Section 7.1 The Feature of Integrating Systems Theorem The kth (k =0,1,...,q)coefficient of a q-order polynomial N(s)is dk N(0)/dsk /k!. Proof. Follows directly from its Taylor series expansion. Theorem Given the transfer function N(s)/M(s).N(s)and M(s)are polynomials,and q=deg[N(s)}<p=deg[M(s)}.Let m<q be any nonnegative integer.Then lim d N(s) s 0 dsk - =0,k=0,1,m holds if and only if the coefficients of the first k(k =0,1,...,m) terms of N(s)are the same as those of M(s),respectively. oac Zhang.W.D..CRC Press.2011 Version 1.0 10/79
Section 7.1 The Feature of Integrating Systems Theorem The kth (k = 0, 1, ..., q) coefficient of a q-order polynomial N(s) is d kN(0)/dsk /k!. Proof. Follows directly from its Taylor series expansion. Theorem Given the transfer function N(s)/M(s). N(s) and M(s) are polynomials, and q = deg{N(s)} ≤ p = deg{M(s)}. Let m ≤ q be any nonnegative integer. Then lim s→0 d k dsk 1 − N(s) M(s) = 0, k = 0, 1, ..., m holds if and only if the coefficients of the first k(k = 0, 1, ..., m) terms of N(s) are the same as those of M(s), respectively. Zhang, W.D., CRC Press, 2011 Version 1.0 10/79
Section 7.1 The Feature of Integrating Systems Proof. Sufficiency is obvious.To prove necessity,assume that N(s)=Bgs9+.+Bks+…+315+Bo, M(s)=apsp+...+aksk+...+as+ao, where Bi(i=0,1,...,q)and ai(i=0,1,...,p)are positive real numbers.Let Fs)=1- N(s) M(s) Then M(s)F(s)=M(s)-N(s). 4口，+@，4定4定90C Zhang.W.D..CRC Press.2011 Version 1.0 11/79
Section 7.1 The Feature of Integrating Systems Proof. Sufficiency is obvious. To prove necessity, assume that N(s) = βqs q + ... + βk s k + ... + β1s + β0, M(s) = αps p + ... + αk s k + ... + α1s + α0, where βi(i = 0, 1, ..., q) and αi(i = 0, 1, ..., p) are positive real numbers. Let F(s) = 1 − N(s) M(s) . Then M(s)F(s) = M(s) − N(s). Zhang, W.D., CRC Press, 2011 Version 1.0 11/79
Section 7.1 The Feature of Integrating Systems Proof ctd.1. The inductive method is used:First,the case k=0 and k =1 are shown to be true;Then the case for the k-order is shown to be true if the case for the (k-1)-order is true. When k=0. ling F(s)=2oB Q0 Let the right-hand side be 0.We have ao=Bo When k=1, M(F) =F6是Me)+MsF句) [M(s)-N(s) d = d 4口，4心4定4生，定QC Zhang.W.D..CRC Press.2011 Version 1.0 12/79
Section 7.1 The Feature of Integrating Systems Proof ctd.1. The inductive method is used: First, the case k = 0 and k = 1 are shown to be true; Then the case for the k-order is shown to be true if the case for the (k − 1)-order is true. When k = 0, lim s→0 F(s) = α0 − β0 α0 Let the right-hand side be 0. We have α0 = β0 When k = 1, d ds[M(s)F(s)] = F(s) d ds M(s) + M(s) d ds F(s) d ds[M(s) − N(s)] = d ds M(s) − d ds N(s) Zhang, W.D., CRC Press, 2011 Version 1.0 12/79
Section 7.1 The Feature of Integrating Systems Proof ctd.2. Since lim F(s)=0, s→0 the derivative of F(s)is = (s) )ds M(s) a1-61 00 Let lims0F(s)=0.This yields a1=B1. 2ac Zhang.W.D..CRC Press.2011 Version 1.0 13/79
Section 7.1 The Feature of Integrating Systems Proof ctd.2. Since lim s→0 F(s) = 0, the derivative of F(s) is lim s→0 d ds F(s) = d ds M(s) − d ds N(s) M(s) = α1 − β1 α0 . Let lims→0 d ds F(s) = 0. This yields α1 = β1. Zhang, W.D., CRC Press, 2011 Version 1.0 13/79
Section 7.1 The Feature of Integrating Systems Proof ctd.3. Now assume that the conclusion holds for k-1.To prove the theorem,it suffices to prove that the conclusion holds for the kth time differentiating.Consider the following fact: sMs)F(s】= or回+cMg日+ dk d C)F())F( ds Mg-Me】=S dsk M(s)- dskN(s), where k! Ck=(k)!' 4口，+@，4定4=定0C Zhang.W.D..CRC Press.2011 Version 1.0 14/79
Section 7.1 The Feature of Integrating Systems Proof ctd.3. Now assume that the conclusion holds for k − 1. To prove the theorem, it suffices to prove that the conclusion holds for the kth time differentiating. Consider the following fact: d k dsk [M(s)F(s)] = d k dsk M(s)F(s) + C 1 k d k−1 dsk−1 M(s) d ds F(s) + ... + C k−1 k d ds M(s) d k−1 dsk−1 F(s) + M(s) d k dsk F(s) d k dsk [M(s) − N(s)] = d k dsk M(s) − d k dsk N(s), where C i k = k! i!(k − i)!. Zhang, W.D., CRC Press, 2011 Version 1.0 14/79