Optimum synthesis of a class of multiple-loop feedback systems

Realization of arbitrary transfer functions by a special class of multiple-loop feedback configuration is investigated. The th-order system consists of nominally identical single-pole active stages with arbitrary but constant interconnections. The constraints on the sensitivity functions with respect to the active stages of such a system are obtained. is shown that the effect of identical perturbations of the gains of active stages, such as those which may be expected due to environmental variations, is invariant for a given transfer function, and hence can be accurately predicted. Subject to these constraints, conditions are derived for the minimum of a suitably defined multiparameter sensitivity index, which may then be used to minimize the effect of mutually independent random perturbations of the gains, such as those expected due to manufacturing tolerances. For second-order systems, the optimal design is obtained analytically. For higher order systems, a general optimization scheme, employing steepest descent from an initial design, is outlined. The optimal design of a fourth-order stagger-tuned bandpass filter is presented as an illustration.