SSTEM MODELLIN In order to solve a control syste proble, the descrptons of the syste and ts coponents ust be put nto a for sutable for analyss and evaluaton. The followng ethods can be used to odel physcal coponents and systes:. Dfferental Equatons (Wll be Explaned n detal wth Laplace Transfors ). Transfer Functons 3. Block Dagras (Ths Lecture) 4. StateSpace Model (Later Weeks) Lnear Systes Our focus wll be on lnear systes. A syste s lnear f t obeys the prncple of superposton and hoogenty,.e. If and then, and u( t) produces y( t) u ( t) produces y ( t) u ( t) u (t) produces y( t) y ( t) au ( t) bu ( t) produces ay ( t) by ( t)
Dfferental Equatons The nput/output relatonshp for a lnear syste takes the for of a lnear dfferental equaton. r(t) Lnear Syste y(t) In general, the dfferental equaton for the nput/output relatonshp s: n a d For physcal systes, causal systes.) y(t) 0 0 dt n. b d dt r(t) () (We say that these systes are We can apply Laplace Transforaton to lnear dfferental equatons to obtan transfer functons that descrbe the behavour of the systes. [Laplace Transfor wll be dscussed n the cong weeks.] 3 Transfer Functons Takng Laplace transfor of (), assung zero ntal condtons, we obtan n a s ( s ) b s ( s ) 0 0 where (s) and (s) are the Laplace transfors of y(t) and r(t). The transfer functon of the syste s gven by T ( s ) ( s ) ( s ) 0 n 0 b a s s The transfer functon descrbes the nputoutput behavour of the syste. In (), the syste s of order n. () In block dagra representaton, we have (s) (s) T(s)(s) T(s) 4
Block Dagras Block dagras are used to gve the functonal representaton of control systes. Three an sybols used are: suer/coparator, eleent block and takeoff pont.. Suer or Coparator E E. Eleent Block E.E E (. ).E 3. Takeoff Pont 5 Exaple: A feedback servoechans (e.g. n the case of the elevator syste) wth negatve feedback s represented as (s) Copensatng network c (s) Aplfer A Motor (s) (s) H(s) Feedback Network (s) ay be the dsplaceent, velocty or acceleraton. To understand the behavour of the syste, we need to derve the relatonshp between (s) and (s). 6
eneralzed Block Dagra (snglenput sngleoutput systes) (s) E(s) (s) (s) B(s) H(s) (s) reference nput (s) controlled output E(s) actuatng error B(s) feedback varable (s) forward transfer functon (FTF) H(s) feedback transfer functon (s)h(s) loop transfer functon or open loop transfer functon (OLTF) ( s) overall transfer functon or closedloop transfer functon (CLTF) ( s) 7 Clearly, ( s ) ( s ) E ( s ) E ( s ) ( s ) B ( s ) ( s ) H ( s ) ( s ) ( s) ( s) ( s) ( s) H ( s) ( s).e. ( s ) ( s ) ( s ) ( s ) H ( s ) FTF OLTF (s)h(s) 0 s referred to as the characterstc equaton of the syste. 8
Exaple: Derve the closedloop transfer functon of feedback syste. a postve (s) E(s) (s) (s) B(s) H(s) Clearly, fro the block dagra ( s) ( s) E( s) E( s) ( s) B( s) ( s) H ( s) ( s).e. ( s) ( s) ( s) ( s) H( s) ( s) ( s ) ( ) s FTF ( s) ( s) H ( s) OLTF 9 Multloop Block Dagras The block dagras of any practcal systes contan several nteractng loops. Such a coplex block dagra can be splfed by block dagra algebra. Two ponts to note when splfyng the block dagras: a) The product of the transfer functons n the loop forward path ust rean the sae. b) The product of the transfer functons around the loop ust rean the sae. Mason s an Forula can also be appled to deterne the overall transfer functon. Interested students can refer to the book by B. C. Kuo. 0
Soe basc and useful block dagra anpulatons are llustrated below. ou verfy the equvalence by checkng the consstency of the sgnals. See Appendx.3 for other standard anpulatons. Z H Z H Z H Z H H H Exaple: Fnd the overall transfer functon for the followng block dagra: 4 x 3 H H ) Move takeoff pont x backwards, we get 4 3 H H
) Cobnng, 3 and 4, we get a 3 4 H H ) Move take off pont a forward, 3 4 H H 3 4 3 v) Splfy the nner block ( 3 4) H ( ) 3 4 H 3 4 v) Splfy further ( H H 3 4 ) ( 3 4 ) Fnally, H ( 3 4) H ( ) 3 NB: For splcty of notaton, we ll drop the arguents where necessary. 4 ( 3 4 ) 4
Alternatvely, we can defne soe auxlary varables as follows: e e e 3 4 3 H H Then, e e e e He3 3 e H ( 4 3 ) e3 After elnatng e e 3, we wll get the sae transfer functon. 5 Fro the block dagra, we have e e e He3 e3 e H ( ) e 4 3 3 (A) (A) (A3) (A4) Sub. (A) nto (A3), we get e3 e H e3 H Elnate e by usng (A), ( H ) e e H 3 ( ) H 6
.e. e 3 ( H) H (A5) Sub. (A5) nto (A4), we get ( H ) ( 4 3 ) H H.e. [ ( 4 3) ] H So, we have 4 3 H ( ) ( ) H ( H )( ) 4 3 4 3 7 Exercse: Consder the block dagra gven below. Show that the closedloop transfer functon s gven by 8
Soluton: a c Move takeoff pont a to c. We get / c Splfyng the outlned feedback loop, we get: / ( ) / Hence, 9 Appendx.3 0
Appendx.3 Appendx.3
3 Suary : Block Dagras The general nput/output relatonshp of a lnear syste s expressed as or n Laplace doan: n a d y(t) 0 0 n dt a s ( s ) b 0 0 s b The transfer functon of the syste defnes the nputoutput relaton and t s gven by b s ( s) 0 T ( s) n ( s) a s 0 d dt ( s ) r(t) 4
Block Dagras Block dagras are used to splfy the representaton of coplex systes. eneralzed Block Dagra (snglenput sngleoutput systes) We want to derve (s) as a functon of (s)..e. (s) E(s) (s) (s) H(s) ( s) ( s) ( s) ( s) H ( s) ( s) ( s ) ( s ) ± ( s ) ( s ) H ( s ) FTF ± OLTF ( s) H ( s) 0 s the characterstc equaton of the syste. 5 Multloop Block Dagras If the block dagra contans several nteractng loops, t can be systeatcally splfed by block dagra algebra. Two ponts to note when splfyng the block dagras: a) The product of the transfer functons n the loop forward path ust rean the sae. b) The product of the transfer functons around the loop ust rean the sae. One can also defne soe auxlary varables and then perfor algebrac anpulatons to obtan the transfer functon. 6