Exercises for lectures 7 Steady state, tracking and disturbance rejection
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1 Exrc for lctur 7 Stady tat, tracng and dturbanc rjcton Martn Hromčí Automatc control
2 Frquncy rpon drvaton Automatcé řízní - Kybrnta a robota W lad a nuodal nput gnal to th nput of th ytm, gvn by a tranfr functon y( ) G( ) u( ), that tady at th bgnnng a u( t) ant u( ) L ant It wll b th mag of th nput gnal a a y( ) G( ) G( ) y pr( ) ( j)( j) j j whr ag( ) ag( j) a G( j) ( j) j j j ag( ) ag( j) a G( j) ( j) j j j y () pr ar partal fracton corrpondng to natural mod of th rpon. Th mod dappar n th tady tat bcau th ytm tabl, o jt y () t Martn Hromčí Pr-ARI jt j j G( j) Im G( j) arctg R G ( j )
3 Frquncy rpon drvaton Automatcé řízní - Kybrnta a robota It follow, that Whr w can u th Eulr formula: jt jt y () t j a G( j) jt a G( j) j j a G( j) j j( t) j( t) jn( t) a G( j) j a G( j) n( t ) Im G( j) G( j) arctg R G ( j ) j jt jx co x j n x jx co x j n x jx jx j n x Martn Hromčí Pr-ARI
4 Automatcé řízní - Kybrnta a robota Exampl: Rpon to nuodal nput A ytm wth quaton y( t) y( t) u( t) and tranfr fun. G( ) ( ) rpond to nuodal nput gnal u( t) n(0 t) a follow 0 y( ) G( ) u( ) y () t yt () Trannt and tady tat y () t 0 t yt ( ) n(0 t ) 0 0 y ( t) y ( t) arctan( 0) rad ut () yt () Martn Hromčí Pr-ARI
5 Automatcé řízní - Kybrnta a robota Frquncy rpon graphcally frquncy rpon (plot G( j) ) can b drawn n two way Bod plot ampltud and pha paratly - functon bod Ampltud uually logarthmc n db, or lnar n ab. valu Pha alway lnar, n dgr (dg) or n rad. Frquncy alway logarthmc, uually n rad/ or n Hz Nyqut plot n complx pac - functon nyqut bnft: how ampltud and pha multanouly drawbac: w don t contrbuton of ndvdual factor Dcbl - namd aftr A.G. Bll, ntroducd by Bll Lab dmnonl unt, orgnally for powr rato 0log( Py Pu) If th powr rato wa xprd by th voltag rato, thn 0log( Vy Vu) Gnrally for th rato of two varabl (n our ca ampltud), n ARI alway x[db] 0 log( x) Martn Hromčí Pr-ARI
6 Exampl: Stady-gan Automatcé řízní - Kybrnta a robota Ex. : For tat-pac modl x( t) Ax( t) Bu( t) y( t) Cx( t) Du( t) G() C I A B D G(0) CA B D Ex. : For xtrnal modl G () b () a () 0 b0 0, a0 0 b(0) b0 G(0) a0 0, b0 0 a(0) a0 c 0, a0 0, b0 0 Martn Hromčí Pr-ARI
7 Exampl: Stady tat Automatcé řízní - Kybrnta a robota Ex. : For a ytm wth th tranfr functon G () Stady-tat valu of th tp rpon Stady-tat valu of th mpul rpon Ex. : Impropr u of th fnal valu thorm Lt hav a ytm gvn by 3 th tranfr functon G () I th tady-tat valu of th tp rpon h 3? No! Th ytm untabl thrfor th tp rpon do not hav any tady-tat valu 3 3 t h() t Martn Hromčí Pr-ARI ( ) g lm G( ) 0 h G(0) lm G( ) 0 0
8 Automatcé řízní - Kybrnta a robota Exampl: Gan at nfnty (VF) Sgnal f() t ha (aftr)ntal valu wth th mag f (0 ) lm f ( ) Obvouly f (0 ) 0 whn th dgr of th numrator at lat two l than th dgr of th dnomnator For (non-trctly) propr tranfr functon n n bn bn n n an an th mpul rpon tart n n n g(0 ) lm ( bn ) ( an ) and th tp rpon n n n h(0 ) lm ( bn ) ( an ) bn an For trctly propr tranfr functon n bn n n an an th mpul rpon tart n g(0 ) bn an b 0 n g(0 ) 0 and th tp rpon n h(0 ) 0 Martn Hromčí Pr-ARI
9 Impul and tp rpon by typ Automatcé řízní - Kybrnta a robota Typ 0 (tatc) F () Typ ( t ordr atatm) F () Typ ( nd ordr atatm) F () Typ 3 (3 rd ordr atatm) F () Martn Hromčí Pr-ARI
10 Automatcé řízní - Kybrnta a robota Stady-tat rror formula tp ramp ( ) lm L( ) K p 0 ( ) lm L( ) Kv 0 ( ) parabola lm L( ) Ka 0 Lmt ar nam a Stady-tat rror contant of tady-tat rror: Contant of poton rror K lm L( ) Contant of vlocty rror K lm L( ) Contant of acclraton rror Th contant dtrmn th tady-tat bhavor and thu omtm ar ud for th dgn pcfcaton. K v a p 0 0 lm L( ) 0 Martn Hromčí Pr-ARI
11 Drvaton of tady-tat rror from Bod plot Automatcé řízní - Kybrnta a robota Typ 0 z z L( ) K K lm L( ) L(0) K p p 0 p Th ntal valu of th ampltud frquncy rpon z 0 log (0) 0 log M K 0 log K p Th ntal valu = poton contant (= tady-tat gan) v db P 0log K p 0log L(0) 0dB 0 0log M( ) Martn Hromčí Pr-ARI-07-05
12 Drvaton of tady-tat rror from Bod plot Automatcé řízní - Kybrnta a robota Typ Th ampltud frquncy rpon tart for mall ω 0 n wth an ntal lop of It can b rplacd by a functon whch ntrct th omga ax whn z z L( ) K K lm L( ) K p p 0 log M( ) 0 log 0 v 0 K 0 z p 0dB dc K Kv p Martn Hromčí Pr-ARI log K z L() K p z 0 p 0dB 0 0log M( ) z 0dB dc K v
13 Drvaton of tady-tat rror from Bod plot Automatcé řízní - Kybrnta a robota Typ Th ampltud frquncy rpon tart for mall ω 0 n wth an ntal lop It can b rplacd by a functon whch ntrct th omga ax whn z z L K K L K p p ( ) a lm ( ) 0 0 log M( ) 0 log 40dB dc K 0 0 z p z L() K p 0log K z 0 p 0dB 0 K 0log M( ) z p K a K a Martn Hromčí Pr-ARI
14 Exampl Automatcé řízní - Kybrnta a robota >> L=(+)/(+)/(3+), M 5dBv=valu(L,0),L=L/v*0^(5/0),K=valu(L,0),bod(L) L = / ^ K = >> KpdB=0*log0(ab(valu(L,j*.0))), Kp=0^(5/0) KpdB = , Kp = >> nfty = /(+Kp) nfty = 0.50 Intal lop 0, o t a ytm of typ 0 (wthout atatm). Intal valu of th aymptot 5dB and thrfor 5 0 K p 5d B Stady-tat rror of tp rpon tp, K p 0.5 L=(+)/(+)/(3+)/,v=valu(coprm(*L),0);L=L/v*0, L = / 6 + 5^ + ^3 Kv=valu(coprm(*L),0),bod(L) Kv = 0 0 Intal lop 0 db/d, o t a ytm of typ. Drawd ntal aymptot cro th zro at frquncy 0 and thrfor Kv 0 Stady-tat rror of mpul rpon ramp, K v 0. Martn Hromčí Pr-ARI
15 Exampl of th tady-tat valu Automatcé řízní - Kybrnta a robota ntgrator r y Attnton: Th clod loop tabl n all ca! ntg. Sgnal coror: rfrnc output rror r 0( 3)( ) y 3 ntg. r 9( 3)( ) 3 y rfrnc tp, ramp parabola Martn Hromčí Pr-ARI
16 How to trac wth a ytm of nuffcnt typ? Automatcé řízní - Kybrnta a robota How to nur propr tracng wth a ytm of nuffcnt typ? ung a dynamc controllr, whch proc th control rror r () () C () u () F () y () G( ) F( ) C( ) r () () G () y () Th rgulator C dgnd o, that G( ) F( ) C( ) ha a rqurd typ and th clod-loop ytm tabl! W want to nur zro rror of tp rp. for ytm of typ 0 ( F(0), F(0) 0) () C( ) F( ), tp C (0) F (0) Choong C( ) w obtanc(0) and thu tp, 0 But thn C ( ) C u () utp, C( ) F( ) th ytm mut b tll "fdd, thr no othr choc. (0) 0 C(0) F(0) F(0) Martn Hromčí Pr-ARI
17 Automatcé řízní - Kybrnta a robota Exampl 0( ) To trac th ramp wth a ytm gvn by th tranfr functon F () w u rgulator. Thn th clod-loop ytm ( ) C( ) ( 3) tabl ( ) 0( )( 3) W achvd th rqurd typ, 0( )( 3) G( ) F( ) C( ) hnc th tady-tat rror zro. ( ) But thn C ( ) u () What th prc? u alo a ramp! C ( ) F ( ) C ( ) jaá j cna? uramp, lm 0 C ( ) F ( ) 0 0 u j taé rampa 3 0( ) ( ) Martn Hromčí Pr-ARI
18 Exampl: Poton tracng wth fnt rror Automatcé řízní - Kybrnta a robota Control of movmnt drcton (LRV, automatc car) 5 G () C() 0 rror ( 0) ( ) r( ) r( ) r( ) C( ) G( ) 5 (0 5 ) 5 0 tady-tat rror ( 0) lm 0 (0 5 ) 5 o, 0 lm ( 0) rampa, 0 (0 5 ) 5 for xampl C () r () () C () u () G () y () >> G=5/(+0),C=(+)/ G = 5 / 0 + C = + / >> root(g.dn*c.dn+... G.num*C.num) an = tracng.mdl Martn Hromčí Pr-ARI
19 Exampl: Poton tracng wth zro rror Automatcé řízní - Kybrnta a robota targt ngagmnt, fr ctrl To trac th ramp, th opn-loop ytm ha to contan doubl-ntgrator, for xampl Error C () ( 0) ( ) r( ) r( ) 3 C( ) G( ) Stady-tat rror to ramp r () () C () u () G () y () ( 0) rampa, 0 3 lm >> C=(*+)/^ C = + / ^ >> root(g.dn*... C.dn+ G.num*C.num) an = tracng.mdl Martn Hromčí Pr-ARI
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