Section 8.1 Controller Parameterization for General Plants Example (ctd.1) where Q1(s)is a stable transfer function satisfying i1-Q(s=0,1-Q(s=0 This is equivalent to Q(s)=1+(s-1)(s-2)Q2(s) 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)= 1+(s-1)(s-2)Q2(s) -Q2(5) (5-10(5-2)- -1 = Q2(s) 24c Zhang.W.D..CRC Press.2011 Version 1.0 15/89
Section 8.1 Controller Parameterization for General Plants Example (ctd.1) where Q1(s) is a stable transfer function satisfying lim s→1 [1 − Q1(s)] = 0, lim s→2 [1 − Q1(s)] = 0 This is equivalent to Q1(s) = 1 + (s − 1)(s − 2)Q2(s) 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) = 1 + (s − 1)(s − 2)Q2(s) −Q2(s) = (s − 1)(s − 2) − −1 Q2(s) Zhang, W.D., CRC Press, 2011 Version 1.0 15/89
Section 8.1 Controller Parameterization for General Plants When the system performance is considered,it is always desirable that the system has asymptotic tracking property.The parameterization can be further developed to cover the requirement on asymptotic tracking Theorem All controllers that make the unity feedback control system internally stable and have asymptotic tracking property for a step input can be parameterized as Q(s) C(s)=1-G(5)Q(S where Q(s)= [1+sQ2(s]M+(s) K 4口，+@4定4生，定分QC Zhang.W.D..CRC Press.2011 Version 1.0 16/89
Section 8.1 Controller Parameterization for General Plants When the system performance is considered, it is always desirable that the system has asymptotic tracking property. The parameterization can be further developed to cover the requirement on asymptotic tracking Theorem All controllers that make the unity feedback control system internally stable and have asymptotic tracking property for a step input can be parameterized as C(s) = Q(s) 1 − G(s)Q(s) where Q(s) = [1 + sQ2(s)]M+(s) K Zhang, W.D., CRC Press, 2011 Version 1.0 16/89
Section 8.1 Controller Parameterization for General Plants Theorem (ctd.1) Q2(s)is any stable transfer function that makes Q(s)proper and satisfies 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. Q(s)should be stable and has zeros wherever G(s)has RHP poles. Such a transfer function can be expressed as Q(s)= Qi(s)M+(s) K 290 Zhang.W.D..CRC Press.2011 Version 1.0 17/89
Section 8.1 Controller Parameterization for General Plants Theorem (ctd.1) Q2(s) is any stable transfer function that makes Q(s) proper and satisfies lim s→sj d k dsk 1 − [1 + sQ2(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. Q(s) should be stable and has zeros wherever G(s) has RHP poles. Such a transfer function can be expressed as Q(s) = Q1(s)M+(s) K Zhang, W.D., CRC Press, 2011 Version 1.0 17/89
Section 8.1 Controller Parameterization for General Plants proof ctd.1. where Q1(s)is stable.If Iim[1-G(s)Q(s】=0 s→0 the closed-loop system possesses the asymptotic tracking property, which implies that Q1(s)=1+sQ2(s) where Q2(s)is a stable transfer function that makes Q(s)proper. This leads to 1-G(s)Q(s)=1-I1+sQ2(s)]N+(s)N-(s)e-0s M-(s) 4口，404注4生定分QC Zhang.W.D..CRC Press.2011 Version 1.0 18/89
Section 8.1 Controller Parameterization for General Plants proof ctd.1. where Q1(s) is stable. If lim s→0 [1 − G(s)Q(s)] = 0 the closed-loop system possesses the asymptotic tracking property, which implies that Q1(s) = 1 + sQ2(s) where Q2(s) is a stable transfer function that makes Q(s) proper. This leads to 1 − G(s)Q(s) = 1 − [1 + sQ2(s)]N+(s)N−(s)e −θs M−(s) Zhang, W.D., CRC Press, 2011 Version 1.0 18/89
Section 8.1 Controller Parameterization for General Plants proof ctd.2. Then Q2(s)should satisfy 1(w-() M-(S) k=0,1,,-1 Feature of the new parameterization:No coprime factorization is used.Instead,the properness of Q(s)and the related constraints must be tested In the design framework of this book,the parameterization is only used to derive the analytical design formula.It is not necessary to compute the parameterization 4口，46,4在4生，定 240 Zhang.W.D..CRC Press.2011 Version 1.0 19/89
Section 8.1 Controller Parameterization for General Plants proof ctd.2. Then Q2(s) should satisfy lim s→sj d k dsk 1 − [1 + sQ2(s)]N+(s)N−(s)e −θs M−(s) = 0 k = 0, 1, ..., lj − 1 Feature of the new parameterization: No coprime factorization is used. Instead, the properness of Q(s) and the related constraints must be tested In the design framework of this book, the parameterization is only used to derive the analytical design formula. It is not necessary to compute the parameterization Zhang, W.D., CRC Press, 2011 Version 1.0 19/89