A. N. Michel

Stability analysis of complex dynamical systems

We present two new algorithms to estimate the domain of attraction of the equilibriumx=0 of a nonlinear systemx=f(x). One of these algorithms utilizes quadratic Lyapunov functions while the second algorithm makes use of norm Lyapunov functions. Both of these procedures yield estimates for the domain of attraction which are comparable to those obtained by existing methods; however, the present algorithms appear to be significantly more efficient than existing algorithms.We also show how sometimes the applicability of the above results can be extended to high order systems by invoking the comparison principle. In doing so, we establish some results for the comparison principle which are of interest in their own right. Specifically, we relate the domain of attraction of a low order comparison system to the domain of attraction of a higher order system and we give an interpretation of the comparison principle in terms of stability preserving maps.In order to demonstrate the applicability of the present results, and in order to compare the present results with existing results, several specific examples are presented.

Input-output stability of time-varying nonlinear multiloop feedback systems

Sufficient conditions for the input-output stability (BIBO stability) of time-varying nonlinear multiloop feedback systems are established. The present objective is to analyze large-scale systems in terms of their lower order subsystems (subloops) and in terms of their interconnecting structure. Both time-domain and frequency-domain results are presented. In order to demonstrate the usefulness of the present approach, two specific examples are considered.

On the status of stability of interconnected systems

In the present paper we report on the status of stability of complex dynamical systems. We consider those systems which may be viewed as an interconnection of lower order subsystems. We emphasize those types of results which are phrased in terms of the stability properties of the subsystems and in terms of the qualitative properties of the system interconnecting structure. We consider both Lyapunov stability and input-output stability results. In our presentation, we concentrate on selected sample results and we show how some of these results are related. We then use these sample results to identify modifications, extensions, generalizations, and applications. In this way, we are able to provide a reasonably good picture of the entire subject on hand.

Stability analysis of composite systems

The stability of composite systems is investigated in terms of their subsystems and their interconnecting structure. Whereas previous investigators utilized vector Lyapunov functions in their approach, scalar functions consisting of weighted sums of scalar Lyapunov functions of subsystems of the composite systems are employed presently. It is shown that the scalar Lyapunov function approach may in general yield stability results which are less conservative than those obtained by the vector Lyapunov function approach. Both methods are applied to specific examples considered previously. These examples demonstrate the improvement of the present results.

Stability of discrete systems over a finite interval of time

In many cases of practical interest there is concern with the behaviour of dynamic systems only over a finite time interval. This concern may arise in one of two ways: In one case the system under consideration is defined over a fixed and finite interval of time, while in the second case the system in question is defined for all time; however, the behaviour of the system is of interest only over a finite time interval. Recently, Weiss and Infante (1965, 1967) treated the problem of system stability over a finite time interval for the ease of continuous systems. In this paper a theory is developed which concerns itself with the stability of discrete systems over a finite interval of time. The dynamic systems which are considered are general enough so as to include unforced systems, systems under the influence of perturbing forces, linear systems, non-linear systems, time invariant systems, time-varying systems, simple systems and composite systems. In the present development various definitions of stabilit...

Stability, transient behaviour and trajectory bounds of interconnected systems†

In practice, many systems are complex and of high dimension. Many such systems may be viewed as being composed of several simpler sub-systems which when connected in an appropriate fashion yield the original composite system. The stability, the transient behaviour and estimates for the trajectory bounds of certain composite systems are analysed in terms of their sub-systems. This is accomplished by defining the stability of sub-systems and of composite systems in terms of certain time-varying sub-sets of the state space which are pre-specified in a given problem. After stating definitions of stability for sub-systems which are under the influence of perturbing forces and for composite systems, theorems are stated and proved which yield sufficient conditions for stability. These theorems involve the existence of Lyapunov-like functions which do not possess any particular definiteness requirements on V and [Vdot]. The time-varying sub-sets of the state space which are utilized in the stability definitions a...

Stability analysis of stochastic composite systems

New results for asymptotic stability and exponential stability with probability one of several classes of continuous parameter and discrete parameter stochastic composite systems are established. In all cases the objective is to analyze composite systems in terms of their lower order subsystems and in terms of their interconnecting structure. The results are applied to three specific examples.

Stability of stochastic composite systems

In a recent paper [15] results for the asymptotic stability and exponential stability (with probability one) of a class of continuous parameter stochastic composite systems, with disturbances confined to the sub-system structure, were established. In this short paper these results are extended to allow stability analysis of systems for which stochastic disturbances may not only enter into the subsystem structure but also into the interconnecting structure of composite systems. As in previous related results, the objective is to analyze composite systems in terms of their lower order subsystems and in terms of their interconnecting structure.

Power system transient stability analysis: Formulation as nearly Hamiltonian systems

AbstractIn this paper we formulate power systems as nonlinear nearly Hamiltonian systems. Using the invariance principle for ordinary differential equations, necessary and sufficient conditions for asymptotic stability are established and a new method of estimating the domain of attraction of the stable equilibrium point is developed. The present results constitute a novel approach to stability analysis and involve the following three steps:a.Given a system with dissipation, the stability of its equilibrium is ascertained by determining the stability of the associated conservative system.b.Attractivity of the stable equilibrium of the entire system (with dissipation) is determined from the system topology.c.An estimate of the domain of attraction of the asymptotically stable equilibrium is obtained by making use of results obtained in (a) and (b).nThe stability criterion developed in this paper sheds new light on the mechanism of instability in power systems and it provides analytical verification to the concept of the potential-energy boundary surface (PEBS). The PEBS is a hypersurface which makes up a part of the boundary of the domain of attraction of the stable equilibrium in a power system. The existence and properties of the PEBS have thus far been deduced primarily via simulations and heuristic methods.

Stability analysis of complex dynamical systems: Some computational methods

We present two new algorithms to estimate the domain of attraction of an isolated asymptotically stable equilibrium and we present a new result (using the comparison principle) which can sometimes be used to extend the applicability of these algorithms to high order systems.