Section 2.1 Process Dynamic Responses The General Plant In theoretical study,the following model may be used: KN+(s)N_(s)e-0s G(s)=M(s)M-(s) K-Real constant denoting the static gain 0-Positive real constant denoting the time delay "+"-Denote that all of the roots are in the closed RHP "-"-Denote that all of the roots are in the open LHP Assumption 1:N+(0)=N-(0)=M+(0)=M-(0)=1,which is made solely to simplify the statement Assumption 2:degiN+deg{N}<deg(M-}+degtM+, with which the plant is proper 定月QC Zhang.W.D..CRC Press.2011 Version 1.0 12/64
Section 2.1 Process Dynamic Responses The General Plant In theoretical study, the following model may be used: G(s) = KN+(s)N−(s) M+(s)M−(s) e −θs K—Real constant denoting the static gain θ—Positive real constant denoting the time delay “+”—Denote that all of the roots are in the closed RHP “-”—Denote that all of the roots are in the open LHP Assumption 1: N+(0) = N−(0) = M+(0) = M−(0) = 1, which is made solely to simplify the statement Assumption 2: deg{N+} + deg{N−} ≤ deg{M−} + deg{M+}, with which the plant is proper Zhang, W.D., CRC Press, 2011 Version 1.0 12/64
Section 2.2 Rational Approximations for Time Delay 2.2 Rational Approximations for Time Delay Why Use Rational Approximations Main reasons: o The time delay is an irrational function,which is of infinite dimension o Most design methods developed so far are based on rational functions.They are only applicable to plants of finite dimension One way to overcome the problem:Approximate the time delay by employing rational functions 4口，+@，4定4定90C Zhang.W.D..CRC Press.2011 Version 1.0 13/64
Section 2.2 Rational Approximations for Time Delay 2.2 Rational Approximations for Time Delay Why Use Rational Approximations Main reasons: The time delay is an irrational function, which is of infinite dimension Most design methods developed so far are based on rational functions. They are only applicable to plants of finite dimension One way to overcome the problem: Approximate the time delay by employing rational functions Zhang, W.D., CRC Press, 2011 Version 1.0 13/64
Section 2.2 Rational Approximations for Time Delay 2.2 Rational Approximations for Time Delay Why Use Rational Approximations Main reasons: o The time delay is an irrational function,which is of infinite dimension o Most design methods developed so far are based on rational functions.They are only applicable to plants of finite dimension One way to overcome the problem:Approximate the time delay by employing rational functions 4口，+@，4定4定90C Zhang.W.D..CRC Press.2011 Version 1.0 13/64
Section 2.2 Rational Approximations for Time Delay 2.2 Rational Approximations for Time Delay Why Use Rational Approximations Main reasons: The time delay is an irrational function, which is of infinite dimension Most design methods developed so far are based on rational functions. They are only applicable to plants of finite dimension One way to overcome the problem: Approximate the time delay by employing rational functions Zhang, W.D., CRC Press, 2011 Version 1.0 13/64
Section 2.2 Rational Approximations for Time Delay Rational Approximations 1.Apprximation with lags e-0s lim 1 n→00 The first-order approximation: e-0s- 1 1+0s 2.Taylor series expansion e-0s=im1-0s+2s2/21+..+(-10s”/nl The first-order approximation: e-s=1-0s 4口，+@，4定4=定0C Zhang.W.D..CRC Press.2011 Version 1.0 14/64
Section 2.2 Rational Approximations for Time Delay Rational Approximations 1. Apprximation with lags e −θs = lim n→∞ 1 1 + θs/n n The first-order approximation: e −θs = 1 1 + θs 2. Taylor series expansion e −θs = lim n→∞ 1 − θs + θ 2 s 2 /2! + ... + (−1)n θ n s n /n! The first-order approximation: e −θs = 1 − θs Zhang, W.D., CRC Press, 2011 Version 1.0 14/64
Section 2.2 Rational Approximations for Time Delay Rational Approximations 1.Apprximation with lags e-0s lim n→00 （1+sn) The first-order approximation: e-0s- 1 1+0s 2.Taylor series expansion e-9s=lim1-0s+02s2/2!+.+(-1)P9s"/nl n→0∞ The first-order approximation: e-9s=1-05 4口，+@，4定4=定0C Zhang.W.D..CRC Press.2011 Version 1.0 14/64
Section 2.2 Rational Approximations for Time Delay Rational Approximations 1. Apprximation with lags e −θs = lim n→∞ 1 1 + θs/n n The first-order approximation: e −θs = 1 1 + θs 2. Taylor series expansion e −θs = lim n→∞ 1 − θs + θ 2 s 2 /2! + ... + (−1)n θ n s n /n! The first-order approximation: e −θs = 1 − θs Zhang, W.D., CRC Press, 2011 Version 1.0 14/64