Section 7.1 The Feature of Integrating Systems The transfer function Q(s)is in fact the IMC controller.Then the transfer matrix H(s)becomes H(s) = G(s)Q(s)[1-G(s)Q(s)]G(s) Q(s) -G(s)Q(s) Since G(s)is not stable,the stability of Q(s)cannot guarantee the stability of the closed-loop system. Theorem Assume that G(s)is an integrating plant.The unity feedback loop shown in Figure is internally stable if and only if ①Q(s)is stable [1-G(s)Q(s)]G(s)is stable 4口+@4定4生，定00 Zhang.W.D..CRC Press.2011 Version 1.0 5/79Section 7.1 The Feature of Integrating Systems The transfer function Q(s) is in fact the IMC controller. Then the transfer matrix H(s) becomes H(s) =  G(s)Q(s) [1 − G(s)Q(s)]G(s) Q(s) −G(s)Q(s)  Since G(s) is not stable, the stability of Q(s) cannot guarantee the stability of the closed-loop system. Theorem Assume that G(s) is an integrating plant. The unity feedback loop shown in Figure is internally stable if and only if 1 Q(s) is stable. 2 [1 − G(s)Q(s)]G(s) is stable. Zhang, W.D., CRC Press, 2011 Version 1.0 5/79