Section 6.1 The Quasi-Ho Smith Predictor Case 1: Consider the following stable rational plant of MP: KN_(s) G(5)=M-(5) where K is the gain,N-(s)and M-(s)are the polynomials with roots in the LHP,N_(0)=M-(0)=1,and deg{N-}<deg{M-}. It is easy to control such a plant.For the Hoo performance index and the weighting function W(s)=1/s we have llW(s)s(s)ll lW(s)[1-G(s)Q(s)]lloo ≥0 The following controller is the optimal one: Qopt(s)= M_(s) KN_(s) 4口，+@，4定4=定0C Zhang.W.D..CRC Press.2011 Version 1.0 5/74
Section 6.1 The Quasi-H∞ Smith Predictor Case 1: Consider the following stable rational plant of MP: G(s) = KN−(s) M−(s) where K is the gain, N−(s) and M−(s) are the polynomials with roots in the LHP, N−(0) = M−(0) = 1, and deg{N−} ≤ deg{M−}. It is easy to control such a plant. For the H∞ performance index and the weighting function W (s) = 1/s we have kW (s)S(s)k∞ = kW (s)[1 − G(s)Q(s)]k∞ ≥ 0 The following controller is the optimal one: Qopt(s) = M−(s) KN−(s) Zhang, W.D., CRC Press, 2011 Version 1.0 5/74
Section 6.1 The Quasi-Ho Smith Predictor Introduce the filter J(s)= 1 (As+1)9 where A is the performance degree.In light of the discussion in Section 5.7,nj is chosen as follows: 可={eM}-eM}日eM产eN} deg{M_}deg{N_} The suboptimal proper controller is M_(s) Q(S)=KN-(5)s+1” The closed-loop transfer function is 1 T(s)=7 s+1)% 4口，+@，4定4=定0C Zhang.W.D..CRC Press.2011 Version 1.0 6/74
Section 6.1 The Quasi-H∞ Smith Predictor Introduce the filter J(s) = 1 (λs + 1)nj where λ is the performance degree. In light of the discussion in Section 5.7, nj is chosen as follows: nj = deg{M−} − deg{N−} deg{M−} > deg{N−} 1 deg{M−} = deg{N−} The suboptimal proper controller is Q(s) = M−(s) KN−(s)(λs + 1)nj The closed-loop transfer function is T(s) = 1 (λs + 1)nj Zhang, W.D., CRC Press, 2011 Version 1.0 6/74
Section 6.1 The Quasi-Ho Smith Predictor Case 2: Consider a bit more complex case.Assume that the plant has a zero in the RHP: Gs)=KW-(s-2s+1) M-(s) where z>0,N_(0)=M_(0)=1,and deg{N-+1<deg{M-).Solve the weighted sensitivity problem again: llW(s)S(s)ll=lIW(s)[1-G(s)Q(s)]ll ≥IW(z)I The optimal controller is obtained as follows: Qopt(s)= M-(s) KN_(s) 4口，+@，4定4=定0C Zhang.W.D..CRC Press.2011 Version 1.0 7/74
Section 6.1 The Quasi-H∞ Smith Predictor Case 2: Consider a bit more complex case. Assume that the plant has a zero in the RHP: G(s) = KN−(s)(−z −1 r s + 1) M−(s) where zr > 0, N−(0) = M−(0) = 1, and deg{N−} + 1 ≤ deg{M−}. Solve the weighted sensitivity problem again: kW (s)S(s)k∞ = kW (s)[1 − G(s)Q(s)]k∞ ≥ |W (zr)| The optimal controller is obtained as follows: Qopt(s) = M−(s) KN−(s) Zhang, W.D., CRC Press, 2011 Version 1.0 7/74
Section 6.1 The Quasi-Ho Smith Predictor Introduce the following filter: J(5)= 1 (As+1)9 where nj=deg{M-}-deg{N_} The suboptimal proper controller is M_(s) Q(s)=KN_(s)(Xs+1)mi The closed-loop transfer function can be written as T(s)= -21s+1 (As+1) 4口，+@，4定4=定0C Zhang.W.D..CRC Press.2011 Version 1.0 8/74
Section 6.1 The Quasi-H∞ Smith Predictor Introduce the following filter: J(s) = 1 (λs + 1)nj where nj = deg{M−} − deg{N−} The suboptimal proper controller is Q(s) = M−(s) KN−(s)(λs + 1)nj The closed-loop transfer function can be written as T(s) = −z −1 r s + 1 (λs + 1)nj Zhang, W.D., CRC Press, 2011 Version 1.0 8/74
Section 6.1 The Quasi-Ho Smith Predictor Case 3: Now,consider the general stable rational plant described by G(s)= KN+(s)N_(s) M-(S) where N-(s)and M-(s)are the polynomials with roots in the LHP,N(s)is a polynomial with roots in the RHP, N+(0)=N_(0)=M_(0)=1,and deg{N++deg{N-}<deg{M-).As this is a rational plant, Go(s)=G(s) Motivated by the foregoing design procedures,the following function is chosen as the desired closed-loop transfer function: T(s)=N4(s)J(s) where J(s)is a filter 4口+@4定4生，定00 Zhang.W.D..CRC Press.2011 Version 1.0 9/74
Section 6.1 The Quasi-H∞ Smith Predictor Case 3: Now, consider the general stable rational plant described by G(s) = KN+(s)N−(s) M−(s) where N−(s) and M−(s) are the polynomials with roots in the LHP, N+(s) is a polynomial with roots in the RHP, N+(0) = N−(0) = M−(0) = 1, and deg{N+} + deg{N−} ≤ deg{M−}. As this is a rational plant, Go(s) = G(s) Motivated by the foregoing design procedures, the following function is chosen as the desired closed-loop transfer function: T(s) = N+(s)J(s) where J(s) is a filter Zhang, W.D., CRC Press, 2011 Version 1.0 9/74