Analytic expressions of transfer function responses and choice of numerator coefficients (zeros)

General analytic expressions of transfer function responses are derived in this paper. The analytic forms include the numerator coefficients of the transfer function, a Vandermonde-like matrix, and a vector containing the transfer function eigenresponses, the latter two only depending on the eigenvalues. Several characteristics of basic transient responses are derived, e.g., their exact number of zeros. An upper bound on the number of extrema of the step response is obtained. An algorithm is suggested for the selection of the numerator coefficients, assuming fixed eigenvalues, using the derived characteristics of the basic transient responses, effectively leading to a suboptimal choice. Furthermore, the analytic formulas are used to calculate the numerator coefficients optimally, by minimizing the step transient, the ramp transient, the parabolic transient, etc. Simulations are presented indicating the properties of the basic transient responses and the transfer function response properties based on optimal and suboptimal choices of the numerator coefficients.