Modeling of Nonlinear Dynamic Systems with Volterra Polynomials

Abstract

The paper presents a review of the studies that were conducted at Energy Systems Institute (ESI) SB RAS in the field of mathematical modeling of nonlinear input-output dynamic systems with Volterra polynomials. The first part presents an original approach to identification of the Volterra kernels. The approach is based on setting special multi-parameter families of piecewise constant test input signals. It also includes a description of the respective software; presents illustrative calculations on the example of a reference dynamic system as well as results of computer modeling of real heat exchange processes. The second part of the review is devoted to the Volterra polynomial equations of the first kind. Studies of such equations were pioneered and have been carried out in the past decade by the laboratory of ill-posed problems at ESI SB RAS. A special focus in the paper is made on the importance of the Lambert function for the theory of these equations.

Integral Equations Related to Volterra Series and Inverse Problems: Elements of Theory and Applications in Heat Power Engineering

“…The identification of the mesh analogs of Volterra kernels tuned to test signals with amplitudes of 25% of the initial values of D 0 = 0.16 kg/s, Q 0 = 100 kW was carried out using the technique [42,44] for T = 40 s, h = 1 s. Figure 3 shows mesh analogs for K 1 , K 2 , K 11 characterizing the influence of ∆D(t) and ∆Q(t) on the change of ∆i(t). Kernels were identified on the discrete mesh with the step h = 1 using responses ∆i(t) to test signals of the form Equation ( 5) with amplitudes equal to 25% of the initial values.…”

“…The technique of using scalar input signals in Equation ( 5) is described in detail in the review [42] (see the section "About one approach to identification of Volterra kernels"). For the case of a signal represented as a vector function of time, the series of articles [36,39,[42][43][44] presents algorithms for choosing the amplitudes of test signals for identifying Volterra kernels, which, in general, can be represented as follows:…”

“…Experimental debugging. Determination of the optimal values of the amplitudes based on the solution of some extremal problems introduced in [42,43].…”

“…The results presented in [42] on the identifying Volterra kernels in integral models Equations ( 1) and ( 3) for the scalar case x(t)…”

Application of a Volterra quadratic polynomial to modeling elements of heat engineering devices

“…The use of the Volterra polynomials in thermal power engineering was demonstrated in [10] for solving the problem of creating a method for the dynamic calculation of boiler units. And in [11,12], using the example of a heat exchanger model with independent heat supply, represented as an object with lumped parameters, the authors showed that an integral model based on a quadratic segment of the Volterra series can be employed to fairly accurately and quickly describe nonlinear transient processes in the heat exchanger. A detailed description of this object is presented in [13].…”