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 kW￾MY NQimk￾  Step  Set J s   s  k where k is just large enough that QimJ is proper and is just small enough that kW￾MY NQimJ k￾  Step  Set Q  QimJ  Step  Set C  X  MQY NQ That Step is feasible follows from the equation W￾MY NQimJ   W￾MY NQimJ  W￾M 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￾  W￾M Y and T  W￾MN 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