Chapter 4 Ho PID Controllers for Stable Plants 4口，+@，4定4=定0C Zhang.W.D..CRC Press.2011 Version 1.0 1/71
Chapter 4 H∞ PID Controllers for Stable Plants Zhang, W.D., CRC Press, 2011 Version 1.0 1/71
Ho PID Controllers for Stable Plants 14.1 Traditional Design Methods 24.2 H PID Controllers for the First-Order Plant 34.3 Hoo PID controller and the Smith Predictor 4.4 Quantitative Performance and Robustness 54.5 H PID Controllers for the Second-Order Plant 64.6 All Stabilizing PID Controllers for Stable Plants 4口，4@，4定4定0C Zhang.W.D..CRC Press.2011 Version 1.0 2/71
H∞ PID Controllers for Stable Plants 1 4.1 Traditional Design Methods 2 4.2 H∞ PID Controllers for the First-Order Plant 3 4.3 H∞ PID controller and the Smith Predictor 4 4.4 Quantitative Performance and Robustness 5 4.5 H∞ PID Controllers for the Second-Order Plant 6 4.6 All Stabilizing PID Controllers for Stable Plants Zhang, W.D., CRC Press, 2011 Version 1.0 2/71
Section 4.1 Traditional Design Methods 4.1 Traditional Design Methods PID Controllers Importance:95%controllers in practice are PID controllers Ideal PID: u0=k[o+元/0+n0] K-Gain Tj-Integral constant e(t)-Error Tp-Derivative constant u(t)-Controller output Assume that C(s)is the transfer function from e(s)to u(s).Using the Laplace transform,we have c=k(++T） 4口+@4定4定0C Zhang.W.D..CRC Press.2011 Version 1.0 3/71
Section 4.1 Traditional Design Methods 4.1 Traditional Design Methods PID Controllers Importance: 95% controllers in practice are PID controllers Ideal PID: u(t) = Kc e(t) + 1 TI Z e(t)dt + TD de(t) dt Kc—Gain TI—Integral constant TD—Derivative constant e(t)—Error u(t)—Controller output Assume that C(s) is the transfer function from e(s) to u(s). Using the Laplace transform, we have C(s) = Kc 1 + 1 TIs + TDs Zhang, W.D., CRC Press, 2011 Version 1.0 3/71
Section 4.1 Traditional Design Methods Practical PID Forms Ideal PID:Has a pure differentiator in it and therefore is not physically realizable An important method for realizing an improper transfer function:Introduce a low-pass transfer function to it Three practical forms: C(s)=Kc1+ TDS 1 TDs+1 C(s)= Ke C(s)= K1+ Usually TF=0.1Tp in a PID 4口，+@，4定4=定0C Zhang,W.D..CRC Press.2011 Version 1.0 4/71
Section 4.1 Traditional Design Methods Practical PID Forms Ideal PID: Has a pure differentiator in it and therefore is not physically realizable An important method for realizing an improper transfer function: Introduce a low-pass transfer function to it Three practical forms: C(s) = Kc 1 + 1 TIs + TDs TF s + 1 C(s) = Kc 1 + 1 TIs TDs + 1 TF s + 1 C(s) = Kc 1 + 1 TIs + TDs 1 TF s + 1 Usually TF = 0.1TD in a PID Zhang, W.D., CRC Press, 2011 Version 1.0 4/71
Section 4.1 Traditional Design Methods Practical PID Forms Ideal PID:Has a pure differentiator in it and therefore is not physically realizable An important method for realizing an improper transfer function:Introduce a low-pass transfer function to it Three practical forms: C(s)= ++） TDS C(s)=Kc(1+ TDs+1 Tis TEs+1 C(s)= TF5+1 Usually TF=0.1Tp in a PID 4口，+@，4定4=定0C Zhang.W.D..CRC Press.2011 Version 1.0 4/71
Section 4.1 Traditional Design Methods Practical PID Forms Ideal PID: Has a pure differentiator in it and therefore is not physically realizable An important method for realizing an improper transfer function: Introduce a low-pass transfer function to it Three practical forms: C(s) = Kc 1 + 1 TIs + TDs TF s + 1 C(s) = Kc 1 + 1 TIs TDs + 1 TF s + 1 C(s) = Kc 1 + 1 TIs + TDs 1 TF s + 1 Usually TF = 0.1TD in a PID Zhang, W.D., CRC Press, 2011 Version 1.0 4/71
Section 4.1 Traditional Design Methods Tuning Rules Assume that the step response model is K G(s)= e-0s Ts+ and the ultimate cycle model is Ku and Tu Table:Frequently used tuning methods. uning methods R-C method C-C method Z-N method KKe 1.2(0/r-1 1.35(0/r)-1+0.27 0.6KK T 2(0/r) 2.5(0/r)[1+(8/r)/5] 1+0.6(0/r) 0.5Tu/T To/T 0.5(0/r) 0.37(0/r) 1+0.2(0/r) 0.125Tu/r 240 Zhang.W.D..CRC Press.2011 Version 1.0 5/71
Section 4.1 Traditional Design Methods Tuning Rules Assume that the step response model is G(s) = K τ s + 1 e −θs and the ultimate cycle model is Ku and Tu Table: Frequently used tuning methods. Tuning methods R-C method C-C method Z-N method KKc 1.2(θ/τ ) −1 1.35(θ/τ ) −1 + 0.27 0.6KKu TI /τ 2(θ/τ ) 2.5(θ/τ )[1 + (θ/τ )/5] 1 + 0.6(θ/τ ) 0.5Tu/τ TD/τ 0.5(θ/τ ) 0.37(θ/τ ) 1 + 0.2(θ/τ ) 0.125Tu/τ Zhang, W.D., CRC Press, 2011 Version 1.0 5/71
Section 4.1 Traditional Design Methods RZN Tuning Z-N method:The most widely used method Disadvantage:The resulting PID controller usually gives excessive overshoot A solution:Refined Z-N (RZN)method.Perhaps the most famous improved method The modified PID controller is ()=()(()dt+T de(t) Band u are determined by extensive simulation studies.Define Normalized gain:K=KK,Normalized time delay:0=0/T 4口，404注4生定分QC Zhang.W.D..CRC Press.2011 Version 1.0 6/71
Section 4.1 Traditional Design Methods RZN Tuning Z-N method: The most widely used method Disadvantage: The resulting PID controller usually gives excessive overshoot A solution: Refined Z-N (RZN) method. Perhaps the most famous improved method The modified PID controller is u(t) = Kc [βr(t) − y(t)] + 1 µTI Z e(t)dt + TD de(t) dt β and µ are determined by extensive simulation studies. Define Normalized gain:Kn = KKu, Normalized time delay:θn = θ/τ Zhang, W.D., CRC Press, 2011 Version 1.0 6/71
Section 4.1 Traditional Design Methods RZN Tuning Z-N method:The most widely used method Disadvantage:The resulting PID controller usually gives excessive overshoot A solution:Refined Z-N(RZN)method.Perhaps the most famous improved method The modified PID controller is k.((t 3 and u are determined by extensive simulation studies.Define Normalized gain:Kn=KKu;Normalized time delay:0n=0/T 定0QC Zhang.W.D..CRC Press.2011 Version 1.0 6/71
Section 4.1 Traditional Design Methods RZN Tuning Z-N method: The most widely used method Disadvantage: The resulting PID controller usually gives excessive overshoot A solution: Refined Z-N (RZN) method. Perhaps the most famous improved method The modified PID controller is u(t) = Kc [βr(t) − y(t)] + 1 µTI Z e(t)dt + TD de(t) dt β and µ are determined by extensive simulation studies. Define Normalized gain:Kn = KKu, Normalized time delay:θn = θ/τ Zhang, W.D., CRC Press, 2011 Version 1.0 6/71
Section 4.1 Traditional Design Methods When 2.25 Kn<15 and 0.16<0n<0.57,for a 10%overshoot: B= 15-K 15+K ,4=1 and for a 20%overshoot: 36 B= 27+5K:“=1 If1.5<Kn<2.25and0.57<0n<0.96, =（信，+u=音k Z-N/RZN usually gives very bad response for the plant with large time delay.Hence.some designers believe that PID cannot be used for the plant with large time delay.Actually.with proper design methods PID can be applied to such systems 4口，48,4注4生，定0C Zhang.W.D..CRC Press.2011 Version 1.0 7/71
Section 4.1 Traditional Design Methods When 2.25 < Kn < 15 and 0.16 < θn < 0.57, for a 10% overshoot: β = 15 − Kn 15 + Kn , µ = 1 and for a 20% overshoot: β = 36 27 + 5Kn , µ = 1 If 1.5 < Kn < 2.25 and 0.57 < θn < 0.96, β = 8 17 4 9 Kn + 1 , µ = 4 9 Kn Z-N/RZN usually gives very bad response for the plant with large time delay. Hence, some designers believe that PID cannot be used for the plant with large time delay. Actually, with proper design methods PID can be applied to such systems Zhang, W.D., CRC Press, 2011 Version 1.0 7/71
Section 4.1 Traditional Design Methods When 2.25 Kn<15 and 0.16<0n<0.57,for a 10%overshoot: B= 15-K 15+K ,4=1 and for a 20%overshoot: 36 B= 27+5K,:4=1 If1.5<Kn<2.25and0.57<0n<0.96, B=吕（信+）u=音k Z-N/RZN usually gives very bad response for the plant with large time delay.Hence,some designers believe that PID cannot be used for the plant with large time delay.Actually,with proper design methods PID can be applied to such systems Zhang.W.D..CRC Press.2011 Version 1.0 7/71
Section 4.1 Traditional Design Methods When 2.25 < Kn < 15 and 0.16 < θn < 0.57, for a 10% overshoot: β = 15 − Kn 15 + Kn , µ = 1 and for a 20% overshoot: β = 36 27 + 5Kn , µ = 1 If 1.5 < Kn < 2.25 and 0.57 < θn < 0.96, β = 8 17 4 9 Kn + 1 , µ = 4 9 Kn Z-N/RZN usually gives very bad response for the plant with large time delay. Hence, some designers believe that PID cannot be used for the plant with large time delay. Actually, with proper design methods PID can be applied to such systems Zhang, W.D., CRC Press, 2011 Version 1.0 7/71