Section 2.1 Process Dynamic Responses Integrating Plants with Time Delays When the original mass or energy equilibrium is upset by a change at the input,the output will increase or decrease with a fixed speed until the physical limit is reached. Feature:Have poles at the origin 2.0 13 1.0 05 0.0 Time 定9QC Zhang.W.D..CRC Press.2011 Version 1.0 9/64
Section 2.1 Process Dynamic Responses Integrating Plants with Time Delays When the original mass or energy equilibrium is upset by a change at the input, the output will increase or decrease with a fixed speed until the physical limit is reached. Feature: Have poles at the origin Zhang, W.D., CRC Press, 2011 Version 1.0 9/64
Section 2.1 Process Dynamic Responses Models of Integrating Plants K 6=gns+ms+.ns+可e K-Real constant denoting the static gain 0-Positive real constant denoting the time delay Ti(i=1,2,...,n)-Have positive real parts and denote time constants m-Integer The first-order model frequently used in practice: G(s)=Ne-0s 4口，+@，4定4定90C Zhang.W.D..CRC Press.2011 Version 1.0 10/64
Section 2.1 Process Dynamic Responses Models of Integrating Plants G(s) = K sm(τ1s + 1)(τ2s + 1)...(τns + 1) e −θs K—Real constant denoting the static gain θ—Positive real constant denoting the time delay τi(i = 1, 2, ..., n)—Have positive real parts and denote time constants m—Integer The first-order model frequently used in practice: G(s) = K s e −θs Zhang, W.D., CRC Press, 2011 Version 1.0 10/64
Section 2.1 Process Dynamic Responses Models of Integrating Plants K 6=gns+ms+.ns+万e K-Real constant denoting the static gain 0-Positive real constant denoting the time delay Ti(i=1,2,...,n)-Have positive real parts and denote time constants m-Integer The first-order model frequently used in practice: G(s)=Ke-0s K 4口：4@，4定42定QC Zhang.W.D..CRC Press.2011 Version 1.0 10/64
Section 2.1 Process Dynamic Responses Models of Integrating Plants G(s) = K sm(τ1s + 1)(τ2s + 1)...(τns + 1) e −θs K—Real constant denoting the static gain θ—Positive real constant denoting the time delay τi(i = 1, 2, ..., n)—Have positive real parts and denote time constants m—Integer The first-order model frequently used in practice: G(s) = K s e −θs Zhang, W.D., CRC Press, 2011 Version 1.0 10/64
Section 2.1 Process Dynamic Responses Categories of Plants Stable plants According to pole positions: Integrating plants Unstable plants According to zero positions: MP plants NMP plants MP plants:Its transfer function does not contain zeros in the closed RHP or a time delay NMP plants:All plants that are not MP 4口，+@，4定4=定0C Zhang.W.D..CRC Press.2011 Version 1.0 11/64
Section 2.1 Process Dynamic Responses Categories of Plants According to pole positions: Stable plants Integrating plants Unstable plants According to zero positions: MP plants NMP plants MP plants: Its transfer function does not contain zeros in the closed RHP or a time delay NMP plants: All plants that are not MP Zhang, W.D., CRC Press, 2011 Version 1.0 11/64
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:deg N+deg(N<deg(M+deg(M. with which the plant is proper 4口，+@4定4定， 2a0 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