Chapter 7 Control of Integrating Plants 4口，+@，4定4=定0C Zhang.W.D..CRC Press.2011 Version 1.0 1/79
Chapter 7 Control of Integrating Plants Zhang, W.D., CRC Press, 2011 Version 1.0 1/79
Control of Integrating Plants 17.1 The Feature of Integrating Systems 27.2 Hoo PID Controllers for Integrating Plants 37.3 H2 PID Controllers for Integrating Plants 47.4 Controller Design for General Integrating Plants 57.5 Maclaurin PID Controllers for Integrating Plants 67.6 Best Achievable Performance of a PID Controllers 4口，4@，4定4定0C Zhang.W.D..CRC Press.2011 Version 1.0 2/79
Control of Integrating Plants 1 7.1 The Feature of Integrating Systems 2 7.2 H∞ PID Controllers for Integrating Plants 3 7.3 H2 PID Controllers for Integrating Plants 4 7.4 Controller Design for General Integrating Plants 5 7.5 Maclaurin PID Controllers for Integrating Plants 6 7.6 Best Achievable Performance of a PID Controllers Zhang, W.D., CRC Press, 2011 Version 1.0 2/79
Section 7.1 The Feature of Integrating Systems 7.1 The Feature of Integrating Systems Assumption:Integrating plants in this book do not have any open RHP poles.Those with poles in the open RHP are included in unstable plants.This assumption is made solely for simplicity of presentation Consider the feedback control loop in Figure,where G(s)is an integrating plant and C(s)is the controller 4口，+@，4定4=定0C Zhang,W.D..CRC Press.2011 Version 1.0 3/79
Section 7.1 The Feature of Integrating Systems 7.1 The Feature of Integrating Systems Assumption: Integrating plants in this book do not have any open RHP poles. Those with poles in the open RHP are included in unstable plants. This assumption is made solely for simplicity of presentation Consider the feedback control loop in Figure, where G(s) is an integrating plant and C(s) is the controller Zhang, W.D., CRC Press, 2011 Version 1.0 3/79
Section 7.1 The Feature of Integrating Systems Internal Stability The closed-loop system is internally stable if and only if all elements in the transfer matrix H(s)are stable: [图]=[周] where G(s)C(s) G(s) H(s)= 1+ G(s)C(s) 1+G(s)C(s) C(s) G(s)C(s) 1+G(s)C(s) 1+G(s)C(s) Since the Youla parameterization for stable plants cannot be used for integrating plants,the following transfer function is defined: Q(s)= C(s) 1+G(s)C(s) 4口，+@，4定4=定0C Zhang.W.D..CRC Press.2011 Version 1.0 4/79
Section 7.1 The Feature of Integrating Systems Internal Stability The closed-loop system is internally stable if and only if all elements in the transfer matrix H(s)are stable: y(s) u(s) = H(s) r(s) d 0 (s) where H(s) = G(s)C(s) 1 + G(s)C(s) G(s) 1 + G(s)C(s) C(s) 1 + G(s)C(s) −G(s)C(s) 1 + G(s)C(s) Since the Youla parameterization for stable plants cannot be used for integrating plants, the following transfer function is defined: Q(s) = C(s) 1 + G(s)C(s) Zhang, W.D., CRC Press, 2011 Version 1.0 4/79
Section 7.1 The Feature of Integrating Systems The transfer function Q(s)is in fact the IMC controller.Then the transfer matrix H(s)becomes H(s) G(s)Q(s)[1-G(s)Q(s)]G(s) Q(s) -G(s)Q(s) Since G(s)is not stable,the stability of Q(s)cannot guarantee the stability of the closed-loop system. Theorem Assume that G(s)is an integrating plant.The unity feedback loop shown in Figure is intemally stable if and only if D Q(s)is stable 1-G(s)Q(sG(s)is stable 4口，+@，4定4=定0C Zhang.W.D..CRC Press.2011 Version 1.0 5/79
Section 7.1 The Feature of Integrating Systems The transfer function Q(s) is in fact the IMC controller. Then the transfer matrix H(s) becomes H(s) = G(s)Q(s) [1 − G(s)Q(s)]G(s) Q(s) −G(s)Q(s) Since G(s) is not stable, the stability of Q(s) cannot guarantee the stability of the closed-loop system. Theorem Assume that G(s) is an integrating plant. The unity feedback loop shown in Figure is internally stable if and only if 1 Q(s) is stable. 2 [1 − G(s)Q(s)]G(s) is stable. Zhang, W.D., CRC Press, 2011 Version 1.0 5/79