Transient response control via characteristic ratio assignment

This note develops an approach to directly control the transient response of linear time-invariant control systems. We begin by considering all-pole transfer functions of order n for which we introduce a set of parameters α_i, i=1,...n called the characteristic ratios. We also introduce a generalized time constant τ. We prove that α₁ and τ can be used to characterize the system overshoot to a step input and the speed of response, respectively. By independently adjusting α₁ and τ in all-pole systems, arbitrarily small or no overshoot as well as arbitrarily fast speed of response can be achieved. These formulas are used to develop a procedure to design feedback controllers with feedforward or two parameter output feedback type for achieving time response specifications. For a minimum phase plant we show that arbitrary transient response specifications, namely one with independently specified overshoot and specified rise time or speed of response can be exactly attained.