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 kWSk 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 S z for z a zero of P in Res S p for p a pole of P in Res The weighted sensitivity function G WS must therefore satisfy G z W z for z a zero of P in Res G p for p a pole of P in Res So the requirement of internal stability imposes interpolation constraints on G The performance spec kWSk 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 WS WM Y 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
1-c3%DESIGN EXAMPLE<FLEXIBLE BEAM 159 Procedure Input:P,Wi Step 1 Do a coprime factorization of P:Find four functions in S satisfying the equations NX +MY =1. Step.Find a stable function Qim such that WiM(Y NQim)oo 1 1. Step 3 Set 1 J(6)=09+可 where k is just large enough that QimJ is proper and 0 is just small enough that IWiM(Yr NQimJ)川o11. Step fi Set Q=QimJ2 Step 5 Set C=(X+MQ)/(Y NQ)2 That Step 3 is feasible follows from the equation WiM(Y NQimJ)=WiM(Y NQim)J+WiMY(1 J). The 4rst term on the righthand side has -norm less than 1 from Step 2 and the fact that 1,whilethe. -norm of the second term goes to 0 as 0 goes to 0 by Lemma 12 Step 2 is the model-matching problem,4nd a stable function Qim to minimize ‖TrT,Qiml‖o, where Ti:=WIMY and T:=WiMN2 Step 2 is feasible iff Yopt,the minimim model-matching error,is 1 12 10.3 Design Example:Flexible Beam This section presents an example to illustrate the procedure of the preceding section2 The example is based on a real experimental setup at the University of Toronto2 The control system depicted in Figure 1021,has the following components:a flexible beam,a high-torque dc motor at one end of the beam a sonar position sensor at the other end,a digital computer as the controller with analog-to digital interface hardware,a power ampli4er to drive the mctor,and an antialiasing 4lter2 The cbjective is to control the position of the sensed end of the beam2 A plant model was obtained as follows2 The beam is pinned to the mctor shaft and is free at the sensed end2 First the beam itself was modeled as an ideal Euler-Bernoulli beam with no damping: this yielded a partial differential equation model,reflecting the fact that the physical model of the
DESIGN EXAMPLE FLEXIBLE BEAM Procedure Input P W Step Do a coprime factorization of P Find four functions in S satisfying the equations P N M NX M Y Step Find a stable function Qim such that kWM Y NQim k Step Set J s s k where k is just large enough that QimJ is proper and is just small enough that kWM Y NQimJ k Step Set Q QimJ Step Set C X MQ Y NQ That Step is feasible follows from the equation WM Y NQimJ WM Y NQim J WM Y J The rst term on the right hand side has norm less than from Step and the fact that kJ k while the norm of the second term goes to as goes to by Lemma Step is the model matching problem nd a stable function Qim to minimize kT TQimk where T WM Y and T WMN Step is feasible i opt the minimum model matching error is Design Example Flexible Beam This section presents an example to illustrate the procedure of the preceding section The example is based on a real experimental setup at the University of Toronto The control system depicted in Figure has the following components a exible beam a high torque dc motor at one end of the beam a sonar position sensor at the other end a digital computer as the controller with analog to digital interface hardware a power amplier to drive the motor and an antialiasing lter The ob jective is to control the position of the sensed end of the beam A plant model was obtained as follows The beam is pinned to the motor shaft and is free at the sensed end First the beam itself was modeled as an ideal Euler Bernoulli beam with no damping this yielded a partial dierential equation model reecting the fact that the physical model of the