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