Linearization

Abstract

Linearization is commonly used to analyze a nonlinear system for small departures from an operating point. Its stability and transient nature can then be quantified for comparison with the system’s requirements. When the perturbations are small enough, their products and powers are arguably negligible, so discarding them reduces the system dynamics to linear equations in the perturbed variables from which the eigenvalues may be found and the motion described. The legacy of small perturbations is just that; their smallness must be assured as otherwise the validity of the approximate linear equations degenerates rapidly, leaving no precise measure of a neighborhood of acceptable solutions. The nature of the nonlinearity plays an important role in determining if there is a neighborhood of validity. Continuous single-valued functions, such as the torque-speed curve of an ac induction motor or the pitching moment curve of an airplane, are typically good applications. Discontinuous functions which are not single-valued, such as gear train backlash or relays, are never the objects of small perturbation analysis. In general, a function expandable in a Taylor series is a candidate for small perturbation analysis, although success requires the truncation error after two terms to be acceptably small. Since each succeeding term in the series contains the corresponding derivative, those functions having discontinuities will have extremely large values in their higher derivatives and, accordingly, their truncation errors cannot be ignored.