Section 4.2 Hc PID Controllers for the First-Order Plant There exist three constraints on Q(s): Q(s)should be stable for internal stability 2 To make the controller physically realizable,Q(s)should be proper 3 To have a finite oo-norm,Q(s)should satisfy lim S(s)=lim[1-G(s)Q(s)]=0 s→0 →0 This constraint is also required for asymptotic tracking Idea:Drop the requirement of properness first and find the optimal Q(s),namely Qopt(s).Then roll Qopt(s)off at high frequencies The minimum of W(s)S(s)is 0/2.This gives the following unique optimal solution W(s)-0/2 (Ts+1)1+0s/2) Qopt(s)= W(s)G(s) 4口，+@，4定4=定0C Zhang.W.D..CRC Press.2011 Version 1.0 12/71
Section 4.2 H∞ PID Controllers for the First-Order Plant There exist three constraints on Q(s): 1 Q(s) should be stable for internal stability 2 To make the controller physically realizable, Q(s) should be proper 3 To have a finite ∞-norm, Q(s) should satisfy lim s→0 S(s) = lim s→0 [1 − G(s)Q(s)] = 0 This constraint is also required for asymptotic tracking Idea: Drop the requirement of properness first and find the optimal Q(s), namely Qopt(s). Then roll Qopt(s) off at high frequencies The minimum of kW (s)S(s)k∞ is θ/2. This gives the following unique optimal solution: Qopt(s) = W (s) − θ/2 W (s)G(s) = (τ s + 1)(1 + θs/2) K Zhang, W.D., CRC Press, 2011 Version 1.0 12/71
Section 4.2 Hc PID Controllers for the First-Order Plant There exist three constraints on Q(s): Q(s)should be stable for internal stability 2 To make the controller physically realizable,Q(s)should be proper 3 To have a finite oo-norm,Q(s)should satisfy lim S(s)=lim [1-G(s)Q(s)]=0 s→0 →0 This constraint is also required for asymptotic tracking Idea:Drop the requirement of properness first and find the optimal Q(s),namely Qopt(s).Then roll Qopt(s)off at high frequencies The minimum of W(s)S(s)is 0/2.This gives the following unique optimal solution: Qopt(s)= W(s)-0/2_(rs+1)(1+0s/2) W(s)G(s) K 是9QC Zhang.W.D..CRC Press.2011 Version 1.0 12/71
Section 4.2 H∞ PID Controllers for the First-Order Plant There exist three constraints on Q(s): 1 Q(s) should be stable for internal stability 2 To make the controller physically realizable, Q(s) should be proper 3 To have a finite ∞-norm, Q(s) should satisfy lim s→0 S(s) = lim s→0 [1 − G(s)Q(s)] = 0 This constraint is also required for asymptotic tracking Idea: Drop the requirement of properness first and find the optimal Q(s), namely Qopt(s). Then roll Qopt(s) off at high frequencies The minimum of kW (s)S(s)k∞ is θ/2. This gives the following unique optimal solution: Qopt(s) = W (s) − θ/2 W (s)G(s) = (τ s + 1)(1 + θs/2) K Zhang, W.D., CRC Press, 2011 Version 1.0 12/71
Section 4.2 Ho PID Controllers for the First-Order Plant Qpt(s)is improper.A low-pass filter must be introduced to roll Qopt(s)off at high frequencies.Choose the following filter: Bo Js)=05+12 Bo-A constant 入一A positive real number The filter should not violate the constraint for the asymptotic property: im[1-G(s)Qopt(s)J(s)]=0 Elementary computations give Bo =1.Then the suboptimal proper Q(s)is Q(s=Qt(s)s)=s+11+0s/2) K(入s+1)2 Zhang.W.D..CRC Press.2011 Version 1.0 13/71
Section 4.2 H∞ PID Controllers for the First-Order Plant Qopt(s) is improper. A low-pass filter must be introduced to roll Qopt(s) off at high frequencies. Choose the following filter: J(s) = β0 (λs + 1)2 β0—A constant λ—A positive real number The filter should not violate the constraint for the asymptotic property: lim s→0 [1 − G(s)Qopt(s)J(s)] = 0 Elementary computations give β0 = 1. Then the suboptimal proper Q(s) is Q(s) = Qopt(s)J(s) = (τ s + 1)(1 + θs/2) K(λs + 1)2 Zhang, W.D., CRC Press, 2011 Version 1.0 13/71
Section 4.2 H PID Controllers for the First-Order Plant A is an adjustable parameter called performance degree.It closely relates to the closed-loop performance: Smallerλ <=>Fast response Larger入 <=Slow response 入→0 <=The optimal lW(s)S(s)llo The controller of the corresponding unity feedback loop is Q(s) 1(rs+1)(1+0s/2) C(s)=1-GQ(阿=K2s2+2+0/2s This is a PID controller An important feature:It cancels two poles of the approximate model,or equivalently,two dominant poles of the original model 是9aC Zhang.W.D..CRC Press.2011 Version 1.0 14/71
Section 4.2 H∞ PID Controllers for the First-Order Plant λ is an adjustable parameter called performance degree. It closely relates to the closed-loop performance: Smaller λ <=> Fast response Larger λ <=> Slow response λ → 0 <=> The optimal kW (s)S(s)k∞ The controller of the corresponding unity feedback loop is C(s) = Q(s) 1 − G(s)Q(s) = 1 K (τ s + 1)(1 + θs/2) λ2s 2 + (2λ + θ/2)s This is a PID controller An important feature: It cancels two poles of the approximate model, or equivalently, two dominant poles of the original model Zhang, W.D., CRC Press, 2011 Version 1.0 14/71
Section 4.2 H PID Controllers for the First-Order Plant Compare the Hoo PID controller with the practical PID controller of the form C(9)=Kc（ ++) 1 TF5+1 Parameters of the Hoo PlD controller are A2 TF=2x+02T1=2+TD=27 ,Kc=K(2X+0/2) If the practical PlD is in the form of parameters of the Hoo PID controller are A2 0 TF- 2X+9121=2+7-T,TD=2 T-TF,Kc=K(2X+0/2) 定0Q0 Zhang.W.D..CRC Press.2011 Version 1.0 15/71
Section 4.2 H∞ PID Controllers for the First-Order Plant Compare the H∞ PID controller with the practical PID controller of the form C(s) = KC 1 + 1 TIs + TDs 1 TF s + 1 Parameters of the H∞ PID controller are TF = λ 2 2λ + θ/2 ,TI = θ 2 + τ,TD = θτ 2TI ,KC = TI K(2λ + θ/2) If the practical PID is in the form of C(s) = KC 1 + 1 TIs + TDs TF s + 1 parameters of the H∞ PID controller are TF = λ 2 2λ + θ/2 ,TI = θ 2 + τ − TF ,TD = θτ 2TI − TF ,KC = TI K(2λ + θ/2) Zhang, W.D., CRC Press, 2011 Version 1.0 15/71