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 5/78Section 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