Control Systems II Lecture 2

Transcription

1 Control Sytem II Lecture 2 <Dr Ahmed El-Shenwy> ١

2 Coure Content Modeling of liner ytem Repreenttion uing phe vrible Stte pce uing cnonicl form Propertie of trnition mtri Pole/zero Eigen vlue Pole Plcement in tte feedbck Introduction to non linerity Common non linerity Decribing function method Limit cycle The phe plne method ٢

3 Lerning Outcome A.4. Principle of deign including element deign, proce nd/or ytem relted to pecific dicipline. A.5. Methodologie of olving engineering problem, dt collection nd interprettion. A.5. Principle of opertion nd performnce pecifiction of electricl nd electromechnicl engineering ytem A.27. Anlyi, deign nd implementtion of vriou method of control uing nlogue nd digitl control ytem A.3. Formulte the problem, relizing the requirement nd identifying the Contrint. B.9 Deign computer progrm to nlyze nd imulte different electricl ytem component nd control ppliction. ٣

4 Outline Linking tte pce repreenttion nd trnfer function Phe vrible cnonicl form Input feedforwrd cnonicl form Phyicl tte vrible model Digonl cnonicl form Jordn cnonicl form ٤

5 The clicl control theory nd method (uch root locu) tht we hve been uing in cl to dte re bed on imple input-output decription of the plnt, uully epreed trnfer function. Thee method do not ue ny knowledge of the interior tructure of the plnt, nd limit u to ingle-input ingle-output (SISO) ytem, nd we hve een llow only limited control of the cloed-loop behvior when feedbck control i ued. Modern control theory olve mny of the limittion by uing much richer decription of the plnt dynmic. The o-clled tte-pce decription provide the dynmic et of coupled firt-order differentil eqution in et of internl vrible known tte vrible, together with et of lgebric eqution tht combine the tte vrible into phyicl output vrible. ٥

6 STATE-SPACE REPRESENTATIONS The concept of the tte of dynmic ytem refer to minimum et of vrible, known tte vrible, tht fully decribe the ytem nd it repone to ny given et of input The tte vrible re n internl decription of the ytem which completely chrcterize the ytem tte t ny time t, nd from which ny output vrible yi(t) my be computed. ٦

7 The Stte Eqution A tndrd form for the tte eqution i ued throughout ytem dynmic. In the tndrd form the mthemticl decription of the ytem i epreed et of n coupled firt-order ordinry differentil eqution, known the tte eqution, in which the time derivtive of ech tte vrible i epreed in term of the tte vrible (t),..., n(t) nd the ytem input u(t),..., ur(t). ٧

8 It i common to epre the tte eqution in vector form, in which the et of n tte vrible i written tte vector (t) = [ (t), 2 (t),..., n (t)] T, nd the et of r input i written n input vector u(t) = [u (t), u 2 (t),..., u r (t)] T. Ech tte vrible i time vrying component of the column vector (t). In thi note we retrict ttention primrily to decription of ytem tht re liner nd time-invrint (LTI), tht i ytem decribed by liner differentil eqution with contnt coefficient. ٨

9 where the coefficient ij nd b ij re contnt tht decribe the ytem. Thi et of n eqution define the derivtive of the tte vrible to be weighted um of the tte vrible nd the ytem input. ٩

10 where the tte vector i column vector of length n, the input vector u i column vector of length r, A i n n n qure mtri of the contnt coefficient ij, nd B i n n r mtri of the coefficient b ij tht weight the input. A ytem output i defined to be ny ytem vrible of interet. A decription of phyicl ytem in term of et of tte vrible doe not necerily include ll of the vrible of direct engineering interet. An importnt property of the liner tte eqution decription i tht ll ytem vrible my be repreented by liner combintion of the tte vrible i nd the ytem input ui. ١٠

11 An rbitrry output vrible in ytem of order n with r input my be written: ١١

12 where y i column vector of the output vrible y i (t), C i n m n mtri of the contnt coefficient c ij tht weight the tte vrible, nd D i n m r mtri of the contnt coefficient d ij tht weight the ytem input. For mny phyicl ytem the mtri D i the null mtri, nd the output eqution reduce to imple weighted combintion of the tte vrible: ١٢

13 Stte Eqution Bed Modeling The complete ytem model for liner time-invrint ytem conit of: (i) et of n tte eqution, defined in term of the mtrice A nd B, nd (ii) et of output eqution tht relte ny output vrible of interet to the tte vrible nd input, nd epreed in term of the C nd D mtrice. The tk of modeling the ytem i to derive the element of the mtrice, nd to write the ytem model in the form: The mtrice A nd B re propertie of the ytem nd re determined by the ytem tructure nd element. The output eqution mtrice C nd D re determined by the prticulr choice of output vrible. ١٣

14 Block Digrm Repreenttion of Liner Sytem Decribed by Stte Eqution Step : Drw n integrtor (S ) block, nd ign tte vrible to the output of ech block. Step 2: At the input to ech block (which repreent the derivtive of it tte vrible) drw umming element. Step 3: Ue the tte eqution to connect the tte vrible nd input to the umming element through cling opertor block. Step 4: Epnd the output eqution nd um the tte vrible nd input through et of cling opertor to form the component of the output. ١٤

15 Emple Drw block digrm for the generl econd-order, ingle-input ingle-output ytem: ١٥

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17 The overll modeling procedure developed in thi chpter i bed on the following tep:. Determintion of the ytem order n nd election of et of tte vrible from the liner grph ytem repreenttion. 2. Genertion of et of tte eqution nd the ytem A nd B mtrice uing well defined methodology. Thi tep i lo bed on the liner grph ytem decription. 3. Determintion of uitble et of output eqution nd derivtion of the pproprite C nd D mtrice. ١٧

18 Emple Find the Stte eqution for the erie R-L-C electric circuit hown in cpcitor voltge v C (t) nd the inductor current i L (t) re tte vrible ١٨

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20 Emple Drw direct form reliztion of block digrm, nd write the tte eqution in phe vrible form, for ytem with the differentil eqution ٢٠

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22 Conider the following RLC circuit We cn chooe tte vrible to be vc( t), 2 il( Alterntively, we my chooe ˆ vc( t), ˆ 2 vl( t). Thi will yield two different et of tte pce eqution, but both of them hve the identicl inputoutput reltionhip, epreed by V( ) R. 2 U( ) LC RC Cn you derive thi TF? t ), ٢٢

23 Linking tte pce repreenttion nd trnfer function Given trnfer function, there eit infinitely mny input-output equivlent tte pce model. We re intereted in pecil formt of tte pce repreenttion, known cnonicl form. It i ueful to develop grphicl model tht relte the tte pce repreenttion to the correponding trnfer function. The grphicl model cn be contructed in the form of ignl-flow grph or block digrm. ٢٣

24 ٢٤ We recll Mon gin formul when ll feedbck loop re touching nd lo touch ll forwrd pth, Conider 4 th- order TF We notice the imilrity between thi TF nd Mon gin formul bove. To repreent the ytem, we ue 4 tte vrible Why? ) ( ) ( ) ( b b U Y G feedbck loop gin um of forwrd pth gin Sum of N q q k k k k k L P P T

25 Signl-flow grph model Thi 4 th -order ytem cn be repreented by Y( ) b 4 ( ) U( ) 3 2 G How do you verify thi ignl-flow grph by Mon gin formul? ٢٥

26 ٢٦ Block digrm model Agin, thi 4 th -order TF cn be repreented by the block digrm hown ) ( ) ( ) ( b b U Y G

27 ٢٧ With either the ignl-flow grph or block digrm of the previou 4 th -order ytem, we define tte vrible then the tte pce repreenttion i,,,, b y b y u

28 ٢٨ Writing in mtri form we hve ) ( ) ( ) ( ) ( ) ( ) ( t t t t t t Du C y Bu A,, 3 2 D b C B A

29 When tudying n ctul control ytem block digrm, we wih to elect the phyicl vrible tte vrible. For emple, the block digrm of n open loop DC motor i ٢٩ We drw the ignl-flow digrph of ech block eprtely nd then connect them. We elect =y(t), 2 =i(t) nd 3 =(/4)r(t)-(/2)u(t) to form the tte pce repreenttion.

30 ٣٠ Phyicl tte vrible model The correponding tte pce eqution i ] [ ) ( y t r

31 Electro Mechnicl Sytem ٣١

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34 Control Flow ٣٤

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