Section 8.1 Controller Parameterization for General Plants Theorem(ctd.1) Q1(s)is any stable transfer function that makes Q(s)proper and satisfies piCoe lim M-(S) =0,k=0,1,,5-1 and all RHP zero-pole cancellations in [1-G(s)Q(s)]G(s)are removed Proof. To guarantee the internal stability of the closed-loop system,first, Q(s)should be stable.This implies that Q(s)should be proper and Q1(s)should be stable. Second,[1-G(s)Q(s)]G(s)should be stable.This condition has three implications: 4口，+@4定4生 定9QC Zhang.W.D..CRC Press.2011 Version 1.0 10/89
Section 8.1 Controller Parameterization for General Plants Theorem (ctd.1) Q1(s) is any stable transfer function that makes Q(s) proper and satisfies lim s→pj d k dsk 1 − Q1(s)N+(s)N−(s)e −θs M−(s) = 0, k = 0, 1, ..., lj − 1 and all RHP zero-pole cancellations in [1 − G(s)Q(s)]G(s) are removed Proof. To guarantee the internal stability of the closed-loop system, first, Q(s) should be stable. This implies that Q(s) should be proper and Q1(s) should be stable. Second, [1 − G(s)Q(s)]G(s) should be stable. This condition has three implications: Zhang, W.D., CRC Press, 2011 Version 1.0 10/89
Section 8.1 Controller Parameterization for General Plants Proof ctd.1. Q(s)must cancel all RHP poles of G(s),1-G(s)Q(s)must cancel all RHP poles of G(s),and all RHP zero-pole cancellations in [1-G(s)Q(s)]G(s)are removed.All stable transfer functions that have zeros wherever G(s)has RHP poles can be expressed as Q)=Q(sM-(的 K where Q1(s)is a stable transfer function that makes Q(s)proper. It follows that 1-G(s)Q(s)=1-Q(s)W+(s)W-(s)e-s M-(⑤) 4口，+@，4定4定90C Zhang.W.D..CRC Press.2011 Version 1.0 11/89
Section 8.1 Controller Parameterization for General Plants Proof ctd.1. Q(s) must cancel all RHP poles of G(s), 1 − G(s)Q(s) must cancel all RHP poles of G(s), and all RHP zero-pole cancellations in [1 − G(s)Q(s)]G(s) are removed. All stable transfer functions that have zeros wherever G(s) has RHP poles can be expressed as Q(s) = Q1(s)M+(s) K where Q1(s) is a stable transfer function that makes Q(s) proper. It follows that 1 − G(s)Q(s) = 1 − Q1(s)N+(s)N−(s)e −θs M−(s) Zhang, W.D., CRC Press, 2011 Version 1.0 11/89
Section 8.1 Controller Parameterization for General Plants Proof ctd.2. That 1-G(s)Q(s)has zeros wherever G(s)has RHP poles is equivalent to dk lim dsk 1- Q(s)N-(s)N_(s)e-0s S→s M-(S) =0,k=0,1,…,-1 Corollary Assume that G(s)is a stable plant.That is,M(s)=1.All controllers that make the unity feedback control system internally stable can be parameterized as Q(s) C(s)=1-G(5)QS) where Q(s)is any stable transfer function. 2ac Zhang.W.D..CRC Press.2011 Version 1.0 12/89
Section 8.1 Controller Parameterization for General Plants Proof ctd.2. That 1 − G(s)Q(s) has zeros wherever G(s) has RHP poles is equivalent to lim s→sj d k dsk 1 − Q1(s)N+(s)N−(s)e −θs M−(s) = 0, k = 0, 1, ..., lj − 1 Corollary Assume that G(s) is a stable plant. That is, M+(s) = 1. 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) is any stable transfer function. Zhang, W.D., CRC Press, 2011 Version 1.0 12/89
Section 8.1 Controller Parameterization for General Plants Example Consider a plant with the transfer function s-2 G(6)=5-1)05+2 The plant has only one simple unstable pole at s=1.Then Q(s)=(s-1)Q1(s) where Q1(s)is a stable transfer function satisfying 1-0(+引=0 This is equivalent to Q1(s)=-3+(s-1)Q2(s) Zhang.W.D..CRC Press.2011 Version 1.0 13/89
Section 8.1 Controller Parameterization for General Plants Example Consider a plant with the transfer function G(s) = s − 2 (s − 1)(s + 2) The plant has only one simple unstable pole at s = 1. Then Q(s) = (s − 1)Q1(s) where Q1(s) is a stable transfer function satisfying lim s→1 1 − Q1(s) s − 2 s + 2 = 0 This is equivalent to Q1(s) = −3 + (s − 1)Q2(s) Zhang, W.D., CRC Press, 2011 Version 1.0 13/89
Section 8.1 Controller Parameterization for General Plants Example (ctd.1) where Q2(s)is any stable transfer function that makes Q(s) proper.All controllers that make the unity feedback system internally stable can be parameterized as C(5)= (s-1)(s+2)[-3+(s-1)Q2(s1 (s+2)-(s-2-3+(s-1)Q2(s] Example Consider the stabilizing problem of the plant 1 G(5)=6-1)(5-2 which has unstable poles at s=1 and s=2,respectively.Then Q(s)=(s-1)(s-2)Q1(s) 24c Zhang.W.D..CRC Press.2011 Version 1.0 14/89
Section 8.1 Controller Parameterization for General Plants Example (ctd.1) where Q2(s) is any stable transfer function that makes Q(s) proper. All controllers that make the unity feedback system internally stable can be parameterized as C(s) = (s − 1)(s + 2)[−3 + (s − 1)Q2(s)] (s + 2) − (s − 2)[−3 + (s − 1)Q2(s)] Example Consider the stabilizing problem of the plant G(s) = 1 (s − 1)(s − 2) which has unstable poles at s = 1 and s = 2, respectively. Then Q(s) = (s − 1)(s − 2)Q1(s) Zhang, W.D., CRC Press, 2011 Version 1.0 14/89