Large-scale dynamical systems: State equations, Lipschitz conditions, and linearization

Abstract

A large-scale dynamical system is given in a block-diagram form. The block diagram is assumed to be composed of blocks (subsystems), each characterized by a state equation and an output equation in the normal form <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\dot{x} =f(x, u, t), y = g(x, u, t)</tex> . Investigations are made on the effect of subsystem characteristics and their interconnections upon properties of the entire system. The first half is concerned with representation of the entire system. Sufficient conditions are given for the entire system to have a representation in the normal form with the state space composed of the direct product of subsystem state spaces. In a linear system these conditions are also necessary. An additional condition is given under which the entire system further satisfies a global Lipschitz condition when each subsystem is also globally Lipschitzian. The second half is concerned with the small-signal behavior of the entire system and a theorem is given which states that if the large linear system resulting from linearization of each subsystem (linearized about a solution of the original nonlinear system) has a state equation, then the large nonlinear system has a normal form in the neighborhood of the solution such that the linearized equation of the normal form is identical with the state equation of the large linear system.