Autonomous second-order nonlinear systems and weighted linearization: Under what conditions are the inherent specifications preserved?

Abstract

Recently, an interesting idea on generalizing the traditional linearization technique by weighting integration of the Jacobian matrix over the state space has been introduced, and its usefulness has been verified in control system design. Noticing the applicability of linearized models in studying the properties of original systems in the literature relevant to nonlinear systems, this paper aims to investigate the aforementioned generalized linearization technique in the viewpoint of preserving the specifications of an autonomous second-order nonlinear system. To this end, by considering a general form for the weight function possessing two adjustable parameters, the regions in the space of tunable parameters of weight function in which the linearized model preserves the specifications of the original nonlinear system are analytically specified. The specifications, whose preservation is investigated, include stability/instability status of the system equilibrium point, the type of this equilibrium point, eigenvalue location of the corresponding Jacobian matrix, and quadratic Lyapunov function corresponding to a stable equilibrium point. The analytical achievements of the paper are verified by various numerical examples.