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 6/78Section 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