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 5/79
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 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
Section 7.1 The Feature of Integrating Systems Proof. Necessity is obvious.Consider sufficiency.Assume that the two conditions hold.It remains to show that G(s)Q(s)is stable.If G(s)Q(s)is unstable,1-G(s)Q(s)is unstable,which implies that [1-G(s)Q(s)]G(s)must be unstable.This contradicts the assumption. The conclusion may not be applicable to other structures. Consider the IMC structure shown in Figure 4口，+回，424生，定9QC Zhang.W.D..CRC Press.2011 Version 1.0 6/79
Section 7.1 The Feature of Integrating Systems Proof. Necessity is obvious. Consider sufficiency. Assume that the two conditions hold. It remains to show that G(s)Q(s) is stable. If G(s)Q(s) is unstable, 1 − G(s)Q(s) is unstable, which implies that [1 − G(s)Q(s)]G(s) must be unstable. This contradicts the assumption. The conclusion may not be applicable to other structures. Consider the IMC structure shown in Figure Zhang, W.D., CRC Press, 2011 Version 1.0 6/79
Section 7.1 The Feature of Integrating Systems When the model is exact,the system is open-loop for G(s)and Q(s).Since G(s)is unstable and G(s)Q(s)is stable,there must exist closed RHP zero-pole cancellation between G(s)and Q(s). In this case,the closed-loop system is not internally stable Consequently,the IMC structure cannot be used for the control of integrating plants 4口，+@，4定4定90C Zhang.W.D..CRC Press.2011 Version 1.0 7/79
Section 7.1 The Feature of Integrating Systems When the model is exact, the system is open-loop for G(s) and Q(s). Since G(s) is unstable and G(s)Q(s) is stable, there must exist closed RHP zero-pole cancellation between G(s) and Q(s). In this case, the closed-loop system is not internally stable Consequently, the IMC structure cannot be used for the control of integrating plants Zhang, W.D., CRC Press, 2011 Version 1.0 7/79
Section 7.1 The Feature of Integrating Systems Steady-state Performance Consider the first-order integrating plant: G(s)-Ke-0s where K is the gain,0 is the time delay.Assume that the disturbance at the plant input is d'(s)=1/s.The effect of d'(s) on the system output can be equivalent to that of a disturbance d(s)at the plant output: 同=d)G=。 K It is seen that the system is in fact of Type 2.Only when the controller is designed for ramps,can the steady-state error caused by d'(s)vanish asymptotically 定9aC Zhang,W.D..CRC Press.2011 Version 1.0 8/79
Section 7.1 The Feature of Integrating Systems Steady-state Performance Consider the first-order integrating plant: G(s) = K s e −θs where K is the gain, θ is the time delay. Assume that the disturbance at the plant input is d 0 (s) = 1/s. The effect of d 0 (s) on the system output can be equivalent to that of a disturbance d(s) at the plant output: d(s) = d 0 (s)G(s) = K s 2 e −θs It is seen that the system is in fact of Type 2. Only when the controller is designed for ramps, can the steady-state error caused by d 0 (s) vanish asymptotically Zhang, W.D., CRC Press, 2011 Version 1.0 8/79
Section 7.1 The Feature of Integrating Systems In general,if the plant has m poles at the origin,the system should be of Type m+1 for asymptotic tracking;or equivalently,the controller has to satisfy lim →0 1-G(s)Q(5=0,k=0,1,,m sk or 0dsx1-G(s)Q(s=0,k=0,1,m lim This conclusion is very important in the design of systems with integrating plants Derivatives of a function are frequently calculated in the design of systems with integrating plants.To avoid complicated computation,two algebra results are given here 4口，+@，4定4=定0C Zhang.W.D..CRC Press.2011 Version 1.0 9/79
Section 7.1 The Feature of Integrating Systems In general, if the plant has m poles at the origin, the system should be of Type m + 1 for asymptotic tracking; or equivalently, the controller has to satisfy lim s→0 1 − G(s)Q(s) s k = 0, k = 0, 1, ..., m or lim s→0 d k dsk [1 − G(s)Q(s)] = 0, k = 0, 1, ..., m This conclusion is very important in the design of systems with integrating plants Derivatives of a function are frequently calculated in the design of systems with integrating plants. To avoid complicated computation, two algebra results are given here Zhang, W.D., CRC Press, 2011 Version 1.0 9/79