Applications of nonlinear Volterra equations of the first kind to the control problem for heat exchange dynamics

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 identification of the mesh analogs of Volterra kernels tuned to test signals with amplitudes of 25% of the initial values of 0 0.16 D  kg/s, 0 100 Q  kW was carried out using the technique [42,44] for Additionally, when drafting the initial data for the identification of kernels, we take into account conditions of the form of Equations ( 8)-( 11) following from the necessary and sufficient conditions for the solvability of the corresponding integral equations in the required functions' classes. Note that in this case, we do not consider conditions 25 Q   kW, on which we will carry out the verification of mesh analogs of Equations ( 12) and (13).…”

“…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:…”

“…Numerical methods for solving polynomial equations for N = 2 based on cubature formulas for middle rectangles are constructed in [47]. Further research was aimed at developing the theory and numerical methods for solving systems of polynomial Equation (1) for p = 2, N = 2, 3 based on the Newton-Kantorovich method [53][54][55], as well as the practical method of using the developed approaches as applied to Equation (3) in the problem of stabilizing the enthalpy ∆i(t) by the formation of the control input action ∆D(t) [42,44,56].…”

Modeling of Nonlinear Dynamic Systems with Volterra Polynomials

“…In this case equation ( 73) is a polynomial Volterra equation of the first kind, and its continuous solution is of a local character. In (Solodusha, 2011b) consideration is given to a numerical scheme of solving the polynomial equation ( 73) for 𝑝 = 2 provided there is no feedback. Extending the research started in (Solodusha, 2011c) we will consider an algorithm for obtaining the control action 𝑥 1 (𝑡) taking into account a posteriori data about deviation of the output variable 𝑦(𝑡) from the sought value 𝑦 * , so that 𝑥 1 (𝑡) = 𝑢(𝑡 − ), 𝑢(𝜉) = 0, 𝜉 ∈ [−, 0],is a known constant delay.…”