Definition
The higher-order sinusoidal input describing functions (HOSIDF) were first introduced b
dr. ir. P.W.J.M. Nuij The HOSIDFs are an extension of the sinusoidal input
describing function In control systems theory, the describing function (DF) method, developed by Nikolay Mitrofanovich Krylov and Nikolay Bogoliubov in the 1930s, and extended by Ralph Kochenburger is an approximate procedure for analyzing certain nonlinear control ...
[Gelb, A., and W. E. Vander Velde: Multiple-Input Describing Functions and Nonlinear System Design, McGraw Hill, 1968.] which describe the response (
gain and
phase) of a system at harmonics of the base frequency of a sinusoidal input signal. The HOSIDFs bear an intuitive resemblance to the classical
frequency response function and define the periodic output of a stable,
causal
Causality (also referred to as causation, or cause and effect) is influence by which one event, process, state, or object (''a'' ''cause'') contributes to the production of another event, process, state, or object (an ''effect'') where the ca ...
,
time invariant nonlinear system to a
sinusoidal input signal:
This output is denoted by
and consists of harmonics of the input frequency:
Defining the single sided spectra of the input and output as
and
, such that
yields the definition of the k-th order HOSIDF:
Advantages and applications
The application and analysis of the HOSIDFs is advantageous both when a nonlinear model is already identified and when no model is known yet. In the latter case the HOSIDFs require little model assumptions and can easily be identified while requiring no advanced mathematical tools. Moreover, even when a model is already identified, the analysis of the HOSIDFs often yields significant advantages over the use of the identified nonlinear model. First of all, the HOSIDFs are intuitive in their identification and interpretation while other nonlinear model structures often yield limited direct information about the behavior of the system in practice. Furthermore, the HOSIDFs provide a natural extension of the widely used sinusoidal describing functions in case nonlinearities cannot be neglected. In practice the HOSIDFs have two distinct applications: Due to their ease of identification, HOSIDFs provide a tool to provide on-site testing during system design. Finally, the application of HOSIDFs to (nonlinear) controller design for nonlinear systems is shown to yield significant advantages over conventional time domain based tuning.
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Electrical engineering
Control theory
Signal processing