Chapter 8( Control of Unstable Plants 4口，+@，4定4=定0C Zhang.W.D..CRC Press.2011 Version 1.0 1/89
Chapter 8 Control of Unstable Plants Zhang, W.D., CRC Press, 2011 Version 1.0 1/89
Control of Unstable Plants 18.1 Controller Parameterization for General Plants 28.2 Ho PID Controllers for Unstable Plants 3 8.3 H2 PID Controllers for Unstable Plants 4 8.4 Performance Limitation and Robustness 58.5 Maclaurin PID Controllers for Unstable Plants 68.6 PID Design for the Best Achievable Performance 78.6 All Stabilizing PID Controllers for Unstable Plants 4口，44定4生，分QC Zhang.W.D..CRC Press.2011 Version 1.0 2/89
Control of Unstable Plants 1 8.1 Controller Parameterization for General Plants 2 8.2 H∞ PID Controllers for Unstable Plants 3 8.3 H2 PID Controllers for Unstable Plants 4 8.4 Performance Limitation and Robustness 5 8.5 Maclaurin PID Controllers for Unstable Plants 6 8.6 PID Design for the Best Achievable Performance 7 8.6 All Stabilizing PID Controllers for Unstable Plants Zhang, W.D., CRC Press, 2011 Version 1.0 2/89
Section 8.1 Controller Parameterization for General Plants 8.1 Controller Parameterization for General Plants Three types of plants:Stable,integrating,and unstable Why categorizing:Controllers can be designed aiming at the reduced scope of plants,so that the design is more effective and simple controllers are easier to obtain Plants considered in this section:Unstable plants with time delay Difficulties in the design: The existence of RHP poles makes the stabilization of the closed-loop system difficult to achieve 2The combined effect of the RHP poles and the time delay greatly limits the achievable performance 4口，+@4定4生 定9AC Zhang.W.D..CRC Press.2011 Version 1.0 3/89
Section 8.1 Controller Parameterization for General Plants 8.1 Controller Parameterization for General Plants Three types of plants: Stable, integrating, and unstable Why categorizing: Controllers can be designed aiming at the reduced scope of plants, so that the design is more effective and simple controllers are easier to obtain Plants considered in this section: Unstable plants with time delay Difficulties in the design: 1 The existence of RHP poles makes the stabilization of the closed-loop system difficult to achieve 2 The combined effect of the RHP poles and the time delay greatly limits the achievable performance Zhang, W.D., CRC Press, 2011 Version 1.0 3/89
Section 8.1 Controller Parameterization for General Plants 8.1 Controller Parameterization for General Plants Three types of plants:Stable,integrating,and unstable Why categorizing:Controllers can be designed aiming at the reduced scope of plants,so that the design is more effective and simple controllers are easier to obtain Plants considered in this section:Unstable plants with time delay Difficulties in the design: The existence of RHP poles makes the stabilization of the closed-loop system difficult to achieve 2 The combined effect of the RHP poles and the time delay greatly limits the achievable performance 定9QC Zhang.W.D..CRC Press.2011 Version 1.0 3/89
Section 8.1 Controller Parameterization for General Plants 8.1 Controller Parameterization for General Plants Three types of plants: Stable, integrating, and unstable Why categorizing: Controllers can be designed aiming at the reduced scope of plants, so that the design is more effective and simple controllers are easier to obtain Plants considered in this section: Unstable plants with time delay Difficulties in the design: 1 The existence of RHP poles makes the stabilization of the closed-loop system difficult to achieve 2 The combined effect of the RHP poles and the time delay greatly limits the achievable performance Zhang, W.D., CRC Press, 2011 Version 1.0 3/89
Section 8.1 Controller Parameterization for General Plants Assumption For the control system with an unstable plant,there exists a limit on the ratio of the time constant to the time delay.If no further explanation is given,it is assumed that the condition is satisfied The design method in this chapter is based on a new parameterization.As a matter of fact,special cases of the new parameterization were already used in the foregoing chapters Why not the Youla parameterization? It cannot be directly used for a plant with time delay 2 To obtain it,one has to compute the coprime factorization of the plant.No analytical methods are available 3 The Q(s)in the general parameterization no longer corresponds to the IMC controller 定9QC Zhang.W.D..CRC Press.2011 Version 1.0 4/89
Section 8.1 Controller Parameterization for General Plants Assumption For the control system with an unstable plant, there exists a limit on the ratio of the time constant to the time delay. If no further explanation is given, it is assumed that the condition is satisfied The design method in this chapter is based on a new parameterization. As a matter of fact, special cases of the new parameterization were already used in the foregoing chapters Why not the Youla parameterization? 1 It cannot be directly used for a plant with time delay 2 To obtain it, one has to compute the coprime factorization of the plant. No analytical methods are available 3 The Q(s) in the general parameterization no longer corresponds to the IMC controller Zhang, W.D., CRC Press, 2011 Version 1.0 4/89
Section 8.1 Controller Parameterization for General Plants Consider the unity feedback loop,in which the transfer function of the plant is given by G5)= KN+(s)N-(s)e-0s M+(s)M-(s) Assume that G(s)has rp unstable poles and the unstable pole Pj is of lj multiplicity (=1,2,...rp):that is, M:(s)=II(s-pp) 1 Define C(s) Q(s)=1+G(5)C(⑤) which corresponds to the IMC controller 4口，+@，4定4=定0C Zhang.W.D..CRC Press.2011 Version 1.0 5/89
Section 8.1 Controller Parameterization for General Plants Consider the unity feedback loop, in which the transfer function of the plant is given by G(s) = KN+(s)N−(s) M+(s)M−(s) e −θs Assume that G(s) has rp unstable poles and the unstable pole pj is of lj multiplicity (j = 1, 2, ...,rp); that is, M+(s) = Yrp j=1 (s − pj) lj Define Q(s) = C(s) 1 + G(s)C(s) which corresponds to the IMC controller Zhang, W.D., CRC Press, 2011 Version 1.0 5/89
Section 8.1 Controller Parameterization for General Plants The closed-loop system is internally stable,if and only if all elements in the transfer matrix H(s)are stable: H(S)= G(s)Q(s)G(s)[1-G(s)Q(s)] Q(s) -G(s)Q(s) Theorem The unity feedback system with a general plant G(s)is internally stable if and only if ①Q(s)is stable, 2[1-G(s)Q(s)]G(s)is stable. Or equivalently, ①Q(s)is stable, 21-G(s)Q(s)has zeros wherever G(s)has unstable poles, 3All RHP zero-pole cancellations in [1-G(s)Q(s)]G(s)are removed. oac Zhang.W.D..CRC Press.2011 Version 1.0 6/89
Section 8.1 Controller Parameterization for General Plants The closed-loop system is internally stable, if and only if all elements in the transfer matrix H(s) are stable: H(s) = G(s)Q(s) G(s)[1 − G(s)Q(s)] Q(s) −G(s)Q(s) Theorem The unity feedback system with a general plant G(s) is internally stable if and only if 1 Q(s) is stable, 2 [1 − G(s)Q(s)]G(s) is stable. Or equivalently, 1 Q(s) is stable, 2 1 − G(s)Q(s) has zeros wherever G(s) has unstable poles, 3 All RHP zero-pole cancellations in [1 − G(s)Q(s)]G(s) are removed. Zhang, W.D., CRC Press, 2011 Version 1.0 6/89
Section 8.1 Controller Parameterization for General Plants Example This example is used to illustrate that the third condition is necessary. Consider the plant with the transfer function 1 G(s)=g-1 G(s)has one simple RHP pole at s=1.Construct a controller s-1 C(5)=e01se-015-0.1s+0.1)-1 Q(s)corresponding with this C(s)is s-1 Q(5)=eo1se-015-0.15+0.d 4口，+@，4它4生·定0C Zhang.W.D..CRC Press.2011 Version 1.0 7/89
Section 8.1 Controller Parameterization for General Plants Example This example is used to illustrate that the third condition is necessary. Consider the plant with the transfer function G(s) = 1 s − 1 G(s) has one simple RHP pole at s = 1. Construct a controller C(s) = s − 1 e 0.1s (e−0.1s − 0.1s + 0.1) − 1 Q(s) corresponding with this C(s) is Q(s) = s − 1 e 0.1s (e−0.1s − 0.1s + 0.1) Zhang, W.D., CRC Press, 2011 Version 1.0 7/89
Section 8.1 Controller Parameterization for General Plants Example (ctd.1) Q(s)is stable.The first condition is satisfied.Furthermore, e0.1(e-0.1s-0.1s+0.1)-1 1-G(s)Q(6)=e0is(e-015-0.1s+0.1 It has zeros where G(s)has unstable poles.The second condition is also satisfied. However,the closed-loop system is internally unstable,because there exists a RHP zero-pole cancellation in [1-G(s)Q(s)]G(s). which cannot be removed Remark 1:The case associated with the third condition occurs only in the system where the plant or the controller contains a time delay.If both the plant and the controller are rational,it is not necessary to consider the third condition 4口，+@4定4定 Zhang.W.D..CRC Press.2011 Version 1.0 8/89
Section 8.1 Controller Parameterization for General Plants Example (ctd.1) Q(s) is stable. The first condition is satisfied. Furthermore, 1 − G(s)Q(s) = e 0.1s (e −0.1 s − 0.1s + 0.1) − 1 e 0.1s (e−0.1s − 0.1s + 0.1) It has zeros where G(s) has unstable poles. The second condition is also satisfied. However, the closed-loop system is internally unstable, because there exists a RHP zero-pole cancellation in [1 − G(s)Q(s)]G(s), which cannot be removed Remark 1: The case associated with the third condition occurs only in the system where the plant or the controller contains a time delay. If both the plant and the controller are rational, it is not necessary to consider the third condition Zhang, W.D., CRC Press, 2011 Version 1.0 8/89
Section 8.1 Controller Parameterization for General Plants Remark 2:In control system design,G(s)Q(s)is always stable. Since [1-G(s)Q(s)]G(s)=C-1(s)Q(s)G(s),the third condition can be achieved by removing the RHP zero-pole cancellation in C(s)through rational approximations Theorem All controllers that make the unity feedback control system internally stable can be parameterized as Q(s) C(s)=1-G(s)Q(S) where Q(s)= Q(s)M+(s) K 4口，+@，4定4=定0C Zhang.W.D..CRC Press.2011 Version 1.0 9/89
Section 8.1 Controller Parameterization for General Plants Remark 2: In control system design, G(s)Q(s) is always stable. Since [1 − G(s)Q(s)]G(s) = C −1 (s)Q(s)G(s), the third condition can be achieved by removing the RHP zero-pole cancellation in C(s) through rational approximations Theorem All controllers that make the unity feedback control system internally stable can be parameterized as C(s) = Q(s) 1 − G(s)Q(s) where Q(s) = Q1(s)M+(s) K Zhang, W.D., CRC Press, 2011 Version 1.0 9/89