The Application of Lyapunov’s Second Method to Interconnected Systems

As many engineering systems are made up of an interconnection of simple subsystems, it is natural to attempt to utilize this interconnection structure in developing system analysis techniques. In this paper interconnection information is used in conjunction with properties of the individual subsystems to obtain sufficient conditions for asymptotic stability-in-the-large. This method may be applied to a broad class of linear, nonlinear and time-varying interconnected systems as indicated in the following three steps. First, the Lyapun-ov functions and comparison equations are found for the individual subsystems. Second, the comparison equations are interconnected, following the interconnections in the original system, into a system of comparison equations. This system is linear, with constant coefficients and of order equal to the number of subsystems in the original interconnected system. Finally, the stability of the null solution of this auxiliary system is examined. If it is asymptotically stable, then the null solution of the interconnected system is asympotically stable-in-the-large. An example illustrates the use of interconnection information in determining the sufficient conditions for asymptotic stability-in-the-large for a ninth-order interconnected system.