Section 6.1 The Quasi-Ho Smith Predictor 1 J(5)=5+1” and nj= deg{M-}-deg{N-}deg{M-}>deg{N_} 1 deg{M_}=deg{N_) The feature of the closed-loop transfer function is that it has the same RHP zeros as the plant. Once the desired T(s)is determined,the controller of the Smith predictor can be analytically derived through T(s) R(s)= G(s)-T(s)Go(s) 1 M_(s) KN-(s)[(As+1)9-N+(S】 Zhang.W.D..CRC Press.2011 Version 1.0 10/74
Section 6.1 The Quasi-H∞ Smith Predictor J(s) = 1 (λs + 1)nj and nj = deg{M−} − deg{N−} deg{M−} > deg{N−} 1 deg{M−} = deg{N−} The feature of the closed-loop transfer function is that it has the same RHP zeros as the plant. Once the desired T(s) is determined, the controller of the Smith predictor can be analytically derived through R(s) = T(s) G(s) − T(s)Go(s) = 1 K M−(s) N−(s)[(λs + 1)nj − N+(s)] Zhang, W.D., CRC Press, 2011 Version 1.0 10/74
Section 6.1 The Quasi-Ho Smith Predictor For rational plants,the unity feedback loop controller C(s)is identical to R(s).The controller has the same order as that of the plant.The corresponding Q(s)is Q(s)= T(s) M-(s) G(s) N-(s) Case 4: When there is a time delay in the plant,the basic idea of designing the Smith predictor is to move the time delay out from the feedback loop,so that the controller can be designed for the rational part of the plant.Along this line,the design procedure for rational plants can be extended to plants with time delays 4口，+@，4定4定90C Zhang.W.D..CRC Press.2011 Version 1.0 11/74
Section 6.1 The Quasi-H∞ Smith Predictor For rational plants, the unity feedback loop controller C(s) is identical to R(s). The controller has the same order as that of the plant. The corresponding Q(s) is Q(s) = T(s) G(s) = M−(s) N−(s) Case 4: When there is a time delay in the plant, the basic idea of designing the Smith predictor is to move the time delay out from the feedback loop, so that the controller can be designed for the rational part of the plant. Along this line, the design procedure for rational plants can be extended to plants with time delays Zhang, W.D., CRC Press, 2011 Version 1.0 11/74
Section 6.1 The Quasi-Ho Smith Predictor For rational plants,the unity feedback loop controller C(s)is identical to R(s).The controller has the same order as that of the plant.The corresponding Q(s)is Q(s)= T(s) M-(s) G(s) N-(s) Case 4: When there is a time delay in the plant,the basic idea of designing the Smith predictor is to move the time delay out from the feedback loop,so that the controller can be designed for the rational part of the plant.Along this line,the design procedure for rational plants can be extended to plants with time delays Zhang,W.D..CRC Press.2011 Version 1.0 11/74
Section 6.1 The Quasi-H∞ Smith Predictor For rational plants, the unity feedback loop controller C(s) is identical to R(s). The controller has the same order as that of the plant. The corresponding Q(s) is Q(s) = T(s) G(s) = M−(s) N−(s) Case 4: When there is a time delay in the plant, the basic idea of designing the Smith predictor is to move the time delay out from the feedback loop, so that the controller can be designed for the rational part of the plant. Along this line, the design procedure for rational plants can be extended to plants with time delays Zhang, W.D., CRC Press, 2011 Version 1.0 11/74
Section 6.1 The Quasi-Ho Smith Predictor Assume that the plant with time delay is KN+s)N-(⑤e-s G(s)= M_(s) where 0 is the time delay.The desired closed-loop transfer function can be chosen as T(s)=N+(s)J(s)e-0s where J(s)is identical to (1).The R(s)and Q(s)corresponding to this desired closed-loop transfer function is the same as those in (1)and (1)respectively,but C(s)contains a time delay: C(5)= Q(s) M_(s) 1-G(5)Q5=KN-5As+1)9-N+(S)e-阿] which is irrational 定)AC Zhang.W.D..CRC Press.2011 Version 1.0 12/74
Section 6.1 The Quasi-H∞ Smith Predictor Assume that the plant with time delay is G(s) = KN+(s)N−(s) M−(s) e −θs where θ is the time delay. The desired closed-loop transfer function can be chosen as T(s) = N+(s)J(s)e −θs where J(s) is identical to (1). The R(s) and Q(s) corresponding to this desired closed-loop transfer function is the same as those in (1) and (1) respectively, but C(s) contains a time delay: C(s) = Q(s) 1 − G(s)Q(s) = 1 K M−(s) N−(s)[(λs + 1)nj − N+(s)e−θs ] which is irrational Zhang, W.D., CRC Press, 2011 Version 1.0 12/74
Section 6.1 The Quasi-Ho Smith Predictor Stability is a basic requirement for control system design.A question associated with the design is whether the closed-loop system is internally stable. Theorem The closed-loop system is interally stable Proof. Follows directly from the Youla parameterization for stable plants The design method here is in fact a pole placement method.Since the method is developed based on special H solutions,it is named quasi-Hoc control 4口，+@，4定4定90C Zhang.W.D..CRC Press.2011 Version 1.0 13/74
Section 6.1 The Quasi-H∞ Smith Predictor Stability is a basic requirement for control system design. A question associated with the design is whether the closed-loop system is internally stable. Theorem The closed-loop system is internally stable Proof. Follows directly from the Youla parameterization for stable plants The design method here is in fact a pole placement method. Since the method is developed based on special H∞ solutions, it is named quasi-H∞ control Zhang, W.D., CRC Press, 2011 Version 1.0 13/74