Chapter 5 H2 PID Controllers for Stable Plants 4口，+@，4定4=定0C Zhang.W.D..CRC Press.2011 Version 1.0 1/78
Chapter 5 H2 PID Controllers for Stable Plants Zhang, W.D., CRC Press, 2011 Version 1.0 1/78
H2 PID Controllers for Stable Plants 15.1 H2 PID Controllers for the First-Order Plant 25.2 Quantitative Tuning of H2 PID Controllers 35.3 H2 PID Controllers for the Second-Order Plant 45.4 Control of Inverse Response Processes 55.5 PID Controllers Based on the Maclaurin Series Expansion 65.6 PID Controllers with the Best Achievable Performance 75.7 Choice of the Filter 4口，+@，4定4定0C Zhang.W.D..CRC Press.2011 Version 1.0 2/78
H2 PID Controllers for Stable Plants 1 5.1 H2 PID Controllers for the First-Order Plant 2 5.2 Quantitative Tuning of H2 PID Controllers 3 5.3 H2 PID Controllers for the Second-Order Plant 4 5.4 Control of Inverse Response Processes 5 5.5 PID Controllers Based on the Maclaurin Series Expansion 6 5.6 PID Controllers with the Best Achievable Performance 7 5.7 Choice of the Filter Zhang, W.D., CRC Press, 2011 Version 1.0 2/78
Section 5.1 H2 PID Controllers for the First-Order Plant 5.1 H2 PID Controllers for the First-Order Plant An analog of Hoo optimal control theory:H2 optimal control theory Assume that the plant is Ke-0s G(s)= Ts+1 Using the Youla parameterization,we have C(s)= Q(s) 1-G(s)Q(s) where Q(s)is a stable transfer function.It is difficult to treat e-s analytically.Approximate it by the 1/1 Pade approximant: G(S)≈K 1-0s/2 (rs+1)(1+0s/2) 4口+@4定4定0C Zhang.W.D..CRC Press.2011 Version 1.0 3/78
Section 5.1 H2 PID Controllers for the First-Order Plant 5.1 H2 PID Controllers for the First-Order Plant An analog of H∞ optimal control theory: H2 optimal control theory Assume that the plant is G(s) = Ke−θs τ s + 1 Using the Youla parameterization, we have C(s) = Q(s) 1 − G(s)Q(s) where Q(s) is a stable transfer function. It is difficult to treat e −θs analytically. Approximate it by the 1/1 Pade approximant: G(s) ≈ K 1 − θs/2 (τ s + 1)(1 + θs/2) Zhang, W.D., CRC Press, 2011 Version 1.0 3/78
Section 5.1 H2 PID Controllers for the First-Order Plant The design procedure for the H2 PID controller is similar to that for the Hoo PID controller.The controller is first designed for the approximate plant and then used to control the original plant The H2 optimal index is minW(s)s(s)2 where W(s)is the weighting function.Assume that the system input is a unit step.In view of the discussion in Section 3.2,the weighting function in H2 optimal control should be chosen so that the input is normalized to an impulse,that is,d(s)/W(s)=1. Then,W(s)=1/s. W(s)has a pole on the imaginary axis.To guarantee a finite 2-norm and to have the asymptotic property.a constraint has to be imposed on the design lim S(s)=lim 1-G(s)Q(s=0 40 540 4口，+@，4定4=定0C Zhang.W.D..CRC Press.2011 Version 1.0 4/78
Section 5.1 H2 PID Controllers for the First-Order Plant The design procedure for the H2 PID controller is similar to that for the H∞ PID controller. The controller is first designed for the approximate plant and then used to control the original plant The H2 optimal index is min kW (s)S(s)k2 where W (s) is the weighting function. Assume that the system input is a unit step. In view of the discussion in Section 3.2, the weighting function in H2 optimal control should be chosen so that the input is normalized to an impulse, that is, kd(s)/W (s)k2 = 1. Then, W (s) = 1/s. W (s) has a pole on the imaginary axis. To guarantee a finite 2-norm and to have the asymptotic property, a constraint has to be imposed on the design: lim s→0 S(s) = lim s→0 [1 − G(s)Q(s)] = 0 Zhang, W.D., CRC Press, 2011 Version 1.0 4/78
Section 5.1 H2 PID Controllers for the First-Order Plant The design procedure for the H2 PID controller is similar to that for the Hoo PID controller.The controller is first designed for the approximate plant and then used to control the original plant The H2 optimal index is min llW(s)S(s)ll2 where W(s)is the weighting function.Assume that the system input is a unit step.In view of the discussion in Section 3.2,the weighting function in H2 optimal control should be chosen so that the input is normalized to an impulse,that is,ld(s)/W(s)2=1. Then,W(s)=1/s. W(s)has a pole on the imaginary axis.To guarantee a finite 2-norm and to have the asymptotic property,a constraint has to be imposed on the design: lim S(s)=lim[1-G(s)Q(s)]=0 s→0 5→0 4口，+心4定4生定分QC Zhang.W.D..CRC Press.2011 Version 1.0 4/78
Section 5.1 H2 PID Controllers for the First-Order Plant The design procedure for the H2 PID controller is similar to that for the H∞ PID controller. The controller is first designed for the approximate plant and then used to control the original plant The H2 optimal index is min kW (s)S(s)k2 where W (s) is the weighting function. Assume that the system input is a unit step. In view of the discussion in Section 3.2, the weighting function in H2 optimal control should be chosen so that the input is normalized to an impulse, that is, kd(s)/W (s)k2 = 1. Then, W (s) = 1/s. W (s) has a pole on the imaginary axis. To guarantee a finite 2-norm and to have the asymptotic property, a constraint has to be imposed on the design: lim s→0 S(s) = lim s→0 [1 − G(s)Q(s)] = 0 Zhang, W.D., CRC Press, 2011 Version 1.0 4/78
Section 5.1 H2 PID Controllers for the First-Order Plant In other words,S(s)must have a zero at the origin to cancel the pole of W(s).This gives 1 Q(0)=G(0)=K It should be emphasized that the constraint is also required for asymptotic tracking.The set of all Q(s)s satisfying the constraint can be written as 1 Q(s)=衣+sQ1(s where Q1(s)is stable.The function to be minimized is lIW(s)s(s)Il2 w-Gw层+o( 0rs/2+(0+T) K(1-0s/2) (1)(0/211)02() 定0QC Zhang.W.D..CRC Press.2011 Version 1.0 5/78
Section 5.1 H2 PID Controllers for the First-Order Plant In other words, S(s) must have a zero at the origin to cancel the pole of W (s). This gives Q(0) = 1 G(0) = 1 K It should be emphasized that the constraint is also required for asymptotic tracking. The set of all Q(s)s satisfying the constraint can be written as Q(s) = 1 K + sQ1(s) where Q1(s) is stable. The function to be minimized is kW (s)S(s)k 2 2 = W (s) 1 − G(s) 1 K + sQ1(s) 2 2 = θτ s/2 + (θ + τ ) (τ s + 1)(θs/2 + 1) − K(1 − θs/2) (τ s + 1)(1 + θs/2)Q1(s) 2 2 Zhang, W.D., CRC Press, 2011 Version 1.0 5/78
Section 5.1 H2 PID Controllers for the First-Order Plant 1-0s/2 0rs/2+(0+T） K 1+0s/2L(rs+1)(1-0s/2) (1-0s/2)/(1+0s/2)in the equation is an all-pass transfer function.With the definition of 2-norm,it is easy to verify that the 2-norm of a transfer function keeps its value after introducing an all-pass transfer function to it.Therefore, wase-西+t动-gao 2 As we known,by partial fraction expansion a strictly proper transfer function without poles on the imaginary axis can always be uniquely expressed as a stable part(which does not have poles in Re s >0)and an unstable part(which does not have poles in Re 5<0: 240 Zhang,W.D..CRC Press.2011 Version 1.0 6/78
Section 5.1 H2 PID Controllers for the First-Order Plant = 1 − θs/2 1 + θs/2 θτ s/2 + (θ + τ ) (τ s + 1)(1 − θs/2) − K τ s + 1 Q1(s) 2 2 (1 − θs/2)/(1 + θs/2) in the equation is an all-pass transfer function. With the definition of 2-norm, it is easy to verify that the 2-norm of a transfer function keeps its value after introducing an all-pass transfer function to it. Therefore, kW (s)S(s)k 2 2 = θτ s/2 + (θ + τ ) (τ s + 1)(1 − θs/2) − K τ s + 1 Q1(s) 2 2 As we known, by partial fraction expansion a strictly proper transfer function without poles on the imaginary axis can always be uniquely expressed as a stable part (which does not have poles in Re s > 0) and an unstable part (which does not have poles in Re s < 0): Zhang, W.D., CRC Press, 2011 Version 1.0 6/78
Section 5.1 H2 PID Controllers for the First-Order Plant 0rs/2+(0+T) 8 (Ts+1)(1-0s/2)=1-0s/2 +5+1 Then W()S()0.( Temporarily relax the requirement on the properness of Q(s).To obtain the minimum,the only choice is Qion(句=R Consequently,the optimal Q(s)is 0m间=5k2 4口：4@4242定9QC Zhang.W.D..CRC Press.2011 Version 1.0 7/78
Section 5.1 H2 PID Controllers for the First-Order Plant θτ s/2 + (θ + τ ) (τ s + 1)(1 − θs/2) = θ 1 − θs/2 + τ τ s + 1 Then kW (s)S(s)k 2 2 = θ 1 − θs/2 2 2 + τ τ s + 1 − K τ s + 1 Q1(s) 2 2 Temporarily relax the requirement on the properness of Q(s). To obtain the minimum, the only choice is Q1opt(s) = τ K Consequently, the optimal Q(s) is Qopt(s) = τ s + 1 K Zhang, W.D., CRC Press, 2011 Version 1.0 7/78
Section 5.1 H2 PID Controllers for the First-Order Plant Q(s)should be proper.Use the following filter to roll the improper solution off: 1 J5)=λ5+1 where A is the performance degree.It is a positive real number. The suboptimal Q(s)is Q(s)=Qopt:(s)J(s)=K(A5+1) Ts+1 Since Q(0)=1/K,Q(s)satisfies the constraint for asymptotic tracking.The unity feedback loop controller is Q(s) 1(Ts+1)(1+0s/2) C(5)=1-G(5)Q(s)-K0Xs2/2+(+0)s 4口：4@4242定9QC Zhang.W.D..CRC Press.2011 Version 1.0 8/78
Section 5.1 H2 PID Controllers for the First-Order Plant Q(s) should be proper. Use the following filter to roll the improper solution off: J(s) = 1 λs + 1 where λ is the performance degree. It is a positive real number. The suboptimal Q(s) is Q(s) = Qopt(s)J(s) = τ s + 1 K(λs + 1) Since Q(0) = 1/K, Q(s) satisfies the constraint for asymptotic tracking. The unity feedback loop controller is C(s) = Q(s) 1 − G(s)Q(s) = 1 K (τ s + 1)(1 + θs/2) θλs 2/2 + (λ + θ)s Zhang, W.D., CRC Press, 2011 Version 1.0 8/78
Section 5.1 H2 PID Controllers for the First-Order Plant Comparing the controller with c=k(++o）n gives that =20+1=+20= 0,Kc=K(X+0) If the following form is chosen: c=k（++) the parameters of the PID controller are 0入 TF- 2+0万=+2TF,TD=2 T-TF,Kc=K(X+0) 定QC0 Zhang.W.D..CRC Press.2011 Version 1.0 9/78
Section 5.1 H2 PID Controllers for the First-Order Plant Comparing the controller with C = KC 1 + 1 TIs + TDs 1 TF s + 1 gives that TF = θλ 2(λ + θ) ,TI = τ + θ 2 ,TD = θτ 2TI ,KC = TI K(λ + θ) If the following form is chosen: C(s) = KC 1 + 1 TIs + TDs TF s + 1 the parameters of the PID controller are TF = θλ 2(λ + θ) ,TI = τ + θ 2 −TF ,TD = θτ 2TI −TF ,KC = TI K(λ + θ) Zhang, W.D., CRC Press, 2011 Version 1.0 9/78