Section 8.1 Controller Parameterization for General Plants Consider the unity feedback loop,in which the transfer function of the plant is given by G5)= KN+(s)N-(s)e-0s M+(s)M-(s) Assume that G(s)has rp unstable poles and the unstable pole Pj is of lj multiplicity (=1,2,...rp):that is, M:(s)=II(s-pp) 1 Define C(s) Q(s)=1+G(5)C(⑤) which corresponds to the IMC controller 4口，+@，4定4=定0C Zhang.W.D..CRC Press.2011 Version 1.0 5/89
Section 8.1 Controller Parameterization for General Plants Consider the unity feedback loop, in which the transfer function of the plant is given by G(s) = KN+(s)N−(s) M+(s)M−(s) e −θs Assume that G(s) has rp unstable poles and the unstable pole pj is of lj multiplicity (j = 1, 2, ...,rp); that is, M+(s) = Yrp j=1 (s − pj) lj Define Q(s) = C(s) 1 + G(s)C(s) which corresponds to the IMC controller Zhang, W.D., CRC Press, 2011 Version 1.0 5/89
Section 8.1 Controller Parameterization for General Plants The closed-loop system is internally stable,if and only if all elements in the transfer matrix H(s)are stable: H(S)= G(s)Q(s)G(s)[1-G(s)Q(s)] Q(s) -G(s)Q(s) Theorem The unity feedback system with a general plant G(s)is internally stable if and only if ①Q(s)is stable, 2[1-G(s)Q(s)]G(s)is stable. Or equivalently, ①Q(s)is stable, 21-G(s)Q(s)has zeros wherever G(s)has unstable poles, 3All RHP zero-pole cancellations in [1-G(s)Q(s)]G(s)are removed. oac Zhang.W.D..CRC Press.2011 Version 1.0 6/89
Section 8.1 Controller Parameterization for General Plants The closed-loop system is internally stable, if and only if all elements in the transfer matrix H(s) are stable: H(s) = G(s)Q(s) G(s)[1 − G(s)Q(s)] Q(s) −G(s)Q(s) Theorem The unity feedback system with a general plant G(s) is internally stable if and only if 1 Q(s) is stable, 2 [1 − G(s)Q(s)]G(s) is stable. Or equivalently, 1 Q(s) is stable, 2 1 − G(s)Q(s) has zeros wherever G(s) has unstable poles, 3 All RHP zero-pole cancellations in [1 − G(s)Q(s)]G(s) are removed. Zhang, W.D., CRC Press, 2011 Version 1.0 6/89
Section 8.1 Controller Parameterization for General Plants Example This example is used to illustrate that the third condition is necessary. Consider the plant with the transfer function 1 G(s)=g-1 G(s)has one simple RHP pole at s=1.Construct a controller s-1 C(5)=e01se-015-0.1s+0.1)-1 Q(s)corresponding with this C(s)is s-1 Q(5)=eo1se-015-0.15+0.d 4口，+@，4它4生·定0C Zhang.W.D..CRC Press.2011 Version 1.0 7/89
Section 8.1 Controller Parameterization for General Plants Example This example is used to illustrate that the third condition is necessary. Consider the plant with the transfer function G(s) = 1 s − 1 G(s) has one simple RHP pole at s = 1. Construct a controller C(s) = s − 1 e 0.1s (e−0.1s − 0.1s + 0.1) − 1 Q(s) corresponding with this C(s) is Q(s) = s − 1 e 0.1s (e−0.1s − 0.1s + 0.1) Zhang, W.D., CRC Press, 2011 Version 1.0 7/89
Section 8.1 Controller Parameterization for General Plants Example (ctd.1) Q(s)is stable.The first condition is satisfied.Furthermore, e0.1(e-0.1s-0.1s+0.1)-1 1-G(s)Q(6)=e0is(e-015-0.1s+0.1 It has zeros where G(s)has unstable poles.The second condition is also satisfied. However,the closed-loop system is internally unstable,because there exists a RHP zero-pole cancellation in [1-G(s)Q(s)]G(s). which cannot be removed Remark 1:The case associated with the third condition occurs only in the system where the plant or the controller contains a time delay.If both the plant and the controller are rational,it is not necessary to consider the third condition 4口，+@4定4定 Zhang.W.D..CRC Press.2011 Version 1.0 8/89
Section 8.1 Controller Parameterization for General Plants Example (ctd.1) Q(s) is stable. The first condition is satisfied. Furthermore, 1 − G(s)Q(s) = e 0.1s (e −0.1 s − 0.1s + 0.1) − 1 e 0.1s (e−0.1s − 0.1s + 0.1) It has zeros where G(s) has unstable poles. The second condition is also satisfied. However, the closed-loop system is internally unstable, because there exists a RHP zero-pole cancellation in [1 − G(s)Q(s)]G(s), which cannot be removed Remark 1: The case associated with the third condition occurs only in the system where the plant or the controller contains a time delay. If both the plant and the controller are rational, it is not necessary to consider the third condition Zhang, W.D., CRC Press, 2011 Version 1.0 8/89
Section 8.1 Controller Parameterization for General Plants Remark 2:In control system design,G(s)Q(s)is always stable. Since [1-G(s)Q(s)]G(s)=C-1(s)Q(s)G(s),the third condition can be achieved by removing the RHP zero-pole cancellation in C(s)through rational approximations Theorem All controllers that make the unity feedback control system internally stable can be parameterized as Q(s) C(s)=1-G(s)Q(S) where Q(s)= Q(s)M+(s) K 4口，+@，4定4=定0C Zhang.W.D..CRC Press.2011 Version 1.0 9/89
Section 8.1 Controller Parameterization for General Plants Remark 2: In control system design, G(s)Q(s) is always stable. Since [1 − G(s)Q(s)]G(s) = C −1 (s)Q(s)G(s), the third condition can be achieved by removing the RHP zero-pole cancellation in C(s) through rational approximations Theorem All controllers that make the unity feedback control system internally stable can be parameterized as C(s) = Q(s) 1 − G(s)Q(s) where Q(s) = Q1(s)M+(s) K Zhang, W.D., CRC Press, 2011 Version 1.0 9/89