Linear Open Loop Systems

Transcription

1 Colordo School of Me CHEN43 Trfer Fucto Ler Ope Loop Sytem Ler Ope Loop Sytem... Trfer Fucto for Smple Proce... Exmple Trfer Fucto Mercury Thermometer... 2 Derblty of Devto Vrble... 3 Trfer Fucto for Proce wth Multple Iput d/or Multple Output... 3 Exmple Trfer Fucto Strred Tk Heter... 5 Trfer Fucto of Proce Sere... 9 Pole & Zero of Trfer Fucto... 9 Exmple Pole & Zero of Trfer Fucto... 2 Trfer Fucto for Smple Proce f t Iput Dymc Proce yt Output f G y Coder the mple proce wth oe put & oe output. The decrbg -th order ODE : 2 d y d y d y dy 2 2 y bf t Let u ume we re ug devto vrble, o tedy tte, o: y, d we re trtg t 2 d y d y dy. 2 t t t Tkg the Lplce trform of th gve: 2 y y 2 y y y bf y b f 2 2 G Joh Jechur (jjechur@me.edu) - - Copyrght 27 Aprl 23, 27

2 Colordo School of Me CHEN43 Trfer Fucto G defed the trfer fucto d the mple dgrm clled the block where dgrm for the proce. Exmple Trfer Fucto Mercury Thermometer Mke the followg umpto bout the redg from mercury thermometer: All retce to het trfer th flm roud the bulb.e., eglect therml retce of gl & mercury. All therml cpcty the mercury. Mercury lwy h uform temperture. The gl wll doe ot expd or cotrct. The eergy blce o thermometer wll be: Joh Jechur (jjechur@me.edu) Copyrght 27 Aprl 23, 27 de d E K P du dh ha T T mc hat T for cott C ˆ p mc T T ha T T where the tme cott : mc ha At tedy tte: T T * * o term of devto vrble: TT where T Tkg the Lplce trform of th ODE gve:

3 Colordo School of Me CHEN43 Trfer Fucto T T o the trfer fucto : G T T So, we would expect the het trfer retce roud thermometer to be t order ytem. Derblty of Devto Vrble If we dd t ue devto vrble the Lplce trform of the ODE would be: T T T T T T T T T T T T T T * Now there re two put & two trfer fucto: oe for the drvg fucto ( T T ) d oe for the tl coo ( T ). * Trfer Fucto for Proce wth Multple Iput d/or Multple Output t or Wht f there re multple put d/or multple output? We would octe trfer fucto wth ech prg of put & output. The block dgrm for 2 put & output : f f2 G y f G f G + + y 2 2 Joh Jechur (jjechur@me.edu) Copyrght 27 Aprl 23, 27

4 Colordo School of Me CHEN43 Trfer Fucto The overll reltohp for y would be: y G f G f 2 2 For put d oe output, the: y G f G f G f G f y G f The block dgrm for 2 put & 2 output : f G, + + y G2, f2 G2, G2,2 + + y2 The overll reltohp for the y fucto would be: y G f G f,,2 2 y G f G f 2 2, 2,2 2 For put d m output, the: y G f G f G f G f,,2 2,3 3,, j j j y G f for,2,3,, m. or mtrx otto : Joh Jechur (jjechur@me.edu) Copyrght 27 Aprl 23, 27

5 Colordo School of Me CHEN43 Trfer Fucto y G f y colum vector of legth m, f colum vector of legth, d G where m rectgulr mtrx. G clled the trfer fucto mtrx. Exmple Trfer Fucto Strred Tk Heter F, T, h, A, T, F, T F, T, The mterl blce o th ytem wll be: dm d h F F A F F umg cott cro-ectol re, A. The eergy blce : de F H ˆ F H ˆ Q F ˆ H F H ˆ Q Remember, wth the tk: So: de d U K P du dh dh d VHˆ F H ˆ FHˆ Q Joh Jechur (jjechur@me.edu) Copyrght 27 Aprl 23, 27

6 Colordo School of Me CHEN43 Trfer Fucto If we ume tht the ethlpy c be expreed : Hˆ Cˆ T T H ˆ p ref ref the wth H ˆ & T : ref ref d VCˆ ˆ ˆ pt F CpT F CpT Q ˆ d C ˆ ˆ p VT F CpT F CpT Q d Q VT F T F T C If we ume cott, the: d: dh A F F d Q A ht F T F T C d F F Q ht T T A A AC TA dh ha F T F T Q C Iertg the mterl blce: Joh Jechur (jjechur@me.edu) Copyrght 27 Aprl 23, 27 Q T F F ha F T F T C Q ha F T T C If we mke the umpto tht Q V F T T C where h h t. dh/ the V ha cott & F F, o:

7 Colordo School of Me CHEN43 Trfer Fucto If we re ug tem for the hetg medum, the we could relte the rte of het dded, Q, to the tem temperture, T, : So: where: Q UA T T. UA T T V FT T Cˆ UA UA V F T F T T Cp Cp ˆ F UA F UA T T T ˆ V VC V ˆ p VCp K T T KT F F T T KT F p F F V, UA K VC, d K. F At tedy tte: o: T T KT * * * F T T KT F where the devto vrble re defed : TT T *, TT T *, d TT T *. Joh Jechur (jjechur@me.edu) Copyrght 27 Aprl 23, 27

8 Colordo School of Me CHEN43 Trfer Fucto Note tht th equto how how the trred tk flud temperture ffected by chge the other temperture. I th Chpter we wll covert th equto to oe volvg trfer fucto. Tkg the Lplce trform of the equto gve: TT T T KT F T T T KT F T T KT K T F T T F Th how tht we hve two trfer fucto: where: TG TG T / F K G d G A block dgrm for the trred tk heter c be drw follow. T T / F K + + T Joh Jechur (jjechur@me.edu) Copyrght 27 Aprl 23, 27

9 Colordo School of Me CHEN43 Trfer Fucto Trfer Fucto of Proce Sere y () y 2 () f() G () G 2 () G () y () If there re ere of trfer fucto, the: G G y2 GG G f G y G y y f Pole & Zero of Trfer Fucto Accordg to defto of the trfer fucto: where: y f G G Q P d where Q d P re uully polyoml (tme dely wll troduce expoetl term, however). I geerl, the order of Q wll be le th tht of P. The root of the umertor Q re referred to the zero of the trfer fucto. At the zero, G become zero. The root of the deomtor P re referred to the pole of the trfer fucto. At the pole, G become fte. We c get qulttve ee of the repoe of ytem by kowg the pole. Let: Joh Jechur (jjechur@me.edu) Copyrght 27 Aprl 23, 27

10 Colordo School of Me CHEN43 Trfer Fucto f r q Sce: G Q P the: y G f Q r P q Let let the root of P be deoted p. The, f P polyoml of order d there re N o-repetg root d M repetg root (ech oe repetg m tme), the: d: N P p p M m y N Q p p M m r q Whe plt to prtl frcto, ech of the fctor the deomtor wll led to eprte term. Splttg up the fctor of the trfer fucto (whle levg the deomtor from the put fucto de for ow) gve: y y m N M j, C j m p D p j r p q *. N M m * C Dj, r. m j p j p q Joh Jechur (jjechur@me.edu) - - Copyrght 27 Aprl 23, 27

11 Colordo School of Me CHEN43 Trfer Fucto Note tht for the repeted root, the umertor c be polyoml of order up to oe le the order of deomtor. Alo, ech repeted root c hve dfferet order. The oly requremet o the umber of root tht they hve to dd up to,.e.: M N m. Whe we vert the Lplce trform, the: C Cexppt. M m m D M j, D j, L m j m j j p j p L N N p L D M m j, exppt L m j j. D m j! M m j, exppt L m j j m j! M m expp Dj, m j t t j m j! Note tht the root p re mportt for the log-tme chrctertc of the oluto. For the rel o-repetg root: p, the exppt t. Th expoetl decy led to zero cotrbuto from th pole. p, the exppt t. Th expoetl growth led to explove cotrbuto from th pole. p, the exppt for ll t. Th cott term hould ot led to y tblty. If If If For the complex o-repetg root (whch wll occur complex cojugte pr), the p c be expreed. Thee root wll gve re to term of the form exptt. Now, the mportt term wth regrd to tblty the rel porto of the root, : If exp t t t. Th expoetl decy led to zero cotrbuto from th pole., the Joh Jechur (jjechur@me.edu) - - Copyrght 27 Aprl 23, 27

12 Colordo School of Me CHEN43 Trfer Fucto If exp t t t. Th expoetl growth led to explove cotrbuto from th pole. If, the expt t t for ll t. Th term wll led to tble ocllto., the For the repetg root, the tuto mlr. The polyoml term wll lwy grow towrd fty t, o the behvor of the expoetl term wll dctte the overll behvor. If p or, the the expoetl term wll go to zero t d the etre term wll lo go to zero. Th expoetl decy led to zero cotrbuto from th pole. p or, the the expoetl term wll grow to fty t d the etre term wll lo grow to fty. Th expoetl growth led to explove cotrbuto from th pole. p or, the the polyoml term wll dctte the behvor for t. Th polyoml term wll led to explove cotrbuto from th pole.. If If So, geerl: If, tble cotrbuto from th pole., utble cotrbuto from th pole., tble cotrbuto oly f o-repeted root utble cotrbuto f repeted root. If If Exmple Pole & Zero of Trfer Fucto Gve the trfer fucto: G Q Q P fd the zero & determe f tble. The followg chrt how the chrctertc of root. P v.. Note tht there re o rel Joh Jechur (jjechur@me.edu) Copyrght 27 Aprl 23, 27

13 Colordo School of Me CHEN43 Trfer Fucto P ( ) C fctor P to get: P From th, we fd tht the root re: 4 3 r r 2 Sce the rel porto of the root re ll egtve, the ytem tble. Joh Jechur (jjechur@me.edu) Copyrght 27 Aprl 23, 27