gain systems, passive systems, and conic systems.

Stability of large-scale nonlinear systems--Quadratic-order theory of composite-system method using M-matrices

The composite-system method for analyzing stability of large-scale system is studied focusing on the quadratic-order theorems using <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">M</tex> -matrices. Here, by the term "composite-system method", we refer to the method to decompose a large-scale system into smaller subsystems and to make two-step analysis (i.e., first to analyze subsystems and second to combine the results to reduce the property of the whole). Theories about Lyapunov stability and about input-output stability are described from a unified standpoint and their mutual relation is clarified. As an application, multi-input multi-output systems. The contents are generally useful for stability analysis of large-scale nonlinear systems.

Relaxed Large-Scale Systems Stability Analysis Applied to Power Systems

The use of standard scalar Liapunov functions for stability investigations of power systems imposed consideration of very simplified mathematical models. Attempts to get rid of this and some other related drawbacks have led to the decomposition-aggregation method, based on vector Liapunov functions. Within this framework, questions of great importance from an engineering point of view are the relaxation of stability requirements and a suitable estimation of the stability domain, The theoretical part of the present paper is essentially devoted to these questions; they are embedded and solved in the general context of the stability of autonomous nonlinear large-scale systems. The second part of the paper deals with the application of the obtained theoretical results to power system transient stability analysis. The power systems are described here by more refined mathematical models than in all previous related studies. A simple numerical example illustrates the theoretical developments.

Exponential Stabilization of Large-Scale Systems: Decentralized and Multilevel Schemes

Exponential Stabilization of Large-Scale Systems: Decentralized and Multilevel Schemes

Multilevel stabilization of large-scale systems: A spinning flexible spacecraft

Multilevel stabilization schemes for high order linear systems are proposed in the framework of the decomposition-aggregation method used so far for stability analysis of large-scale systems by vector Liapunov functions. In one scheme, the output feedback is applied and the control parameters are determined for low order subsystems to produce a stable aggregate model for the overall system. When system inputs are decentralized such that each subsystem has a distinct input, and states are available for control, another scheme is proposed which has a multilevel structure with local and global controllers. Both stabilization schemes are aimed at the maximization of structural parameters which couple the subsystems. An 11th-order linear model of a spinning flexible spacecraft is stabilized by the proposed decomposition-aggregation method using both the passive and the active stabilization devices. The method can take advantage of the special structural features of the model to provide the best estimate of the stability region for a salient parameter which couples the wobble and the spin motion of the spacecraft.

Practical Stability of Large-Scale Systems

In this paper are given sufficient conditions for practical stability (in the sense of La Salle and Lefschetz) of large-scale systems defined over a finite time interval. Such conditions take the form of existence of vector Lyapunov-like functions whose components may depend, in general, on the whole state vector of the composite system. Sufficient conditions are also established into an aggregate form, in which the trajectory bounds play the role of aggregation variables.

Stabilization of Large-Scale Systems: A Spinning Flexible Spacecraft

Stabilization of high-order constant systems by multilevel control is proposed in the framework of the decomposition-aggregation method for stability analysis of large-scale systems. An 11-order linear model of a spinning flexible spacecraft is stabilized by the method using both the passive and the active stabilization devices. The method can take advantage of the special structural features of the model to provide the best estimate of the stability region for an important parameter which couples the wobble and the spin motion of the spacecraft.

Uniform practical and finite-time stability of large-scale systems

Abstract The purpose of the paper is to develop algebraic conditions for uniform practical stability and uniform finite-time stability of large-scale systems. The stability properties are related to the prespecified time-varying sets, which yield information about trajectory bounds of the systems. The conditions guarantee a stability property of the overall system to be implied by the corresponding stability property of all subsystems. The aggregation-decomposition approach used in the paper reduces the dimensionality of an aggregate matrix of the system to the number of its subsystems. Several examples are presented to illustrate application of the obtained stability criteria.