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 4/78Section 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