Section 9.1 The 2DOF Structure for Stable Plants C1(s):The design of Cl(s)is the same as that for the controller in a 1DOF system C2(s):After C1(s)is designed,the loop consisting of C1(s)and G(s)is viewed as an augmented plant,of which the transfer function is denoted by T(s).The system that consists of C2(s) and T(s)forms the IMC structure with an exact model. Accordingly,C2(s)can be directly designed To illustrate the design procedure,consider the following plant: G(s)=-k +1 The system is required to track a step reference,and at the same time to reject the disturbance that is in the form of a ramp at the plant output 4口，404它4生定0C Zhang,W.D..CRC Press.2011 Version 1.0 8/95
Section 9.1 The 2DOF Structure for Stable Plants C1(s): The design of C1(s) is the same as that for the controller in a 1DOF system C2(s): After C1(s) is designed, the loop consisting of C1(s) and G(s) is viewed as an augmented plant, of which the transfer function is denoted by T(s). The system that consists of C2(s) and T(s) forms the IMC structure with an exact model. Accordingly, C2(s) can be directly designed To illustrate the design procedure, consider the following plant: G(s) = K τ s + 1 e −θs The system is required to track a step reference, and at the same time to reject the disturbance that is in the form of a ramp at the plant output Zhang, W.D., CRC Press, 2011 Version 1.0 8/95
Section 9.1 The 2DOF Structure for Stable Plants First of all,design Ci(s)for disturbance rejection.By utilizing the IMC controller Q(s),C1(s)can be expressed as Q(s) G(s)=1-G(5)Q(5) The rational part of the plant is MP.Utilizing (??)we have Qopt(s)=Ts+1 K Since the disturbance is a ramp,a Type 2 filter should be introduced.In light of the discussion in Section 5.7, J5)= 2λ1s+1 (1s+1)2 where A1 is the performance degree for disturbance rejection. Simple computations give Q(s)=Qop:(s)J(s)=(T5+1)(2A15+1) K(1s+1)2 定9aC Zhang.W.D..CRC Press.2011 Version 1.0 9/95
Section 9.1 The 2DOF Structure for Stable Plants First of all, design C1(s) for disturbance rejection. By utilizing the IMC controller Q(s), C1(s) can be expressed as C1(s) = Q(s) 1 − G(s)Q(s) The rational part of the plant is MP. Utilizing (??), we have Qopt(s) = τ s + 1 K Since the disturbance is a ramp, a Type 2 filter should be introduced. In light of the discussion in Section 5.7, J(s) = 2λ1s + 1 (λ1s + 1)2 where λ1 is the performance degree for disturbance rejection. Simple computations give Q(s) = Qopt(s)J(s) = (τ s + 1)(2λ1s + 1) K(λ1s + 1)2 Zhang, W.D., CRC Press, 2011 Version 1.0 9/95
Section 9.1 The 2DOF Structure for Stable Plants Next,design C2(s)for tracking.The loop that consists of C1(s) and G(s)is regarded as an augmented plant,whose transfer function is 2λ15+1 T(s)-G()Q()- Obviously.T(s)is stable.According to the design procedure for the H2 controller,the optimal C2(s)is the inverse of the rational part of T(s).For a step reference the suboptimal controller is (15+1)2 C2(9)=2x15+1)25+ where A2 is the performance degree for disturbance rejection Let T(s)denote the transfer function from the reference to the system output,and Td(s)denote the transfer function from the disturbance at the plant input to the system output 290 Zhang.W.D..CRC Press.2011 Version 1.0 10/95
Section 9.1 The 2DOF Structure for Stable Plants Next, design C2(s) for tracking. The loop that consists of C1(s) and G(s) is regarded as an augmented plant, whose transfer function is T(s) = G(s)Q(s) = 2λ1s + 1 (λ1s + 1)2 e −θs Obviously, T(s) is stable. According to the design procedure for the H2 controller, the optimal C2(s) is the inverse of the rational part of T(s). For a step reference the suboptimal controller is C2(s) = (λ1s + 1)2 (2λ1s + 1)(λ2s + 1) where λ2 is the performance degree for disturbance rejection Let Tr(s) denote the transfer function from the reference to the system output, and Td (s) denote the transfer function from the disturbance at the plant input to the system output Zhang, W.D., CRC Press, 2011 Version 1.0 10/95
Section 9.1 The 2DOF Structure for Stable Plants It is easy to verify that the response of the reference loop is 1 T(s)=λ25+1 -0s of which the time domain response is T0={9-e-9 t<0 The reference response can be independently adjusted by the performance degree X2 The response of the disturbance loop can be written as --品 4口：4@4242定90C Zhang.W.D..CRC Press.2011 Version 1.0 11/95
Section 9.1 The 2DOF Structure for Stable Plants It is easy to verify that the response of the reference loop is Tr(s) = 1 λ2s + 1 e −θs of which the time domain response is Tr(t) = 0 t < θ 1 − e −(t−θ)/λ2 t ≥ θ The reference response can be independently adjusted by the performance degree λ2 The response of the disturbance loop can be written as Td (s) = K τ s + 1 e −θs 1 − 2λ1s + 1 (λ1s + 1)2 e −θs Zhang, W.D., CRC Press, 2011 Version 1.0 11/95
Section 9.1 The 2DOF Structure for Stable Plants The corresponding time domain response is 0 t<0 K(1-e-(-)/r) 0≤t<20 Ta(t)= K e-(t-2)/A1_ e--29-e--9r 为-T t≥20 The disturbance response can be independently adjusted by employing the performance degree X1 The disturbance loop of a 2DOF system cannot provide better disturbance rejection ability than a 1DOF system.Nevertheless, since the disturbance response and the reference response can be adjusted independently,better disturbance response and reference response can be reached simultaneously in a 2DOF system 9a0 Zhang.W.D..CRC Press.2011 Version 1.0 12/95
Section 9.1 The 2DOF Structure for Stable Plants The corresponding time domain response is Td (t) = 0 t < θ K(1 − e −(t−θ)/τ ) θ ≤ t < 2θ K " λ1 λ1−τ e −(t−2θ)/λ1− τ λ1−τ e −(t−2θ)/τ − e −(t−θ)/τ # t ≥ 2θ The disturbance response can be independently adjusted by employing the performance degree λ1 The disturbance loop of a 2DOF system cannot provide better disturbance rejection ability than a 1DOF system. Nevertheless, since the disturbance response and the reference response can be adjusted independently, better disturbance response and reference response can be reached simultaneously in a 2DOF system Zhang, W.D., CRC Press, 2011 Version 1.0 12/95