13.3.DESIGN EXAMPLERFLEXIBLE BEAM C60 A commonway to specify desired closed-lop peformarceis by astep responsetest2 For this flexiblebeam thespe isthat astep refererceirput (r)should produceaplart output (y)satisfy ing settlirg time≈8s0 overshoot≤00%. We will accomplish this by shapirg T(s),the trarsfer furctionfrom r to y,so that it approximates astandad second-orde system:Theideal T(s)is 4n a(s):-8-+2:4ns+4n A settlirg timeof 8 srequires 4.6 :4n ≈8 and anovershoot of 0%requires =0.0 Thesolutiors are:=0.5902 and 4n=0.958.2 Lets rourd to =0.6 and 4n=C2 So theideal T(s)is 0 Tid(s)= s-+02s+0 Then the ideal sensitiv ity furtionis Sa(s)=0-a(6)=s6+02) s-+02s+0° Now take the weightirg furctionwi(s)to be sia(s),that is W(s)=s+02s+0 s(s+02), Theratiorale for this choice is arough agumert that goes as follows2 Corsider Step 2 of the procedurein the precedirg section from it thefurtion F:=WiM(Y-NQim) equals a constart times an al-pass furction2 The procedure then rolls off Qim to result in the weighted sersitivity furction WiS :=WiM(Y-NQimJ). So WiSF except at high frequerey,that is S≈FSid. Now F behaves approximately like a time delay except at high frequercy (this is a propety of all passfurctions)2So wearrive at therough approximation S≈(time delay）×Sid. DESIGN EXAMPLE FLEXIBLE BEAM  A common way to specify desired closed loop performance is by a step response test For this exible beam the spec is that a step reference input r should produce a plant output y satisfying settling time  s overshoot   We will accomplish this by shaping T s the transfer function from r to y so that it approximates a standard second order system The ideal T s is Tids  n s   ns  n  A settling time of  s requires  n   and an overshoot of  requires exp  p   The solutions are   and n   Lets round to   and n   So the ideal T s is Tids  s  s   Then the ideal sensitivity function is Sids  Tids  ss   s  s   Now take the weighting function W￾s to be Sids￾  that is W￾s  s  s  ss    The rationale for this choice is a rough argument that goes as follows Consider Step  of the procedure in the preceding section from it the function F  W￾MY NQim equals a constant times an all pass function The procedure then rolls o Qim to result in the weighted sensitivity function W￾S  W￾MY NQimJ  So W￾S  F except at high frequency that is S  F Sid Now F behaves approximately like a time delay except at high frequency this is a property of all pass functions So we arrive at the rough approximation S  time delay  Sid