Generation of a Class of Equivalent Networks and its Sensitivities
B. Leon, Charles F. Yokomoto · IEEE Transactions on Circuit Theory · 1972
This paper presents a method of generating a class of equivalent networks from an initial network\eta^{(0)}such that a transfer functionH(p)kept invariant throughout the transformation. The class of networks to be considered is chosen so that the sensitivities ofH(p)to changes in the network elements of the initial network given byS_{e_{k}}H(p)= \frac{\partial H (p)}{\partial e_{k}}\frac{e_{k}}{H(p)}and the sensitivities of all of the generated equivalent networks can be obtained by simple matrix multiplication. Partial differentiation is avoided. The equivalent networks are generated from\eta^{(0)}by transforming the vector of network variables and the input vector in the equilibrium equations defined below. The transformation is performed in such a manner that the equivalent networks\eta^{(1)}are generated by congruence, transforming the matricesMandNthat appear in the system of equationsM \dot{x} = -Nx + bu_{in}w = d^{t}x. The single-output transfer functionH(p)is given byd^{t} (pM + N)^{-1} b. Provided that the equations possess specified properties, the sensitivities are easily obtained and can be applied to the problem of sensitivity minimization. Furthermore, if\phiis the sum\sum |S_{e_{k}}|^{2}, then the partial derivatives of\phiwith respect to the transformation parameters are readily obtained. Again only matrix multiplication is required.