Section 5.1 H2 PID Controllers for the First-Order Plant When the PID controller is in the form of co=kc(+点) parameters are Terorg).To )e 0入 The optimal performance,approximately,is min llW(s)S(s)ll2 =V0 1-0s/2 4口，+@，4定4定90C Zhang.W.D..CRC Press.2011 Version 1.0 10/78
Section 5.1 H2 PID Controllers for the First-Order Plant When the PID controller is in the form of C(s) = KC 1 + 1 TIs TDs + 1 TF s + 1 parameters are TF = θλ 2(λ + θ) ,TI = τ (or θ 2 ),TD = θ 2 (or τ ),KC = TI K(λ + θ) The optimal performance, approximately, is min kW (s)S(s)k2 = θ 1 − θs/2 2 = √ θ Zhang, W.D., CRC Press, 2011 Version 1.0 10/78
Section 5.1 H2 PID Controllers for the First-Order Plant When the PID controller is in the form of co=kc(+点) parameters are T=1万=o3n=如)kK4西 0入 The optimal performance,approximately,is minlW(a)s()i =V 4口，+@，4定4定90C Zhang.W.D..CRC Press.2011 Version 1.0 10/78
Section 5.1 H2 PID Controllers for the First-Order Plant When the PID controller is in the form of C(s) = KC 1 + 1 TIs TDs + 1 TF s + 1 parameters are TF = θλ 2(λ + θ) ,TI = τ (or θ 2 ),TD = θ 2 (or τ ),KC = TI K(λ + θ) The optimal performance, approximately, is min kW (s)S(s)k2 = θ 1 − θs/2 2 = √ θ Zhang, W.D., CRC Press, 2011 Version 1.0 10/78
Section 5.2 Quantitative Tuning of H2 PID Controllers 5.2 Quantitative Tuning of H2 PID Controllers Nominal stability:The larger the time delay,the more difficult to stabilize the closed-loop system As long as the performance degree is greater than a lower bound, the closed-loop system is stable Nominal performance:The existence of time delays adversely affects the performance of the closed-loop system.The performance is worse and worse with the increase of the time delay The performance degree of the H2 PID controller has a similar function to that of the Ho PlD controller: When there is no modeling error,the performance degree can be used to tune the response shape of the nominal closed-loop system quantitatively 4口，+@4定生 定)QC Zhang.W.D..CRC Press.2011 Version 1.0 11/78
Section 5.2 Quantitative Tuning of H2 PID Controllers 5.2 Quantitative Tuning of H2 PID Controllers Nominal stability: The larger the time delay, the more difficult to stabilize the closed-loop system As long as the performance degree is greater than a lower bound, the closed-loop system is stable Nominal performance: The existence of time delays adversely affects the performance of the closed-loop system. The performance is worse and worse with the increase of the time delay The performance degree of the H2 PID controller has a similar function to that of the H∞ PID controller: When there is no modeling error, the performance degree can be used to tune the response shape of the nominal closed-loop system quantitatively Zhang, W.D., CRC Press, 2011 Version 1.0 11/78
Section 5.2 Quantitative Tuning of H2 PID Controllers 5.2 Quantitative Tuning of H2 PID Controllers Nominal stability:The larger the time delay,the more difficult to stabilize the closed-loop system As long as the performance degree is greater than a lower bound, the closed-loop system is stable Nominal performance:The existence of time delays adversely affects the performance of the closed-loop system.The performance is worse and worse with the increase of the time delay The performance degree of the H2 PID controller has a similar function to that of the Hoo PID controller: When there is no modeling error,the performance degree can be used to tune the response shape of the nominal closed-loop system quantitatively 24c Zhang.W.D..CRC Press.2011 Version 1.0 11/78
Section 5.2 Quantitative Tuning of H2 PID Controllers 5.2 Quantitative Tuning of H2 PID Controllers Nominal stability: The larger the time delay, the more difficult to stabilize the closed-loop system As long as the performance degree is greater than a lower bound, the closed-loop system is stable Nominal performance: The existence of time delays adversely affects the performance of the closed-loop system. The performance is worse and worse with the increase of the time delay The performance degree of the H2 PID controller has a similar function to that of the H∞ PID controller: When there is no modeling error, the performance degree can be used to tune the response shape of the nominal closed-loop system quantitatively Zhang, W.D., CRC Press, 2011 Version 1.0 11/78
Section 5.2 Quantitative Tuning of H2 PID Controllers 0.4 0.3 0.2 0.1 0.0 0.0 0.3 0.6 0.9 1.2 0 Figure:Relationship between the performance degree and the overshoot e.g.,12%overshoot ->A=0.30 4口，4@4主4生定分QC Zhang,W.D..CRC Press.2011 Version 1.0 12/78
Section 5.2 Quantitative Tuning of H2 PID Controllers Figure: Relationship between the performance degree and the overshoot e.g., 12% overshoot − > λ = 0.3θ Zhang, W.D., CRC Press, 2011 Version 1.0 12/78