Order Reduction of Linear High-Order Discrete Time Systems Using Polynomial Differentiation Technique in w-domain and PID Controller Design

Transcription

1 Ittiol Joul of Eltoi Eltil Egiig ISSN 97-7 Volum 5, Num, pp 7-5 Ittiol Rsh Pulitio Hous O Rutio of Li High-O Dist Tim Systms Usig Polyomil Difftitio Thiqu i -Domi PID Cotoll Dsig B Stish Niu, JVBJyothi Assistt Pofsso, Bullyy Collg of Egiig fo Wom, Vishptm, AP, INDIA Pojt Stut, Ail Nuo Istitut of Thology Sis, AP, INDIA stishiuol@gmilom, jvjyothi@gmilom Astt I this pp, mtho fo th utio of high o ist timsystms is itou Th poul stps of th popos mtho isuss This mtho is us vi ovl mol o utio -omi tsfomtio Th popos mtho gts stility of lo o mols fo th giv oigil stl high o systms This mtho is vy simpl i implmttio os ot hug mmoy sp fo stog of t Kyos: Difftitio thiqu, o utio, PID otoll IINTRODUCTION Egiig physil systms glly qui high o mthmtil mols fo thi psttio Th lysis otoll sig fo ths is of systms om tious This givs iss to th fo gtig lo o mols fo th oigil systms hih sis th hvio of th oigil systm losly My thiqus hv vlop i th lst f s to ppoximt th high systms y lo o mols i fquy omi Th xistig mthos li Cotiu Ftio Expsio mtho lssil P Appoximtio mthos lim to omputtiolly simpl But,it is stlish i littu tht ths mthos my ot gut th stility of u mols though th oigil systm is stl Som of th fmili mthos vill i littu fo th utio of high o ist tim li Stility Equtio mtho, Bis Cotiu Ftio Expsio mtho quis mo omputtios Routh Appoximtio mtho

2 8 B Stish Niu JVB Jyothi tis th iitil tim momts oly To ovom th ov mtio s of th xistig mthos of ist tim systms utio, mtho is popos fo th utio of high o ist tim systms I th popos mtho, th ist tsf futio is ovt to W- omi fo lttig it ito mol o utio stg This mtho ot oly guts th stility of th oigil high-o systms i u o mols, ut lso psvs th iitil tim momts s ll s Mov pmts Th popos mtho is xpli i th folloig stios IISTATEMENT OF PROBLEM Cosi th o tsf futio i -omi s, B B B B B A A A A D N Apply ili tsfomtio to, th = Th u th o mol is fi s Applyig th Ivs ili tsfomtio is ppli t qutio th u o mol is oti, hih tis th impott htistis of th oigil mol Poul stps: Stp : Apply ili tsfomtio to to oti l th o u mol tsf futio i -omi is 5

3 O Rutio of Li High-O Dist Tim Systms 9 Stp : Dtmitio of th u o omito ostt tm: Th u o omito polyomil ostt tm is oti y usig th ifftitio mtho I this mtho th u o omito polyomil ostt tm is oti fom th q6 Th gl psttio of th o omito is giv y 6 i i i i D Wh =,, - Fo = D A fo = D o Rsptivly Th ostt tm vlu i qutio 5 is ssig to is us i stp to oti th u o mol Stp : Dtmitio of u o mol y usig th ostt tm Th th o oigil systm i qutio is qul to th u th o u mol pst y th qutio 5 = 7 O oss multiplyig gig th qutio 7 8 By qutig th offiits of th sm po of o oth sis i th qutio 8 th folloig ltios oti

4 B Stish Niu JVB Jyothi By solvig ov qutio, th uo pmts of u o mol i qutio 5 lult ith th vlu oti i stp Stp : Apply ivs ili tsfomtio to To III PID CONTROLLER DESIN To mt th sig s spifitios, th PID otoll is sig PID otoll osists of typs of otol, Popotiol, Itgl, Divtiv otol Th tsf futio of th PID otoll is fi s follos: s s s K p K I T KD s Ts Wh KP = th popotiol gi K I = th itgl gi KD = th ivtiv gi Fig: PID otoll i los loop Fist, lt us t loo t th fft of PID otoll o th los loop systm usig th ov shmti figu To gi, th vil is th tig o o th iff t th si f vlu th tul output Th otoll ts this o sigl omputs oth its ivtiv its itgl Th sigl, hih is st to th tuto, is o qul to th popotiol gikp tims th

5 O Rutio of Li High-O Dist Tim Systms mgitu of th o plus th itgl gi K I tims th itgl of th o plus th ivtiv gi tims K D th ivtiv of th o fig IV ALORITHM FOR THE DESIN OF PID CONTROLLER Stp : R th op loop tsf futio of th giv high o systm Stp : Fom th los loop tsf futio Stp : Oti th stp spos of los loop systm Stp : Ch th spos fo th qui spifitios Stp 5: If th spifitios ot mt, gt u o mol y usig th popos utio mtho sig otoll fo th u o mol Stp 6: Oti th iitil vlus of th pmt K P,K I K D y pol o lltio mtho Stp 7: Cs th otoll ith u o mol gt th los loop spos ith th iitil vlus of th otoll pmts Stp 8: Fi th optimum vlus fo th otoll pmts, hih stisfy th qui spifitios Stp 9: By pplyig th optimum vlus, s this otoll ith th oigil systm Stp : Oti th los loop stp spos of th systm ith th otoll Stp : If th spifitio is mt, xit; ls tu th pmts of th otoll till it mts th qui spifitios V ILLUSTRATIVE EXAMPLE Cosi th th o ist tim systm giv y its tsf futio [] APPLICATION OF PROPOSED METHOD: It is popos to pply th popos mtho to oti th so o u mol fo th ov systm Stp: By pplyig th ili tsfomtio is giv , th tsf futio Stp : Dtmitio of u o omito ostt tm By usig th popos mtho, th u o omito ostt tm is oti s 7 is us i stp Stp : Equt th tsf futio ith th gl so o tsf

6 B Stish Niu JVB Jyothi futio Coss multiplyig & gig ov gmt fo th sm po of, th st of qutios oti s : 7 5 i ii iii iv v vi 65 vii By solvig ths qutios ith th vlu = 8759 th vlus,, & lult s Th o u tsf futio usig popos mtho is: Th stp sposs of oigil systm its u o mol omp i Fig 9 Stp Rspos Amplitu 5 Oigil Popos Tim s Fig Exompiso of stp sposs of

7 O Rutio of Li High-O Dist Tim Systms COMPARISION WITH RPRASAD METHOD : Th o u tsf futio usig popos mtho is : Popos mtho Th o mol oti y Stility qutio mtho of RPs is []: Stility qutio mtho Th stp spos of oigil systm u mols y RPs th popos mtho omp i Fig 9 Stp Rspos Amplitu 5 Oigil RPs Popos 5 6 Tim s Fig Ex ompiso of stp sposs of, Applitio of th Popos Rutio mtho Fo th Dsig of PID Cotoll: Th iitil vlus of K P, K I K D oti s K P = -766 ; K I = 8896 ; K D = 8979 Th tu vlus oti usig th igitl omput simultios : K P = 85, K I = O6 ; K D = Th los loop tsf futio of High o systm ith PID Cotoll is oti s: CH Th los loop tsf futio of u o systm ith PID Cotoll is

8 B Stish Niu JVB Jyothi oti s: CR Th stp sposs of th Systm its u o mol, ith ithout th PID Cotoll sig usig th popos mthos sho i Figs sptivly Stp Rspos high o ithout PID high o ith PID Amplitu Tim s Fig Ex ompiso of Clos loop stp sposs of ith PID ithout PID Stp Rspos u o ithout PID u o ith PID Amplitu Tim s Fig Ex ompiso of Clos loop stp sposs of ith PID ithout PID VI CONCLUSION I this pp osv mtho fo th utio of high o ist-tim systms to ovom th mtio s of th xistig mthos of ist tim systms utio Th popos mtho is s o Polyomil Difftitio

9 O Rutio of Li High-O Dist Tim Systms 5 Thiqu pplitio of -omi ili tsfomtio fo otiig oth omito umto of th u o mols Th popos mol utio thiqu is us fo th stility lysis sig of PID otoll fo high-o ist tim systms Th popos mtho ot oly guts th stility of th oigil high-o systms i th u o mols, ut lso psvs th iitil tim momts s ll s Mov pmts REFERENCES [] RPs, O Rutio of Dist tim systm usig stility qutio mtho ight tim momts, IE Joul EL, 99 [] Moh Jmshii, Lg sl Systms Mollig Cotol, Noth- Holl Pulitios, 98 [] Shmsh, Routh P Appoximtio, It Joul of Cotol, 975 [] Y P Shih W T Wu, Simplifitio of -tsf futios y otiu ftios, Ittiol Joul of Cotol systms, Vol 7,97, p 89 [5] T C Ch, C Y Chg K W H, Rutio of Tsf futios y th Stility-Equtio mtho, Joul of Fli Ist, Vol 8, 979, p89 [6] KRmsh, ANiml Kum uusmy, Dsig of Dist Cotoll vi Novl Mol O Rutio Thiqu, Ittiol joul of Eltil Po Egiig :6-68,9 ISSN: [7] Fsi tl Stl u o mols fo Dist tim systms, IEE poigs, vol, Pt D, 986

10 6 B Stish Niu JVB Jyothi