ET 438a Automatic Control Systems Technology. After this presentation you will be able to:

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1 8/7/5 ESSON 8: MODEING PHYSI SYSEMS WIH INE DIFFEENI EQUIONS E 438a uoaic onrol Syses echnology earng Objecives fer his presenaion you will be able o: Expla wha a ifferenial equaion is an how i can represen ynaics physical syses. Ienify lear an non-lear ifferenial equaions. Ienify hoogeneous an non-hoogenous ifferenial equaions. Wrie pu/pu equaions usg erivaives an egrals for elecrical an echanical syses.

2 8/7/5 ear Dynaic Syses 3 Defiion ear Differenial Equaion - a lear cobaion of erivaives of an unknown funcion an he unknown funcion. Derivaives capure how syse variables change wih ie. ear syses - represene wih lear ifferenial equaions Solvg a ifferenial equaion eans f a funcion ha changes wih ie ha saisfies he equaion. he resul is a funcion an no a nuber. his funcion escribes how a quaniy changes wih respec o he epenen variable, usually ie a conrol syse. his can be one usg analyic or nuerical ehos. ear Dynaic Syses 4 Exaple 8- Series circui wih a curren esablishe iially. Wha oes curren o over ie? i() Wrie KV equaion aroun he circui + v r () i() v () i() v () i() + v() v () v () i() i() o solve - f i() ha saisfies he above equaion wih iial curren of. an o analyically or nuerically.

3 8/7/5 ear Differenial Equaions 5 More oplex Differenial Equaions an have higher orer Derivaives, 3, 4 ec i() i() 7i() bove is a n orer lear ODE (orary ifferenial equaion) Iplie funcion of ie. i=i() i i i bove is n orer, non-lear ODE Square i akes i non-lear v s(v) v n orer, non-lear ODE Se of unknown funcion v() When righ-han sie (HS) is, equaion calle hoogeneous. Iplies no sie siulaion 6 Physical Syse Exaples he followg circui creaes a hoogenous ifferenial equaion. Esablish curren =k No I source > 5 = i()= 5 H urren can no change sananeously ucor. urren varies ie base he iial value of i()= 5 3

4 8/7/5 Physical Syse Exaples 7 he followg circui creaes a non-hoogenous ifferenial equaion. Esablish curren =k Iniial curren = i(-)= H 5 urren buils fro iial value. Pracical exaple: Energize a relay coil wih he curren source. urren source rives he syse. Ienify Exaple Equaions 8 Ienify which of he followg equaions as lear/non-lear an hoogeneous/non-hoogeneous. 3. x( ) x( ) v s( v). i( ). i( ) 6. v. v V. s( ) 4. i. i. 7 i. I o e 4

5 8/7/5 Differenial Equaions For onrol Syses 9 Equaions have consan coefficiens an are lear. Sgle pu siulaion r() Sgle pu variable x() General for x()... a x() a x() a x() b r() n a n n Where a n...a, a a an b are consans Wha can r() be? Differenial Equaions For onrol Syses ypical pu funcions r() onsan ap Susoi -5 3 u() s() -5cos(+q) Where an q are consans Uni-sep (square wave) u(- ) = afer before 5

6 8/7/5 haracerisics of ear Syses.) Muliplyg by consan is reflece hough syse. r() K (r()) ear syse x() K (x()) If pu r() gives pu x() hen, K (r()) gives K (x()) I/O proporional.) Superposiion fro circuis hols r () r () ear syse y () y () If pu r () gives y () an pu r () gives y () hen oal pu y ()+y () oal pu is he su of he iviual pu responses. Fro circuis, ransiens an se seay-sae Dynaic Equaions Inpu/pu relaionships Dynaics represene by egrals an erivaives wih respec o ie Elecrical eleen: esisance Elecrical Eleen: Inucance i() Ohs i () Henrys Defg Equaions v() i() i() v() or i() G v() + v() Where G Defg Equaions v () i() + v () Iniial curren a = i () v ( ) i () 6

7 8/7/5 3 Dynaic Equaions Inpu/pu relaionships Elecrical Eleen: apaciance i () + Defg Equaions i () v () Faras v () Iniial volage a = ll laws fro circui heory hol for he analysis of circuis wih ynaic equaions. K, KV, esh analysis, noal analysis can all be perfore. Subsiue he appropriae egral or erivaive o he esh or noal forulaion. v () i ( ) v () Dynaic Elecrical Equaions 4 When he curren or volage a circui eleen volves wo currens or volages he erivaive or egral, ake he ifference of he volages or currens Exaple 8-: Wrie esh equaions for he circui below usg he lupe circui eleen represenaions. H oop v() i () 5 i ().5 H v() v() v () v() v () v() i() i () i(). i () i () 5 i () v() 7

8 8/7/5 Exaple 8- Soluion () 5. H v() i () 5 i ().5 H oop v () v() v() i( ) v () 5 i() i() i( ) v().5 i () Exaple 8- Soluion () 6 Mesh equaions for a syse of egral-ifferenial equaions unknown funcionsi () an i (). oop 5 i () i (). i () v() () /.5= oop 5 i () i () i ( ) v () i () () Soluion echniques: conver all equaions o erivaives only. pproxiae erivaives usg aheaical ehos an calculae approxiae erivaive values for soe sall creen ie. esuls are a lis of copue pos ha approxiae variable over a ie erval. Graph hese pos o see syse response 8

9 8/7/5 Exaple 8-3 OP MP Differeniaors 7 Use rules of circui analysis an ieal OP MPs o f he pu/pu relaionship for he circui below. i v - v + Defe i c () ers of volage i() v() v v () v () () i f ules of OP MPS.) No curren flows o OP MP.) V - = V + Use noal analysis a OP MP verg noe. Su currens a verg pu i ()+i f ()= so i ()=-i f () i ()=i () Feeback curren vo() v () if () f 8 Exaple 8-3 Soluion () oplee erivaion i () i f v v f v () o () f () vo() v () () v () f v () v () o V + ()=V - ()= Posiive eral groune Inpu ircui akes erivaive of pu volage Oupu 9

10 8/7/5 Mechanical Syse Moels 9 Self-regulag ank syse Nee relaionship for how level changes wih ie. Derive ifferenial equaion. V (Q Q ) Wrie level change ers of ank volue V h Q >Q h creases Q <Q h ecreases V (Q h Q ) Mechanical Syse Moels Self-regulag ank syse F he average level change over ie erval. V (Q h h (Q Q Q ) ) ake lii as ie erval goes o zero h li h (Q Q ) Wrie Q ers of syse paraeers

11 8/7/5 Self-egulag ank ssue laar flow for sipliciy. Q eere by pressure a boo of ank. p Q obe hese wo equaions an solve p g h for Q p g h Q Brg all ers wih h or erivaive of h o one sie Q h g h h Q g h h g Q h g g g Self-egulag ank Siplifie resul g h h Q g Q is epenen of he liqui heigh. consier i along wih ensiy, area an flow resisance o be consan e g G g h h G Q Non-hoogeneous equaion ha eeres how heigh of liqui ank varies wih ie. Shug off Q fs hoogeneous response of ank heigh

12 8/7/5 Mechanical Moels 3 Non-Self-egulag ank (Pupe Draage) Oupu flow is fixe by pup flow rae. I is epenen of he liqui heigh ank. V (Q h Q Defe: = - h() = h( ) - h( ) ) h (Q Q ) h li h Mechanical Moels 4 Non-Self-egulag ank (Pupe Draage) h (Q Q ) Inepenen of h h Noe: igh han sie is no a funcion of. I is all a consan an can be aken of he egral. o solve his ifferenial equaion, egrae boh sies of he equaion wih respec o. h Q h() h( ) h( ) Q Q Q Q Q

13 8/7/5 Non-Self-egulag ank (Pupe Draage) 5 If Q is no a consan bu changes wih ie, he forula below oels he response of he ank syse h( ) h( ) h( ) Q () Q Q () Q h( ) Fal ank heigh Defie egral fro calculus Iniial ank heigh For pupe ank wih consan pu an pu flows, ank ras learly wih ie base on he ifference beween he flow raes. Fal ank heigh epens on he pup flow rae an he ie he pup operaes. 6 heral Syses Heag characerisic of a liqui fille heroeer a a = flui eperaure = easure eperaure = heral capaciance of heroeer Defiion of heral capaciance Q How oes easure eperaure change wih ie? Hea ransferre o heroeer epens on, an ie erval Q a Q Increenal conucion hea flow 3

14 8/7/5 heral Syses 7 Derive he heral equaion Q a a Divie boh sies by Use efiion of fro las slie Deere he average change eperaure for a ake lii approaches a li a 8 heral Syses Ge all an is erivaives on one sie of he equaion a a a a Move o sae sie as erivaive of a his is siilar for o he self-regulag ank equaion. his is a non-hoogeneous ifferenial equaion ha escribes how he easure eperaure changes wih ie 4

15 8/7/5 Mechanical Syses 9 Pneuaic onrol Value Posiion /K= x= B= F a v Deere how he posiion of a air acuae conrol value changes wih ie afer pressure is applie. Free boy iagra- ll forces us su o zero F I () F () M F a F () F a = P a () = pu air force ll forces us balance a each sance ie so: F I () = erial force F () = sprg force F () = viscous fricion force Mechanical Syses- onrol Valves 3 ll forces us su o zero- ssue own is posiive irecion F I () F () F a F F a F F a () F () F I () F () F I () () Nee equaion ha relaes posiion, x, o ie M Fricion force F () F F () () v() x() eeber v() x() Viscous fricion is proporional o velociy. Velociy = rae of change of posiion 5

16 8/7/5 Mechanical Syses- onrol Valves 3 Sprg force F () x() K x() Force fro sprg is proporional o is lengh, x. is sprg capaciance, K = sprg consan. So /K=. Inerial force () F I a() M a() v() v() x() a() x() x() x() F I() M a() onrol Valve Moel 3 obe iviual ers F F a () F () F I () F a x() x() M x() F a is consan. Equaion escribes how posiion changes wih ie. Secon orer equaion, non-hoogeneous. 6

17 8/7/5 Suary of Mechanical Moels 33 Self-egulag ank h h G Q Hea ransfer a g G g Non-Self-egulag ank (Pupe Draage) onrol Valve Posiion x() Fa M x() x() h() Q () Q h( ) 34 E 438a uoaic onrol Syses echnology En esson 8: Moelg Physical Syses Wih ear Differenial Equaions 7