Resonance peaks in third-order type 1 control systems

It is well known that a stable second-order system with transfer function G/(s2 + 2¿¿ns + ¿n2) exhibits a peak in the modulus of the frequency-response locus for ¿ < 1/¿2. Conditions for resonance peaks in third-order systems with open-loop transfer function F(s)= G/s(s2 + 2¿¿ns + ¿n2) and closed-loop transfer function G(s) = F(s)/1 + F(s) are less well known. In this paper the resonance-peak conditions for the open- and closed-loop systems are presented and inequalities are given which enable such conditions to be determined.