EE Control Systems LECTURE 8

Transcription

1 Coyright F.L. Lewi 999 All right reerved Udted: Sundy, Ferury, 999 EE 44 - Control Sytem LECTURE 8 REALIZATION AND CANONICAL FORMS A liner time-invrint (LTI) ytem cn e rereented in mny wy, including: differentil eqution (ODE) tte vrile (SV) form trnfer function imule reone lock digrm (BD) or flow grh Ech decrition cn e converted to the other. In the lecture on Mon' Formul we w how to rereent ytem in term of lock digrm, nd how to determine the trnfer function of lock digrm ytem uing Mon' Formul. In thi lecture we hll ee how to mke lock digrm from given trnfer rolem. Thi i the invere rolem from the one Mon' Formul olve. Then, we hll ee how to ocite tte-ce eqution with ny lock digrm. The reltion etween SV ytem nd BD i very trnrent ince the outut of ech integrtor i tte. The rolem of finding SV or BD rereenttion given recried trnfer function i clled the reliztion rolem. BD REALIZATION OF TRANSFER FUNCTIONS IN SERIES FORMS A trnfer function cn e relized BD in erie form or rllel form. Here we introduce two erie form tht re very convenient for olving the BD reliztion rolem for ingle-inut/ingle-outut (SISO) ytem.

2 Conider the illutrtive third-order trnfer function H ( ). () Thi i rtionl function (e.g. rtio of two olynomil in ). For reliztion, it i imortnt to enure tht the trnfer function i monic, tht i, the highet order term in the denomintor h coefficient of. If not, divide through y thi coefficient to ut the trnfer function in monic form. The trnfer function mut lo hve reltive degree of or more. If the reltive degree i zero (e.g. me ower of in the numertor the denomintor), then divide the denomintor into the numertor in one te of long diviion to write H() contnt term lu term whoe reltive degree i t let one. The contnt term i direct feedthrough term, nd the rocedure elow my e crried out to relize the reminder term. A trnfer function i id to e roer if it reltive degree i greter thn or equl to zero, nd trictly roer if the reltive degree i greter thn or equl to one. We ue third-order ytem to illutrte the roch, which work for ny n-th order rtionl, monic, trictly roer trnfer function. To find BD reliztion of H(), divide y the highet ower of to otin ( H ). () ( ) Now think of Mon' Formul. To drw BD we cn ue three feedforwrd th nd three loo if we elect the correct trnmiion nd loo tructure. We give two erie form tht hve convenient tructure for relizing SISO ytem. Note rticulrly tht Mon' Formul i very ey to ue if there re no dijoint loo, nd ll loo touch ll feedforwrd th. Then, the determinnt () i imly minu the um of the loo gin, nd ll cofctor re equl to one. Rechle Cnonicl Form (RCF) The rule ued for RCF i feedck to the left feedforwrd to the right.

3 Thi men tht ll feedck loo hould join t ummer on the left, nd ll feedforwrd th hould join t ummer on the right. A BD tifying thi condition i drwn elow. u(t) / / / y(t) Rechle Cnonicl Form Note tht ll loo nd ll feedforwrd th hve the left-hnd integrtor in common, o ll cofctor re equl to nd the determinnt h no higher-order term. Alying Mon' Formul to thi BD give the trnfer function (). Determine the loo gin nd th gin nd mke ure you elieve thi. Ech integrtor outut i leled tte. The rule ued in thi coure for leling tte will e: Lel the tte from right to left, from to to ottom. We will ee ome emle of thi to clrify it. With the tte leled hown, one my write down directly the tte eqution & u y [ ] C A Bu () A n eercie, one my find the trnfer function H ( ) C( I A) B D nd verify tht it i the me the one we trted with.

4 Thi develoment give very ey wy to relize SISO trnfer function in SV form. One note tht it i ey to write down () directly from () without hving to drw the BD. In fct, imly tke the denomintor of H(), turn the coefficient ckwrd, mke them negtive, nd lce them into the ottom row of the A mtri. Tke the coefficient of the numertor, turn them ckwrd, nd lce them into the C mtri. The A mtri in () i known ottom comnion mtri for the chrcteritic olynomil ( ). The uerdigonl ' in A nd the lower in B men imly tht the three integrtor re connected in erie. Emle. Relize Trnfer Function RCF SV Sytem Let there e recried H ( ). 4 The SV eqution re directly written down & u A Bu 4 y [ ] C Now one my nlyze the ytem including imultion, finding outut given n inut nd IC, etc. Oervle Cnonicl Form (OCF) The rule ued for OCF i feedck from the right feedforwrd from the left A BD tifying thi condition i drwn elow. 4

5 u(t) / / / y(t) Oervle Cnonicl Form Note tht ll loo nd ll feedforwrd th hve the right-hnd integrtor in common, o ll cofctor re equl to nd the determinnt h no higher-order term. Alying Mon' Formul to thi BD give the trnfer function (). Determine the loo gin nd th gin nd mke ure you elieve thi. With the tte leled from right to left hown, one my write down directly the tte eqution & y u [ ] C A Bu (4) A n eercie, one my find the trnfer function H ( ) C( I A) B D nd verify tht it i the me the one we trted with. Thi develoment give very ey wy to relize SISO trnfer function in SV form. One note tht it i ey to write down (4) directly from () without hving to drw the BD. In fct, imly tke the denomintor of H(), tck the coefficient on end, mke them negtive, nd lce them into the firt column of the A mtri. Tke the coefficient of the numertor, tck them on end, nd lce them into the B mtri. Note tht thi OCF tte-ce form i not the me RCF, though oth hve the me trnfer function. In fct, RCF nd OCF re relted y tte-ce trnformtion, which we hll not dicu in thi coure (it i dicued in EE 57, Liner Sytem). 5

6 6 The A mtri in (4) i known left comnion mtri for the chrcteritic olynomil ) (. The uerdigonl ' in A nd the left-hnd in C men imly tht the three integrtor re connected in erie. Emle. Relize Trnfer Function OCF SV Sytem Let there e recried 4 ) ( H. The SV eqution re directly written down [ ] C y Bu A u 4 & Now one my nlyze the ytem including imultion, finding outut given n inut nd IC, etc. BD REALIZATION OF TRANSFER FUNCTIONS IN PARALLEL FORM To relize ytem in rllel form, one erform PFE on the trnfer function to otin ) ( K K K H (5) where the ole re t,, nd the reidue re,, K K K. Now note tht ingle term of thi form cn e relized uing the imle BD hown. Ue Mon' Formul to determine the trnfer function to mke ure you elieve thi.

7 / K The comlete trnfer function with three rllel th cn e relized hown. / K u(t) / K y(t) / K Thi reliztion i known rllel form. If there re reeted ole, then the trnfer function h higher-order ole in the PFE. In thi event, ome rllel th will contin multile integrtor. We do not need to know the detil for thi coure. A ytem which h PFE with no higher-order ole i clled imle. The rllel form i known Jordn Norml Form in mthemtic. The ce of higher-order ole fctor, correonding to multile integrtor in ome th, correond to wht i known eigenvector chin in thoe th. The detil, though fcinting, re not needed in thi coure. With the tte leled from to to ottom hown, one my write down directly the tte eqution & y [ K K K ] C u A Bu (6) A n eercie, one my find the trnfer function H ( ) C( I A) B D 7

8 nd verify tht it i the me the one we trted with. Note tht for clr ytem (e.g. n) one my write c H ( ) d. Thi how tht the reidue cn e lced on the inut th in the figure ove. In fct, long c K in ech th, one cn lit the reidue etween inut nd outut th. Thi develoment give very ey wy to relize SISO trnfer function in SV form. One note tht it i ey to write down (6) directly from (5) without hving to drw the BD. Note tht thi rllel tte-ce form i not the me RCF or OCF, though ll three hve the me trnfer function. In fct, RCF, OCF, nd the Jordn form re relted y tte-ce trnformtion, which we hll not dicu in thi coure. The A mtri in (6) i known rllel form mtri for the chrcteritic olynomil ) ( )( )( ). ( If the A mtri i digonl with the ole ering on the digonl it i clled imle. If the ytem i not imle, then there will e ome off digonl ' in A correonding to Jordn eigenvector chin of length greter thn one. You will her more out thi in nother coure (EE 57). 8