6.3 Stochatic Etimatio ad Cotrol, Fall 004 Lecture 7 Lat time: Groud rule for filterig ad cotrol ytem deig Gral ytem Sytem parameter are cotaied i w( t ad w ( t. Deired output i grated by takig the igal through the deired operator. The differece betw the actual output ad the deired output i the error, whoe mea quared value we wat to miimize. We require a table ad realizable ytem. The error et ( i give by: e( t w ( τ ( t τ dτ w ( τ ( t τ dτ + w ( τ ( t τ dτ d w ( τ ( t τ dτ + w ( τ ( t τ dτ e where w ( τ w ( τ w ( τ. e d Uig the replacemet w e τ reduce the umber of term i R( τ from ie to four. Page of 7
6.3 Stochatic Etimatio ad Cotrol, Fall 004 R ( τ e( t e( t+ τ dτ dτ w ( τ w ( τ R ( τ + τ τ e e + dτ dτ w ( τ w ( τ R ( τ + τ τ e + dτ dτ w ( τ w ( τ R ( τ + τ τ e + dτ dτ w ( τ w ( τ R ( τ + τ τ jωτ S ( R ( τ e dτ Fe( Fe S + Fe( F S + F( Fe( S( + F( F( S( where F ( F ( F (. e d We kow the cofiguratio o we ca work out the form of each of the traform F. Computatio of the trafer fuctio Fe ( ad F ( For may ytem cofiguratio epecially with mior fdback loop, maipulatig the diagram ito a tadard iput-output form i very timecoervig ad ubject to error. Uually jut tracig igal aroud the loop i eaiet. Example: Fid F ( Page of 7
6.3 Stochatic Etimatio ad Cotrol, Fall 004 Do t try to maipulate thi by block diagram algebra ito the form: Rather: F I y P y+ y+ ( τ + Now the ret i jut algebra, I τ y+ y F y P y+ Multiply by ad collect term. 3 τ + + + y y F P F I F F 3 τ + + FP+ FI Now itegrate to get e S( d π j j j Two method: Cauchy Theorem: e, uig jω : e π j Re pole of S ( i LHP π j Re pole of ( S i LHP Tabulated Itegral (applicable oly to ratioal fuctio: S S + S + S + S I j cc ( d π j d d( j Page 3 of 7
6.3 Stochatic Etimatio ad Cotrol, Fall 004 Jut work out the coefficiet c ( ad d( ad ue the table to fid the itegral I. S + S mut be itegrated together. For the ret of the emeter, we will oly deal with igal ad oie which are ucorrelated, o th term will be zero. S ω itegrated to get a expreio for Havig e p, p,..., p, where p i are the deig parameter, we mut determie the optimum et of value of the p i. Differetiate with repect to other variable tha the parameter. e p e p 0 0 If you have oe parameter, jut plot ad fid the miimum. For multiple parameter, the optimizatio become quite complex. If the igal, oie ad diturbace ca be coidered ucorrelated, the e e + e + e d ad we ca expre th compoet of e i ay direct or coveiet way. Example: A ervo Sigal t ( i a member of a tatioary mble: A S ω ω + a Diturbace dt ( i a member of the mble of cotat fuctio: dt ( D Noie t ( i a member of a tatioary white oie mble: S ( ω N Page 4 of 7
6.3 Stochatic Etimatio ad Cotrol, Fall 004 Deired output i the igal. Fid which miimize the teady-tate mea quared error. The iput t (, dt (, t ( are all idepedet. ad have zero mea. Therefore, the thr iput are ucorrelated. e e + e + e d We ll d the error (igal miu deired trafer fuctio ad the oie trafer fuctio. For the igal: F ( + + Fdeired ( Fe ( F ( Fdeired ( + + For the oie: ( F + j e e π j j e F ( F S ( d S ( A A A The itegrad: a a+ a F ( F ( S ( e e A A ( + ( ( a+ ( a A A( ( + ( a+ ( ( a A A( + ( a+ + a + ( a+ ( + a c ( c( d( d( Page 5 of 7
6.3 Stochatic Etimatio ad Cotrol, Fall 004 e 0 0 I ddd 0 dd A ( a+ cd + cd c For the diturbace: Steady tate repoe O to cotat iput d O d + + d O ( + O lim O O 0 d d lim 0 + d D F ( F ( F ( e deired + + F ( + S( N + Uig Cauchy Theorem: e Re pole at N N ( + Page 6 of 7
6.3 Stochatic Etimatio ad Cotrol, Fall 004 e de d A D + + N + a A D + N 0 3 ( + a 3 3 A D( + a + N ( + a 0 th 5 order polyomial i 0 You hould check the tability of your olutio at your olutio poit. For tability i thi example, > 0. Semi-Fr Cofiguratio Deig (Fr cofiguratio deig i a pecial cae of thi. Fixed trafer you have to deal with. You mut drive the plat, F. ( We will deig a compeator C, ( which ca be i a cloed loop cofiguratio, perhap with omethig i the fdback path, B(. Page 7 of 7