An algorithm for modeling the sinusoidal input/steady-state response behavior of nonlinear systems over a set of frequencies and amplitudes

An algorithm for constructing a black box model of the sinusoidal input/steady-state response behavior of nonlinear time-invariant systems over a set of frequencies and amplitudes is presented. It is assumed that the steady-state response is periodic of the same fundamental frequency as the excitation, and that the Fourier coefficients are continuous functions of amplitude and square-integrable functions of frequency. The algorithm converges, in a mean-square sense, to an exact representation of the first N harmonics of the steady-state response minus its d.c. component. The model constructed by the algorithm admits a relatively simple physical realization characterized by 2NM+1 linear dynamic elements, and N(2M+1)+1 nonlinear static elements. The underlying mathematical structure of the model is an orthogonal series expansion relative to time whose coefficients are themselves truncated orthogonal expansions relative to frequency. Here M, the number of harmonics used for frequency interpolation, is determined by the algorithm. Of the N(2M+1)+1 memoryless nonlinearities which characterize the model, N of these are specified ahead of time (Tchebysheff polynomials), and 2NM+1 are parameters which mold the representation to the specific system being modeled. Each of these functions of a single variable can be obtained in a pointwise manner directly from steady-state measurements. The algorithm was implemented on a digital computer, and forced versions of the classic equations of van der Pol and Duffing were run as examples. An additional analytic example of a frequency multiplier of prescribed bandwidth was also presented.