Chapter 10 Time-Domain Analysis and Design of Control Systems

Transcription

1 ME 43 Sytm Dynamic & Control Sction 0-5: Stady Stat Error and Sytm Typ Chaptr 0 Tim-Domain Analyi and Dign of Control Sytm 0.5 STEADY STATE ERRORS AND SYSTEM TYPES A. Bazoun Stady-tat rror contitut an xtrmly important apct of th ytm prformanc, for it would b maningl to dign for dynamic accuracy if th tady output diffrd ubtantially from th dird valu for on raon or anothr. Th tady tat rror i a maur of ytm accuracy. Th rror ari from th natur of th input, ytm typ and from nonlinariti of ytm componnt uch a tatic friction, backlah, tc. Th ar gnrally aggravatd by amplifir drift, aging or dtrioration. Th tady-tat prformanc of a tabl control ytm i gnrally judgd by it tady tat rror to tp, ramp and parabolic input. Conidr a unity fdback ytm a R, hown in th Figur. Th input i th output i C ( ), th fdback ignal H ( ) and th diffrnc btwn input and output i th rror ignal E( ). R( ) E( ) H( ) G( ) C( ) From th abov Figur On th othr hand C R G () + G E( ) G ( ) C () Subtitution of Equation () into () yild E R( ) (3) + G Th tady-tat rror may b found by u of th Final Valu Thorm (FVT) a follow: lim t lim SE lim t 0 0 R + G (4) Equation (4) how that th tady tat rror dpnd upon th input R( ) and th forward tranfr function G( ). Th xprion for tady-tat rror for variou typ of tandard tt ignal ar drivd nxt. /5

2 ME 43 Sytm Dynamic & Control Sction 0-5: Stady Stat Error and Sytm Typ. Unit Stp (Poitional) Input. Input r ( t) ( t) or R( ) L r ( t) From Equation (4) R lim lim lim 0 + G 0 + G 0 + G + G 0 + ( ) p r( t) t whr G( 0) p i dfind a th poition rror contant.. Unit Ramp (Vlocity) Input. Input r ( t) t or r ( t) or R( ) L r ( t) From Equation (4) R ( ) lim lim lim lim 0 + G 0 + G 0 + G 0 G v r( t) t whr lim G( ) v i dfind a th vlocity rror contant Unit Parabolic (Acclration) Input. Input r ( t) t or r ( t) or R( ) L r ( t) 3 From Equation (4) 3 R ( ) lim lim lim lim 0 + G 0 + G 0 + G 0 G a r( t) t whr lim G ( ) a i dfind a th acclration rror contant. 0 /5

3 ME 43 Sytm Dynamic & Control Sction 0-5: Stady Stat Error and Sytm Typ 0.6 TYPES OF FEEDBAC CONTROL SYSTEMS Th opn-loop tranfr function of a unity fdback ytm can b writtn in two tandard form: Th tim contant form G ( z + )( z + ) ( zj + ) T T T ( p + )( z + ) ( zk + ) n T T T (8) whr and T ar contant. Th ytm typ rfr to th ordr of th pol of Equation (8) i of typ n. Th pol-zro form G at 0. G ( + )( + ) ( + j ) ' z z z n p p p ( + )( + ) ( + ) k (9) Th gain in th two form ar rlatd by ' j k z p j k (0) with th gain rlation of Equation (0) for th two form of G( ), it i ufficint to obtain tady tat rror in trm of th gain of any on of th form. W hall u th tim contant form in th dicuion blow. n Equation (8) involv th trm in th dnominator which corrpond to numbr of intgration in th ytm. A 0, thi trm dominat in dtrmining th tady-tat rror. Control ytm ar thrfor claifid in accordanc with th numbr of intgration in th opn loop tranfr function G( ) a dcribd blow.. Typ-0 Sytm. If n 0, G ( ) th tady-tat rror to variou tandard input, obtaind from 0 Equation (5), (6), (7) and (8) ar ( Poition) ( Vlocity) + G + + ( 0) lim lim 0 G 0 Acclration lim lim 0 0 G p () 3/5

4 ME 43 Sytm Dynamic & Control Sction 0-5: Stady Stat Error and Sytm Typ Thu a ytm with n 0, or no intgration in G( ) ha a contant poition rror, infinit vlocity rror and infinit acclration rror. Typ- Sytm. If n, G( ), th tady-tat rror to variou tandard input, obtaind from Equation (5), (6), (7) and (8) ar ( Poition) ( Vlocity) ( Acclration) lim 0 + G( 0) lim lim 0 G( ) 0 v lim lim 0 G ( ) 0 0 () Thu a ytm with n, or with on intgration in G( ) ha a zro poition rror, a contant vlocity rror and infinit acclration rror 3. Typ- Sytm. If n, G ( ), th tady-tat rror to variou tandard input, obtaind from Equation (5), (6), (7) and (8) ar ( Poition) ( Vlocity) ( Acclration) lim 0 + G( 0) lim lim 0 0 G( ) 0 lim lim 0 G( ) 0 a (3) Thu a ytm with n, or with on intgration in G( ) ha a zro poition rror, a zro vlocity rror and a contant acclration rror 4/5

5 ME 43 Sytm Dynamic & Control Sction 0-5: Stady Stat Error and Sytm Typ TABLE. Stady-tat rror in clod loop ytm Typ-0 ytm r( t) A A + p r( t) At Typ- ytm 0 A v Typ- ytm 0 0 r( t) At A a p lim G( ) 0 v lim G( ) a 0 0 lim G( ) whr p G( 0) whr lim G( ) v i dfind a th poition rror contant. i dfind a th vlocity rror contant. 0 whr lim G ( ) a i dfind a th acclration rror contant. 0 5/5

6 ME 43 Sytm Dynamic & Control Sction 0-5: Stady Stat Error and Sytm Typ 0.7 STEADY STATE ERROR FOR NON-UNITY FEEDBAC SYSTEMS R( ) E( ) G( ) C( ) H( ) Add to th prviou block two fdback block H ( ) and H R( ) E( ) G( ) C( ) H( ) Paralll block. Can b combind in on R( ) E( ) G( ) C( ) H( ) R( ) E G( ) + G( ) H( ) G( ) G ( ) C( ) 6/5

7 ME 43 Sytm Dynamic & Control Sction 0-5: Stady Stat Error and Sytm Typ Exampl For th ytm hown blow, find Th ytm typ Appropriat rror contant aociatd with th ytm typ, and Th tady tat rror for unit tp input R( ) E( ) 00 ( + 0) C( ) +5 Solution 7/5

8 ME 43 Sytm Dynamic & Control Sction 0-5: Stady Stat Error and Sytm Typ 8/5

9 ME 43 Sytm Dynamic & Control Sction 0-5: Stady Stat Error and Sytm Typ 9/5

10 ME 43 Sytm Dynamic & Control Sction 0-5: Stady Stat Error and Sytm Typ 0/5

11 ME 43 Sytm Dynamic & Control Sction 0-5: Stady Stat Error and Sytm Typ /5

12 ME 43 Sytm Dynamic & Control Sction 0-5: Stady Stat Error and Sytm Typ /5

13 ME 43 Sytm Dynamic & Control Sction 0-5: Stady Stat Error and Sytm Typ 3/5

14 ME 43 Sytm Dynamic & Control Sction 0-5: Stady Stat Error and Sytm Typ 4/5

15 ME 43 Sytm Dynamic & Control Sction 0-5: Stady Stat Error and Sytm Typ 5/5