Section 9.2 2DOF Structure for Unstable Plants The first-order unstable plant can be expressed as K G(s)= e-0s T5- As the main goal of this section is to discuss how to depress the excessive overshoot,it is assumed that the system is required to track a step reference,and at the same time to reject the effect of a step disturbance at the plant output. Consider Structure I.First,design C1(s).The rational part of the plant is MP.Based on the discussion in Section 8.4,we have 0间=R When the disturbance at the plant output is a step and the plant has only one RHP pole,the filter can easily be determined Zhang.W.D..CRC Press.2011 Version 1.0 20/95
Section 9.2 2DOF Structure for Unstable Plants The first-order unstable plant can be expressed as G(s) = K τ s − 1 e −θs As the main goal of this section is to discuss how to depress the excessive overshoot, it is assumed that the system is required to track a step reference, and at the same time to reject the effect of a step disturbance at the plant output. Consider Structure I. First, design C1(s). The rational part of the plant is MP. Based on the discussion in Section 8.4, we have Qopt(s) = τ s − 1 K When the disturbance at the plant output is a step and the plant has only one RHP pole, the filter can easily be determined Zhang, W.D., CRC Press, 2011 Version 1.0 20/95
Section 9.2 2DOF Structure for Unstable Plants The H2 suboptimal controller with the filter is Q(s)(Ts-1)f(A/+1)2eof-1s+1) K(1s+1)2 Then the controller for the disturbance loop is Q(s) C1(s)= 1-G(S)Q(s) 1(rs-1){r[(1/r+1)2er-1s+1} K(1s+1)2-{r[(A1/r+1)2e8/r-1]s+1}e-s Since Iim{(1s+1)2-{r[(1/r+1)2er-1]s+1}e-s}=0 s→1/T There exists a RHP zero-pole cancellation in C1(s).A rational approximation has to be used to remove it 2ac Zhang.W.D..CRC Press.2011 Version 1.0 21/95
Section 9.2 2DOF Structure for Unstable Plants The H2 suboptimal controller with the filter is Q(s) = (τ s − 1){τ [(λ1/τ + 1)2 e θ/τ − 1]s + 1} K(λ1s + 1)2 Then the controller for the disturbance loop is C1(s) = Q(s) 1 − G(s)Q(s) = 1 K (τ s − 1){τ [(λ1/τ + 1)2 e θ/τ − 1]s + 1} (λ1s + 1)2 − {τ [(λ1/τ + 1)2e θ/τ − 1]s + 1}e−θs Since lim s→1/τ {(λ1s + 1)2 − {τ [(λ1/τ + 1)2 e θ/τ − 1]s + 1}e −θs } = 0 There exists a RHP zero-pole cancellation in C1(s). A rational approximation has to be used to remove it Zhang, W.D., CRC Press, 2011 Version 1.0 21/95
Section 9.2 2DOF Structure for Unstable Plants This can be achieved in many ways.For example,the controller can be chosen as a PID controller in the form of G=K:(+六s+TD) With the Maclaurin series expansion,the following result was obtained in Section 8.5: 7=-T+A-2+0-2 2入+0-B1 Kc T -K(2入+0-B1) TD -T31-(03/6-32/2)/(2λ+0-31) T X2+310-02/2 2入+0-51 4口，+@，4定4=定0C Zhang.W.D..CRC Press.2011 Version 1.0 22/95
Section 9.2 2DOF Structure for Unstable Plants This can be achieved in many ways. For example, the controller can be chosen as a PID controller in the form of C1(s) = KC 1 + 1 TIs + TDs With the Maclaurin series expansion, the following result was obtained in Section 8.5: TI = −τ + β1 − λ 2 + β1θ − θ 2/2 2λ + θ − β1 KC = TI −K(2λ + θ − β1) TD = −τ β1 − (θ 3/6 − β1θ 2/2)/(2λ + θ − β1) TI − λ 2 + β1θ − θ 2/2 2λ + θ − β1 Zhang, W.D., CRC Press, 2011 Version 1.0 22/95
Section 9.2 2DOF Structure for Unstable Plants Second,design C2(s).Regard the feedback loop consisting of C1(s)and G(s)as an augmented plant.The transfer function of the augmented plant is G(s)Ci(s) T(S)=1+G(s)G(9 The optimal C2(s)should be the inverse of T(s)after the time delay in its numerator is removed.However,such a design procedure is tedious.Since Ci(s)is an approximation of the ideal controller,C2(s)can be chosen as the inverse of the ideal T(s) after the time delay in its numerator is removed.Then (A15+1)2 Ca(5)=[r/T+1)7-1s+1)(A2s+1) Zhang,W.D..CRC Press.2011 Version 1.0 23/95
Section 9.2 2DOF Structure for Unstable Plants Second, design C2(s). Regard the feedback loop consisting of C1(s) and G(s) as an augmented plant. The transfer function of the augmented plant is T(s) = G(s)C1(s) 1 + G(s)C1(s) The optimal C2(s) should be the inverse of T(s) after the time delay in its numerator is removed. However, such a design procedure is tedious. Since C1(s) is an approximation of the ideal controller, C2(s) can be chosen as the inverse of the ideal T(s) after the time delay in its numerator is removed. Then C2(s) = (λ1s + 1)2 {τ [(λ1/τ + 1)2e θ/τ − 1]s + 1}(λ2s + 1) Zhang, W.D., CRC Press, 2011 Version 1.0 23/95
Section 9.2 2DOF Structure for Unstable Plants Now consider Structure Il.In Structure Il,C3(s)equals Ci(s)in Structure I.The optimal C4(s)can be chosen as the inverse of the rational part of the plant: Ts-1 C4(s)=K25+I C3(s)is an approximation of the ideal controller.Hence,the reference response of the system with C3(s)approximates to the reference response of the system with the ideal controller: 1 T(⑤)=25+1 The corresponding time domain response is y0={--8 t<0 4口：4@4242定90C Zhang.W.D..CRC Press.2011 Version 1.0 24/95
Section 9.2 2DOF Structure for Unstable Plants Now consider Structure II. In Structure II, C3(s) equals C1(s) in Structure I. The optimal C4(s) can be chosen as the inverse of the rational part of the plant: C4(s) = τ s − 1 K(λ2s + 1) C3(s) is an approximation of the ideal controller. Hence, the reference response of the system with C3(s) approximates to the reference response of the system with the ideal controller: Tr(s) = 1 λ2s + 1 e −θs The corresponding time domain response is y(t) = 0 t < θ 1 − e −(t−θ)/λ2 t ≥ θ Zhang, W.D., CRC Press, 2011 Version 1.0 24/95