Section 7.1 The Feature of Integrating Systems Proof ctd.4. With the assumption,we have dk dk dk zF(s) = M(s）- N(s) M(s) = Ok-Bk a0 The left-hand side should be 0.Therefore, ak Bk. This completes the proof. 4口，+@，4定4定90C Zhang.W.D..CRC Press.2011 Version 1.0 15/79
Section 7.1 The Feature of Integrating Systems Proof ctd.4. With the assumption, we have lim s→0 d k dsk F(s) = d k dsk M(s) − d k dsk N(s) M(s) = αk − βk α0 . The left-hand side should be 0. Therefore, αk = βk . This completes the proof. Zhang, W.D., CRC Press, 2011 Version 1.0 15/79
Section 7.1 The Feature of Integrating Systems Corollary Given the transfer function N(s)/M(s). 51-MG d =0nd期 N(s) =0 hold if and only if the coefficients of the first 2 terms of N(s)are the same as those of M(s),respectively. If a quasi-polynomial that contains a time delay is encountered when using these results,the time delay should be substituted by its Taylor series expansion. 4口，+@，4定4定90C Zhang.W.D..CRC Press.2011 Version 1.0 16/79
Section 7.1 The Feature of Integrating Systems Corollary Given the transfer function N(s)/M(s). lim s→0 1 − N(s) M(s) = 0 and lim s→0 d ds 1 − N(s) M(s) = 0 hold if and only if the coefficients of the first 2 terms of N(s) are the same as those of M(s), respectively. If a quasi-polynomial that contains a time delay is encountered when using these results, the time delay should be substituted by its Taylor series expansion. Zhang, W.D., CRC Press, 2011 Version 1.0 16/79
Section 7.1 The Feature of Integrating Systems Example There are two polynomials:N(s)=(1-0s/2)(B1s+1)and M(s)=(1+0s/2)(As+1)2.Compute the constant B1 that makes the following hold: N(s) =0 and I N(s) =0 According to Corollary,the zeroth-order and the first-order coefficients of N(s)and M(s)should equal,respectively.Both the zeroth-order coefficients of N(s)and M(s)are 1.The first-order coefficient of N(s)is B1-0/2 and the first-order coefficient of M(s)is 2+0/2.This yields B1=2入+0 4口：+@，4定4生定9QC Zhang,W.D..CRC Press.2011 Version 1.0 17/79
Section 7.1 The Feature of Integrating Systems Example There are two polynomials: N(s) = (1 − θs/2)(β1s + 1) and M(s) = (1 + θs/2)(λs + 1)2 . Compute the constant β1 that makes the following hold: lim s→0 1 − N(s) M(s) = 0 and lim s→0 d ds 1 − N(s) M(s) = 0 According to Corollary, the zeroth-order and the first-order coefficients of N(s) and M(s) should equal, respectively. Both the zeroth-order coefficients of N(s) and M(s) are 1. The first-order coefficient of N(s) is β1 − θ/2 and the first-order coefficient of M(s) is 2λ + θ/2. This yields β1 = 2λ + θ Zhang, W.D., CRC Press, 2011 Version 1.0 17/79
Section 7.1 The Feature of Integrating Systems Example It is known that N(s)=(B1s+1)e-0s and M(s)=(As+1). Compute the constant B1 that makes 0 hold. Again,the zeroth-order and the first-order coefficients of N(s)and M(s)should equal,respectively.Both the zeroth-order coefficients of N(s)and M(s)are 1.The first-order coefficient of N(s)is B1-0 and the first-order coefficient of M(s)is njA.Let them equal.One readily obtains B月=n入+0 4口，+@，4定4=定0C Zhang.W.D..CRC Press.2011 Version 1.0 18/79
Section 7.1 The Feature of Integrating Systems Example It is known that N(s) = (β1s + 1)e −θs and M(s) = (λs + 1)nj . Compute the constant β1 that makes lim s→0 1 − N(s) M(s) = 0 and lim s→0 d ds 1 − N(s) M(s) = 0 hold. Again, the zeroth-order and the first-order coefficients of N(s) and M(s) should equal, respectively. Both the zeroth-order coefficients of N(s) and M(s) are 1. The first-order coefficient of N(s) is β1 − θ and the first-order coefficient of M(s) is njλ. Let them equal. One readily obtains β1 = njλ + θ Zhang, W.D., CRC Press, 2011 Version 1.0 18/79
Section 7.2 Hoc PID Controllers for Integrating Plants 7.2 Ho PID Controllers for Integrating Plants One Way to Design Hoo PID Controllers Assume that the coprime factorization of G(s)is G(s)=V(s)/U(s),where U(s)and V(s)are stable proper real rational.According to the discussion in Section 3.3,all stabilizing controllers for integrating plants can be expressed as C() X(s)+U(s)Q(s) Y(s)-V(s)Q(s) where Q(s)is stable,and X(s)and Y(s)are stable proper real rational functions that satisfy the equation V(s)X(s)+U(s)Y(s)=1 4口，+@，4定4=定0C Zhang.W.D..CRC Press.2011 Version 1.0 19/79
Section 7.2 H∞ PID Controllers for Integrating Plants 7.2 H∞ PID Controllers for Integrating Plants One Way to Design H∞ PID Controllers Assume that the coprime factorization of G(s) is G(s) = V(s)/U(s), where U(s) and V(s) are stable proper real rational. According to the discussion in Section 3.3, all stabilizing controllers for integrating plants can be expressed as C(s) = X(s) + U(s)Q(s) Y (s) − V(s)Q(s) where Q(s) is stable, and X(s) and Y (s) are stable proper real rational functions that satisfy the equation V(s)X(s) + U(s)Y (s) = 1 Zhang, W.D., CRC Press, 2011 Version 1.0 19/79