[ ] 1+ lim G( s) 1+ s + s G s s G s Kacc SYSTEM PERFORMANCE. Since. Lecture 10: Steady-state Errors. Steady-state Errors. Then

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1 SYSTEM PERFORMANCE Lctur 0: Stady-tat Error Stady-tat Error Lctur 0: Stady-tat Error Dr.alyana Vluvolu Stady-tat rror can b found by applying th final valu thorm and i givn by lim ( t) lim E ( ) t 0 providd that E() do not hav pol on th jω-axi or in th right half of -plan. Th ytm rror i dfind a th diffrnc btwn th rfrnc input and th output: (t) r(t) y(t) ; or E() R() Y() in th -domain Conidr th unity fdback ytm: R() E()E a () ) Y() Sinc Thn That i, Hnc w hav G ( ) Y ( ) ir( ) G ( ) ) E( ) R( ) Y( ) R( ) G ( ) R( ) E( ) ) R( ) lim E( ) lim 0 0 ) Lctur 0: Stady-tat Error Dr.alyana Vluvolu For a gnral non-unity fdback control ytm hown blow, w nd to b carful about ytm rror. Indd, it may not mak n to dfin R() -Y(). Why? R() E a () ) H() Y() Error Contant (for unity-fdback ytm) Lctur 0: Stady-tat Error Dr.alyana Vluvolu W xamin th tady-tat rror w.r.t th input function R(). Rcall that R( ) lim E( ) lim 0 0 ) () Unit-Stp Input: whr po 0 lim ) R( ) lim 0 [ ) ] lim ) 0 po i th poition rror contant. () Unit-Ramp (ocity) Input: lim 0 [ ) ] R ( ) 0 lim 0 ) lim ) whr lim ) i th ocity rror contant. 0 () Unit Parabolic (lration) Input: R( ) lim 0 ) whr 0 [ ] lim ( ) lim ( ) lim ) 0 G G 0 i th lration rror contant. Lctur 0: Stady-tat Error Dr.alyana Vluvolu

2 Lctur 0: Stady-tat Error Dr.alyana Vluvolu Exampl: Th clod-loop tranfr function of a unity-fdback control ytm i givn by Y ( ) b R ( ) a b whr a, b > 0 (why?). W want to find th tady-tat rror to a unitramp input. W know that Y ( ) R ( ) G ( ) G ( ) a b So, or b ) ( a ) Thu th tady-tat rror to a unit-ramp input i ( a ) a lim lim ) 0 ( b) b b ( a b) ) ( b)[ )], 0 5 Exrci: Givn th unity fdback ytm whr R() E() ) 0 G ( ) Y() A tudnt calculatd that th poition rror contant i po lim G ( ) 0 0 Lctur 0: Stady-tat Error Dr.alyana Vluvolu H thn concludd that th tady-tat rror with rpct to a tp input i /. I th tudnt concluion corrct? (S Nxt Slid.) 6 Exampl Givn 0 ) thn for th unity-fdback ytm, R( ) E( ) R( ) ) ) 0 ( 0) Lctur 0: Stady-tat Error Dr.alyana Vluvolu Clarly, E() i not an analytic function in th clod RHP. Thi i bcau ha root in th RHP. That i, th clod-loop ytm i untabl and hnc tady-tat rror i undfind. 7 Typ of Control Sytm Lctur 0: Stady-tat Error Dr.alyana Vluvolu Lt u now xamin th tady-tat rror w.r.t th ytm OLTF ). For a unity-fdback ytm, ) can alway b xprd a: G ( ) ) () j whr G () ha no pol nor zro at th origin. For xampl, ) ; G ( ) 0( ) G ( ) 0( ) ( ) 0 G ( ) 0 G ) ; ( ) R() E() ) Y() For a unity fdback ytm, th typ of a control ytm w.r.t. R() i dfind by th ordr of th pol of ) at th origin of th -plan (i.. th valu of j {0,,, } ). Th ytm dcribd by qn () i of typ-j. 8 It ha a dirct ffct on th tady-tat rror.

3 Typ 0 Sytm: ( j 0): ( ) ) G ; G 0 (0) ( 0 or ) Thn lim ) lim G ( ) po 0 0 lim ) lim G ( ) lim ) 0 0 Lctur 0: Stady-tat Error Dr.alyana Vluvolu Typ Sytm: ( j ): ( ) ) G ; G (0) ( 0 or ) Thn G ( ) po lim ) lim 0 0 lim ) lim G ( ) 0 0 lim ) 0 0 Lctur 0: Stady-tat Error Dr.alyana Vluvolu Thu, ( unit tp) po ( unit ramp) ( unit parabolic) 9 Thu, ( unit tp) 0 po ( unit ramp) ( unit parabolic) 0 Typ Sytm: ( j ): ( ) ) G ; G (0) ( 0 or ) Thn G ( ) po lim ) lim 0 0 G ( ) lim ) lim 0 0 lim ) lim G ( ) Thu, 0 0 ( unit tp) 0 po ( unit ramp) 0 ( unit parabolic) Lctur 0: Stady-tat Error Dr.alyana Vluvolu Lctur 0: Stady-tat Error Dr.alyana Vluvolu Th tady-tat rror for variou input and ytm typ ar ummarizd a follow: Sytm Typ Unit-tp input, u(t) Unit-ramp input, t Parabolic input, 0 po t Sytm of typ highr than ar not commonly ud in practic bcau: (i) (ii) Thy ar mor difficult to tabiliz. Th dynamic rror tnd to b largr.

4 Stady-tat Error in th Prnc of Diturbanc Lctur 0: Stady-tat Error Dr.alyana Vluvolu In th prnc of diturbanc, thr ar componnt in th tady-tat rror: an rror du to inability of th output to follow th rfrnc input an unwantd contribution from th diturbanc It important to not that th output du to th diturbanc i th unwantd output! It i dirabl that ach of th tady-tat componnt ar mall, if not zro. Lctur 0: Stady-tat Error Dr.alyana Vluvolu Conidr th following fdback control ytm with diturbanc: Th output ha componnt, i.. Y( ) YR ( ) YD ( ), whr Y R () i du to R() alon and Y D () du to D() alon. Th ytm rror alo ha componnt: i.. whr R() E() G ( ) Y() E( ) R( ) ( Y ( ) Y ( )) ( R( ) Y ( )) Y ( ) E ( ) E ( ) R D R D R D E ( ) R( ) Y ( ); and E ( ) Y ( ) D() G ( ) R R D D R D lim E ( ) R 0 R lim E ( ) lim Y ( ) D D D 0 0 R lim ) lim 0 0 Lctur 0: Stady-tat Error Dr.alyana Vluvolu Exampl: Conidr th fdback control ytm a hown blow: D() R() E() Y() Calculat th tady-tat rror whn R() i a unit-ramp input and D() i a unit-tp diturbanc. Sinc E( ) E ( ) E ( ), th tady-tat rror i givn by whr and Now, R D 5 For th diturbanc, w hav YD ( ) ; D( ) D( ) Thrfor, yd lim YD ( ) lim 0 0 But D yd. So, 0!! R D!! R D Lctur 0: Stady-tat Error Dr.alyana Vluvolu Howvr, unlik th rfrnc input, th diturbanc can t b known xactly. In thi xampl, it could b a ngativ tp rathr than a poitiv tp, in which ca If i mall, thn w would hav a problm! Hnc, w hould nur that both R and D ar zro or mall! 6

5 Lctur 0: Stady-tat Error Dr.alyana Vluvolu Exampl: Conidr th fdback control ytm a hown blow: D() E() R() Y() G c () Dign a uitabl Gc ( ) to liminat (at tady tat) th rror du to th diturbanc D(), and at th am tim nur tability of th clod-loop control ytm. Stting R() to zro and w hav D() J G c () J Y D () For a unit-tp diturbanc input, Hnc D( ), and Y ( ) D J D ( ) J ( ) J lim ( ) lim yd YD 0 0 J 0 if 0 Th clod-loop tranfr function w.r.t. R() i givn by. Gc ( ) YR ( ) J Gc ( ) R( ) J J Lctur 0: Stady-tat Error Dr.alyana Vluvolu Thu for tability, w nd (It aum that J > 0.) > 0 and > 0. And th tady-tat rror with rpct to a ramp rfrnc input i zro a th ytm i typ. whr Y D () dnot th output du to th diturbanc. 7 8 Summary 9. Stady-Stat Error Stady-tat rror, lim ( t) lim E( ) t po 0 0 For tabl unity fdback ytm ytm: R( ) R() lim E( ) lim 0 0 ) Error Contant (for unity-fdback ytm) Poition Error Contant, Vlocity Error Contant, Acclration Error Contant, Hnc, lim ) lim ) (unit-tp) (unit-ramp) 0 ( po ) (unit-parabolic) lim ) 0 E() Lctur 0: Stady-tat Error Dr.alyana Vluvolu ) Y() 9 Typ of Control Sytm po v l Lctur 0: Stady-tat Error Dr.alyana Vluvolu Th typ of a control ytm with rpct to an input i pcifid by it tady-tat rror prformanc. Stady-tat rror (for a unity fdback ytm) for variou input and ytm typ ar ummarizd a follow: Sytm Typ Unit-tp input, u(t) Unit-ramp input, t Parabolic input, 0 (finit) 0 (finit) t 0 0 (finit) Stady-tat Error in th Prnc of Diturbanc: Whn calculating th tady-tat rror in th prnc of diturbanc, w want both th following componnt to b mall: rror du to inability of th output to follow th rfrnc input 0 unwantd contribution from th diturbanc