Stability analysis of delayed system using Bode Integral

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1 Stblty nlyss of elye system usng Boe Integl Ansh Ahy, Debt Mt. Detment of Instumentton n Eletons Engneeng, Jvu Unvesty, Slt-Lke Cmus, LB-8, Seto 3, Kolkt-798, In.. Detment of Eletons n ommunton Engneeng, Futue Insttute of Engneeng n Mngement Eml: nshhy9@gml.om Kushk Hle 3 3. Detment of Eletons n Instumentton Engneeng, Ntonl Insttute of Sene n Tehnology,Behmu,In Abstt The PID ontolle metes n be juste n suh mnne tht t gves the ese fequeny esonse n the esults e foun usng the Boe s ntegl fomul n oe to just the sloe of the nyqust uve n ese mnne. The sme e s le fo lnts wth tme ely. The sme hs lso been one n new oh. The ely tem s oxmte s tnsfe funton usng Pé oxmton n then the Boe ntegl s use to etemne the ontolle metes. Both the methoologes e emonstte wth MATLAB smulton of eesenttve lnts n omnyng PID ontolles. A oe omson of the two methoologes s lso one. The PID ontolle metes e lso tune usng el oe Genet Algothm (GA) n oe omson s one between the thee methos. Keywos- PID ontolle, Zegle-Nhols metho, uto-tunng, Pé oxmton, Boe s ntegl I. INTRODUCTION Moe thn 9% of the ontolles use n oess nusty e PID ontolles[]. Howeve the efomne of the PID ontolle lke stsftoy tkng, stubne ejeton, obustness et eens lgely on the tunng methoology. Zegl-Nhols methos[] whee the ontolle metes e tune fom the knowlege of tl fequeny n gn of the lnt. In 984 n utomt tunng metho ws oose n [3] usng smle ely feebk test whh gves usng the esbng funton nlyss. Howeve n oe to obtn the ese hse n gn mgn the lose loo system shoul osllte t the ese ossove fequeny whh n be obtne usng ely wth hysteess[4]o tme ely whh shoul slowly hnge u to obtn lmt yle t the ossove fequeny. Howeve the exement s moe tme onsumng s ome to the evous one. A lose loo ely test ws oose n[5]whee the ossove fequeny s etly entfe. In ths metho the lnt oetes n lose loo wth the ontolle n the outut s fe bk to the efeene of the lose loo system. Km et. l.[6,7,8] use the Boe s ntegl[9] to oxmte the evtves of mltue n hse of the system wth eset to fequeny t gven fequeny. Then these evtves hve been use n the mofe Zegle-Nhols metho n oe to just the sloe of the Nyqust uve t the gven fequeny. The mn objetve of ths e to fn the ontolle metes n oe to obtn ese fequeny esonse fo lnts wth ue tme ely. The ontolle metes hve been tune usng the oxmte evtves of mltue n hse wth eset to fequeny t the gven fequeny obtne usng Boe s ntegl[9] s oose by Km et. l.[6,7,8]. In ths e new metho s beng oose whee the ely tem s oxmte usng Pé oxmton whh eles the ely tem by ue tnsfe funton n then the ontolle metes e tune n oe to obtn the ese fequeny esonse usng the oxmte evtves of mltue n hse foun etly fom the Boe s ntegl fomul. The esonses of both the methos e ome wth mle smulton exmles. The PID ontolle metes e lso foun usng el oe Genet Algothm (GA). It s qute unestnble tht though nvul vton n the ontolle metes s smooth the ombne lnse my be qute omlte onsstng of lol mnm. Ths motvtes the lton of oulton bse ntellgent lgothm lke GA ove the lssl gent bse methos n oe to obtn the tue globl mnm n the otmzton oess. The ue boun n lowe boun e hosen n suh mnne tht t ensues muh bette lose loo efomne wth less eo n the esultnt sloe of the Nyqust uve. Bslly te off esgn s beng oose hee between the lose loo efomne n the esultnt Nyqust sloe. Thus ths e ms n oosng new GA bse PID ontolle tunng metho fo tme ely systems esultng n muh bette lose loo esonse s ome to the one oose by Km et. Al. [6-8] The est of the e s ognze s follows. Seton II esbes justment of loo sloe usng Boe s gn hse

2 eltonsh. Seton III esbes mete tunng of the PID ontolle n oe to obtn the ese esonse. In Seton IV the effet of tme ely on the metes of the Contolle foun by etly lyng the Boe s ntegl s esbe n seton III. Seton V esents bef ovevew of Pé oxmton whh oxmte the ely tem. Amle smulton exmles usng MATLAB [] e gven n seton VI fo both the methos. Pe ens wth the onluson n soe of futue wok n seton VII followe by efeenes. II. LOOP SLOPE ADJUSTMENT USING BODE S GAIN-PHASE RELATION The obustness n efomne of lose loo system eens mmensely on the sloe of the nyqust uve t the ossove fequeny. A elton between T n T of the ontolle n be eve n oe to obtn the ese sloe of the Nyqust uve of the loo tnsfe funton t gven fequeny. Conse the loo tnsfe funton s L( j) G( j) K( j) () Whee the PID ontolle s gven by K( j) KP( jt) () jt The sloe of the nyqust uve of L( j) t s efne s whh s equl to the hse of the evtve of L( j) t. Then t n be eve tht, ( T T ) ( T T ) s ( ) s ( ) T tn ( ) s ( ) T ( TT ) s( ) (3) Whee, G( j ) (4) Futhe, s ( ) n s ( ) e efne s ( ) ln G( j) s (5) ( ) G( j) s (6) s ( ) s ( ) tn( ) T ( s ( ) s ( ) tn( )) T T ( s ( ) s( ) tn( )) (7) Now, s ( ) n s ( ) e oxmte etly usng the hse gn eltonsh eote by Boe[] n 945.The esults e bse on Cuhy s Resue theoem n hve been use extensvely n netwok nlyss. Thee e two ntegls esente []. The fst ntegl shows the elton between the hse of the system t eh fequeny s funton of the evtve of ts mgntue t n the seon ntegl shows the eltonsh between mltue of the system t ny fequeny to the evtve of hse n the stt gn of the system. The fst eltonsh s gven s: ln G( j) v G j v (8) ( ) ln oth v Whee, v evtes fom ln G( j) v ln sne ln oth eeses ly s hene the ntegl eens mostly on Hene t n be seen tht, v ln G( j) G( j ) v v (9) Whh s etly followe by the onluson tht s ( ) ( ) G j () Anothe eltonsh between gn n hse of stble non mnmum hse system s gven by, ( G( j) / ) ln G(j ) = ln K g - ln oth v () Usng () t n be onlue tht, s( ) G( j ) ln K g -ln G(j ) () III. PID CONTROLLER TUNING FOR DESIRED RESPONSE In oe to obtn ese hse mgn t the ossove fequeny the ontolle metes n be foun by justng the hse mgn n the sloe of the nyqust uve t ese the oss ove fequeny. G j + K( j )= - (3) G( j ) K( j ) = (4) The metes of the PID ontolle eque n oe to just the sloe of the Nyqust uve n the ese wy fo move system efomne etly follows by solvng (3), (4) n (7). The PID ontolle metes thus foun to be: K (os( )) (5) G(j )

3 mltue T [( s ( ) s ( ) tn( )) tn( ) ( s ( )) tn( ) s ( )] T= (T -tn( - )) IV. EFFECT OF TIME DELAY: (6) (7) Suose the system wth ue tme ely s esbe s j G ( j ) G( j ) e (8) Whee G( j ) s stble non mnmum hse system. Thus ffeenttng the mltue n hse wth eset to the fequeny t n be shown tht, ln G ( j) ln G( j) (9) G ( j) G( j) () Thus t n be obtne usng the Boe s gn hse eltonsh (9) tht, ln G( j) G ( j ) () Thus s ( ) n s ( ) n be foun to be, s ( ) ( ( ) ) G j () s( ) G( j ) ln K g -ln G (j ) (3) VI. SIMULATION AND RESULTS As eesenttve se the lnt s hosen to be.s G() s e (7) 5 ( s ) Hee the seftons e set s.4/se s the ese ossove fequeny n 5 s the ese hse mgn. Thus t s eque to fn the metes of the PID ontolle whh s ble of movng the lose loo efomne of the system n the ese mnne. As the ontolle moves the ont G(.4 j) of the nyqust uve to ont of K(jω)G(jω) on the unt le whh hs hse 3.e. the ese hse mgn of 5 s heve. In oe to move the lose loo efomne the ese sloe of the oen loo Nyqust uve t the ossove fequeny s set s 65 thus eung the esent sloe by 5 whh ensues gete stne of the nyqust uve fom the tl ont t hgh fequenes. The ontolle metes e foun n two ffeent ohes. At fst the metes e foun usng (5),(6),(7) whee the sloes s ( ), s ( ) e oxmte usng (),(3) whh etly follows fom boe ntegl. An, n ths metho the ontolle s foun to be K(s) =.398(+.37 s) 3.4s (8) The ste esonse of the lose loo system thus foun s shown n fg.().4..8 V. PADE APPROXIMATION FOR TIME DELAY SYSTEM Pé oxmton s fequently use to oxmte ue tme ely by tonl tnsfe funton. Thus on eung the tme ely tem to tonl tnsfe funton PID ontolle metes my be juste usng (5),(6),(7) fo obtnng the ese esonse. The Pé oxmton fo the tem sl e e sl s gven by, N (4) D Whee N n D e esetvely gven s, ( k)! k N k k!( k)! (5) ( k)! k D k!( k)! (6) k tme (se) Fgue. System esonse wth ely etly usng Boe ntegl A seon metho bse on Pe oxmton s beng oose whee the system wth ue tme ely n be oxmte by hghe oe tnsfe funton fee fom ely tem. Ths ely fee tnsfe funton s then use to fn the ste esonse of the lose loo system. On oxmtng the lnt une onseton (7) usng the fst oe e oxmton the oxmte tnsfe funton s foun to be s Gs () (9) s 5s s s 5s s

4 mltue mltue mltue On vng t tonl tnsfe funton fee fom tme ely the ontolle metes n esly be foun usng (5),(6),(7) etly to (9) n thus obtnng the metes ensues move efomne n the ese mnne. The sloes s ( ), s ( ) e foun fom (),() etly. Howeve these oxmtons emloye n oe to obtn the sloes le to esultnt Nyqust sloe of 74.e. lmost 3% eo. Howeve, the esult emonsttes sgnfnt movement n the lose loo efomne. The metes of the PID ontolle eque to mnulte the system seftons lke ossove fequeny, hse mgn, Nyqust sloe to the lose loo esonse to eetemne ese vlues n oe to obtn bette lose loo efomne. The PID ontolle thus foun hee s: K(s) =.376(+.337 s).86s (3) The ste esonse of thus s shown n fg.() tme (se) Fgue. System esonse wth ely by Pe oxmton wth PID ontolle v Boe ntegl eh teton to gve se to bette oulton of soluton vetos n futhe tetons [] Inste of smle eo mnmzton te fo PID ontolle tunng the well known Integl of Tme multle Absolute Eo (ITAE) hs been tken s the efomne nex. J t e t t t t y t t Tunng of the PID ontolle gns hve been one n ths stuy usng the wely use oulton bse otmze known s Genet Algothm. Hee, the numbe of oulton membes n GA s hosen to be 5. The ossove n mutton fton e hosen to be.9 n.3 esetvely. The ue boun n lowe boun of PID ontolle metes e hosen n wy tht the eo n the esultnt sloe of the Nyqust uve oes not exee %. As fo the metes obtne by GA the eo s lmost 7% whh s 4% nese s ome to the metho esbe n [6-8] ue to the oxmtons of the sloes s ( ), s ( ). Howeve s n be foun fom the omtve stuy tht t les to muh move lose loo efomne. The ontolle metes foun by the oulton bse GA emloye hee e s follows, K =.3473, K =.3758, K =.875 The PID ontolle thus foun s: K(s) =.3473(+.396 s) 3.58s (3) An the oesonng ste esonse s gven n fg.(4) Fgue 3. Comson of lose loo System esonses n Fg n Fg The system efomne foun usng both the methos e ome wth the esonse usng el oe Genet lgothm(ga) n oe to omng to oe omson fo the best efomne une the e-hosen esgn seftons. A soluton veto of the eson vbles K, K, K s ntlly nomly hosen fom the seh tme (se) se n unegoes eouton, ossove n mutton, n tme (se) Fgue 4. lose loo System esonse wth ely bse on GA A oe omson of the ste esonses of ll the thee mentone PID tunng methoologes e shown n fg.(5)

5 mltue tme (se) Fgue5. Close loosystem esonse wth ely by Boe ntegl s n Km s Pe, Pe oxmton n Genet lgothm bse otmzton VII. CONCLUSION System esonse wth PID ontolle v GA bse lgothm lose loo System esonse s n Km's e Close loo System esonse wth Pe oxmton Fom the omson of the thee methos t n be obseve tht the PID ontolle metes tune usng the GA gves muh bette efomne thn both of the othe two methos oose hee. The PID ontolle metes foun fom the e oxmte system shows muh moe oveshoot s ome to the othe two methos. Hene, t my be onlue tht the GA bse otmzton esults to muh move lose loo efomne of the system. Howeve t s te off esgn beuse of the ft tht move lose loo efomne le to n nese eo n the sloe of the esultnt Nyqust uve. Futhe wok nlues eveloment of n lgothm whh moves the system efomne wth less mount of eo n the esultng sloe of the Nyqust uve. REFERENCES [] Comnos P, Muno N. PID ontolles: Reent tunng methos n esgn to sefton. IEE Poeengs: Contol Theoy & Alton ;49(): [] J. G. Zegle n N. B. Nhols. Otmum settngs fo utomt ontolles. Tnstons ASME, (64): , 94 [3] Astom KlJ, Hgglun Toe. PID ontolle: Theoy, esgn, n tunng. Instument Soety of Ame; 995. [4] K. J. Ast om n T. H gglun. Automt tunng of smle egultos wth seftons on hse n mltue mgns. Automt, (5):645 65, 984 [5] T. S. She. Close-loo tunng of PID ontolles. In ACC, FA, ges , 99. [6] Km A, G D, Longhm R. PID ontolle esgn usng Boe's ntegls.poeengs of the Amen Contol Confeene ;6(8):57. [7]Km Alez. Dnel G n Roln Longhm, Itetve ontolle tunng usng boe's ntegls. In: Poeengs of the 4st IEEE onfeene on eson n ontol, vol. 4, [8]Km Alez, G Dnel, Longhm Roln. PID ontolle tunng usng Boe's ntegls. IEEE Tnstons on Contol Systems Tehnology 3; (6):8-. [9] H.W. Boe. Netwok Anlyss n Feebk Amlfe Desgn. New Yok, Vn Nostn, 945. [] MATLAB otmzton toolbox use's gue, The Mthwoks, In. [] Stsh Ds, Innl Pn, Shntnu Ds, n Amtv Gut, A novel ftonl oe fuzzy PID ontolle n ts otml tme omn tunng bse on ntegl efomne nes, Engneeng Altons of Atfl Intellgene, vol. 5, no., , Mh.

ME306 Dynamics, Spring HW1 Solution Key. AB, where θ is the angle between the vectors A and B, the proof

ME306 Dynamics, Spring HW1 Solution Key. AB, where θ is the angle between the vectors A and B, the proof ME6 Dnms, Spng HW Slutn Ke - Pve, gemetll.e. usng wngs sethes n nltll.e. usng equtns n nequltes, tht V then V. Nte: qunttes n l tpee e vets n n egul tpee e sls. Slutn: Let, Then V V V We wnt t pve tht: