158 CHAPTER 10.DESIGN FOR PERFORMANCE 10.2 P-1 Unstable We come now to the first time in this book that we need a nonclassical met hod,namely,interpolation theory.To simplify matters we will assume in this section that P has no poles or zeros on the imaginary axis,only distinct poles and zeros in the right half-plane,and at least one zero in the right half-plane (i.e.,P-is unstable). Wi is stable and strictly proper. It would be possible to relax these assumptions,but the development would be messier. To motivate the procedure to follow,let's see roughly how the design problem of finding an internally stabilizing C so that WiS<1 can be translated into an NP problem.The definition of S is 1 S- 1+PC For C to be internally stabilizing it is necessary and sufficient that S E S and PC have no right half-plane pole-zero cancellations (Theorem 3.2).Thus,S must interpolate the value 1 at the right half-plane zeros of P and the value 0 at the right half-plane poles (see also Section 6.1);that is,S must satisfy the conditions S(z)=1 for z a zero of P in Res >0, S(p)=0 for p a pole of P in Res >0. The weighted sensitivity function G:=WiS must therefore satisfy G(z)=Wi(z)for z a zero of P in Res >0, G(p)=0 for p a pole of P in Res >0. So the requirement of internal stability imposes interpolation constraints on G.The performance spec WiSoo<1 translates into Go<1.Finally,the condition S E S requires that G be analytic in the right half-plane. One approach to the design problem might be to find a function G satisfying these conditions, then to get S,and finally to get C by back-substitution.This has a technical snag because the requirement that C be proper places an additional constraint on G not handled by our NP theory of the Chapter 9.For this reason we proceed via controller parametrizat ion. Bring in again a coprime factorization of P: P=N M NX +MY =1. The controller parametrization formula is C X+MQ Y-NQ Q∈S, and for such C the weighted sensitiv ity function is WIS=WM(Y-NQ). The parameter Q must be both stable and proper.Our approach is first to drop the properness requirement and find a suitable parameter,say,Qim,which is improper but stable,and then to get a suitable Q by rolling Qim off at high frequency.The reason this works is that Wi is strictly proper, so there is no performance requirement at high frequency.The method is out lined as follows: CHAPTER  DESIGN FOR PERFORMANCE ￾ P ￾￾ Unstable We come now to the rst time in this book that we need a nonclassical method namely interpolation theory To simplify matters we will assume in this section that P has no poles or zeros on the imaginary axis only distinct poles and zeros in the right half plane and at least one zero in the right half plane ie P ￾ is unstable W￾ is stable and strictly proper It would be possible to relax these assumptions but the development would be messier To motivate the procedure to follow lets see roughly how the design problem of nding an internally stabilizing C so that kW￾Sk￾ can be translated into an NP problem The denition of S is S   P C  For C to be internally stabilizing it is necessary and sucient that S  S and P C have no right half plane pole zero cancellations Theorem  Thus S must interpolate the value at the right half plane zeros of P and the value at the right half plane poles see also Section  that is S must satisfy the conditions Sz  for z a zero of P in Res Sp  for p a pole of P in Res  The weighted sensitivity function G  W￾S must therefore satisfy Gz  W￾z for z a zero of P in Res Gp  for p a pole of P in Res  So the requirement of internal stability imposes interpolation constraints on G The performance spec kW￾Sk￾ translates into kGk￾  Finally the condition S  S requires that G be analytic in the right half plane One approach to the design problem might be to nd a function G satisfying these conditions then to get S and nally to get C by back substitution This has a technical snag because the requirement that C be proper places an additional constraint on G not handled by our NP theory of the Chapter  For this reason we proceed via controller parametrization Bring in again a coprime factorization of P P  N M NX  M Y   The controller parametrization formula is C  X  MQ Y NQ Q  S and for such C the weighted sensitivity function is W￾S  W￾MY NQ The parameter Q must be both stable and proper Our approach is rst to drop the properness requirement and nd a suitable parameter say Qim which is improper but stable and then to get a suitable Q by rolling Qim o at high frequency The reason this works is that W￾ is strictly proper so there is no performance requirement at high frequency The method is outlined as follows