Chapter 6 Control of Stable Plants 4口，+@，4定4=定0C Zhang.W.D..CRC Press.2011 Version 1.0 1/74
Chapter 6 Control of Stable Plants Zhang, W.D., CRC Press, 2011 Version 1.0 1/74
Control of Stable Plants 16.1 The Quasi-Ho Smith Predictor 26.2 The H2 Optimal Controller and the Smith Predictor 36.3 Equivalents of the Optimal Controller 46.4 The PID Controller and High-Order Controllers 56.5 Choice of Weighting Functions 66.6 Simplified Tuning for Quantitative Robustness 4口，+@，4定4定90C Zhang.W.D..CRC Press.2011 Version 1.0 2/74
Control of Stable Plants 1 6.1 The Quasi-H∞ Smith Predictor 2 6.2 The H2 Optimal Controller and the Smith Predictor 3 6.3 Equivalents of the Optimal Controller 4 6.4 The PID Controller and High-Order Controllers 5 6.5 Choice of Weighting Functions 6 6.6 Simplified Tuning for Quantitative Robustness Zhang, W.D., CRC Press, 2011 Version 1.0 2/74
Section 6.1 The Quasi-Ho Smith Predictor 6.1 The Quasi-Ho Smith Predictor Chapter 4 and Chapter 5:The controller is analytically designed by minimizing the weighted sensitivity function This section:The controller is analytically designed by specifying the desired closed-loop response Actually,a simplified version of this method was already used in Sections 5.5 and 5.6 Consider the diagram of the Smith predictor in Figure,where G(s) is the plant,G(s)is its model,and Go(s)is the delay-free part of G(s).If the closed-loop transfer function T(s)is known,the controller of the Smith predictor is T(s) R(s)= G(s)-T(s)Go(s) 4口，+@，4定4=定0C Zhang.W.D..CRC Press.2011 Version 1.0 3/74
Section 6.1 The Quasi-H∞ Smith Predictor 6.1 The Quasi-H∞ Smith Predictor Chapter 4 and Chapter 5: The controller is analytically designed by minimizing the weighted sensitivity function This section: The controller is analytically designed by specifying the desired closed-loop response Actually, a simplified version of this method was already used in Sections 5.5 and 5.6 Consider the diagram of the Smith predictor in Figure, where G˜ (s) is the plant, G(s) is its model, and Go(s) is the delay-free part of G(s). If the closed-loop transfer function T(s) is known, the controller of the Smith predictor is R(s) = T(s) G(s) − T(s)Go(s) Zhang, W.D., CRC Press, 2011 Version 1.0 3/74
Section 6.1 The Quasi-Ho Smith Predictor 6.1 The Quasi-Ho Smith Predictor Chapter 4 and Chapter 5:The controller is analytically designed by minimizing the weighted sensitivity function This section:The controller is analytically designed by specifying the desired closed-loop response Actually,a simplified version of this method was already used in Sections 5.5 and 5.6 Consider the diagram of the Smith predictor in Figure,where G(s) is the plant,G(s)is its model,and Go(s)is the delay-free part of G(s).If the closed-loop transfer function T(s)is known,the controller of the Smith predictor is T(s) R(6)=G(3)-T(s)G.(5) 4口，+@，4定4=定0C Zhang.W.D..CRC Press.2011 Version 1.0 3/74
Section 6.1 The Quasi-H∞ Smith Predictor 6.1 The Quasi-H∞ Smith Predictor Chapter 4 and Chapter 5: The controller is analytically designed by minimizing the weighted sensitivity function This section: The controller is analytically designed by specifying the desired closed-loop response Actually, a simplified version of this method was already used in Sections 5.5 and 5.6 Consider the diagram of the Smith predictor in Figure, where G˜ (s) is the plant, G(s) is its model, and Go(s) is the delay-free part of G(s). If the closed-loop transfer function T(s) is known, the controller of the Smith predictor is R(s) = T(s) G(s) − T(s)Go(s) Zhang, W.D., CRC Press, 2011 Version 1.0 3/74
Section 6.1 The Quasi-Ho Smith Predictor R G G-G Figure:Diagram of the Smith predictor Key of the design How to choose the desired closed-loop transfer function To introduce the idea clearly,the simplest case is considered first. The general result will be inductively derived 定00 Zhang.W.D..CRC Press.2011 Version 1.0 4/74
Section 6.1 The Quasi-H∞ Smith Predictor Figure: Diagram of the Smith predictor Key of the design How to choose the desired closed-loop transfer function To introduce the idea clearly, the simplest case is considered first. The general result will be inductively derived Zhang, W.D., CRC Press, 2011 Version 1.0 4/74
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
