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X H n ðjω1 , Á Á Á, jωn Þ all the permutations of f1; 2; . . 4), asymmetric GFRFs suffice. 3 The GFRFs for a Single Input Double Output Nonlinear System In many practical cases, nonlinear system models are usually described by a nonlinear state equation with a general nonlinear output function of system states. Sometimes, the output function of interest can also be a nonlinear objective function to optimize. Therefore, the computation of the GFRFs for nonlinear systems in this form would be more relevant in practice.

4 Nonlinear Effect in Each Frequency Generation Period Fig. 1 (Jing et al.

There are also three categories of nonlinearities in terms of input u(t) and output x(t) (similar to those from input u(t) to output x(t)), and there is one linear output y(t). Note that there are no nonlinearities in terms of y(t), and all the nonlinearities come from input u(t) and output x(t). For this reason, the GFRFs from u(t) to y(t) are dependent on the GFRFs from u(t) to x(t). 28b) have no relationship with y(t) but x(t). 28a,b). Similar results can be obtained for the following SIDO NDE system M1 X m K X X cp, mÀp ðk1 ; Á Á Á; km Þ m¼1 p¼0 k1 , km ¼0 M2 X m K X X m¼1 p¼0 k1 , km ¼0 p m Y dki xðtÞ Y d ki uðtÞ ¼0 dtki i¼pþ1 dtki i¼1 p m Y d ki xðtÞ Y d ki uðtÞ e ¼ yðtÞ c p, mÀp ðk1 ; Á Á Á; km Þ dtki i¼pþ1 dtki i¼1 ð2:34aÞ ð2:34bÞ where x(t), y(t), u(t) 2 ℝ.

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Simulation, Analysis of GMODS - Processing Pu Surplus Mtls


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