Order Reduction of Continuous LTI Systems using Harmony Search Optimization with Retention of Dominant Poles

Transcription

1 Ode Reducton of Contnuous LTI Systems usng Hamony Seach Optmzaton wth Retenton of Domnant Poles Ode Reducton of Contnuous LTI Systems usng Hamony Seach Optmzaton wth Retenton of Domnant Poles a Akhlesh K. Gupta, b Deepak Kuma and c Paulson Samuel a,b,c Authos ae wth Motlal Nehu Natonal Insttute of Technology, Allahabad, Inda E-mal: a akhleshgupta08@ gmal.com; b deepak_kuma@mnnt.ac.n; c paul@mnnt.ac.n Abstact: In ths pape, a mxed appoach s pesented fo the ode educton of a hghe ode contnuous lnea tme nvaant (LTI) systems n fequency doman. Ths appoach utlzes the domnant pole etenton method to synthesze the poles of educed ode model (ROM) wheeas the Hamony seach (HS) optmzaton technque s used to fnd the numeato polynomal of ROM. The esultng educed ode models ae stable as the domnant poles ae etaned n educed ode model. On the othe hand, Hamony seach optmzaton s adapted though H nom mnmzaton. Futhe, two numecal examples ae consdeed n ode to pesent the effectveness of poposed appoach ove wellknown exstng methods avalable n lteatue. Keywods: Domnant pole etenton, Ftness functon, Hamony Seach, ISE. 1. INTRODUCTION Model ode educton (MOR) s an actve eseach aea whch deals wth the geneaton of smple model fo a hghe ode system wth pesevaton of nput-output behavo. Conventonal fequency doman MOR methods ae contnued facton expanson (CFE) method [1-3], moment matchng technque [4], Padè appoxmaton [5], etc. but sometmes nstablty ssue ases wth these methods. Theefoe, seveal stablty pesevaton methods such as stablty equaton [6], Huwtz polynomal appoxmaton [7-8], Routh appoxmaton [9] etc ae developed. Recently, the combnatoal methods [10-1], have ganed much attenton ove conventonal appoaches n whch the denomnato of educed model s obtaned by stablty guaanteed method [6-9] and the numeato polynomal of ROM s detemned by conventonal appoach [1-5]. Snce last few yeas, seveal optmzaton technques [13] shown n Fgue. 1, ae successfully mplemented by seveal eseaches n dffeent aeas of engneeng. Hence, the optmzaton methods such as genetc algothm (GA) [14-15], patcle swam optmzaton (PSO) [16-17], Bg bang-bg cunch (BB-BC) [18-0] cuckoo seach (CS) [1], Dffeental evoluton (DE) [] and Hamony seach (HS) [3] ae used n MOR fo numeato polynomal calculaton fo bette appoxmaton. In ths atcle, the Hamony seach algothm and domnant pole etenton [4] methods ae combned to fnd out the numeato polynomal and denomnato polynomal of educed ode model espectvely. 69 Intenatonal Jounal of Contol Theoy and Applcatons

2 Akhlesh K. Gupta, Deepak Kuma and Paulson Samuel The contents of pape ae explaned nto fve sectons. The ntoducton pat s gven n secton 1. Secton ncopoates the poblem fomulaton and the denomnato and numeato polynomal educton methods ae explaned n secton 3. The valdaton of poposed method s gven by numecal examples n secton 4 and fnally, the concluson s gven n secton 5.. PROBLEM STATEMENT The n th ode contnuous LTI sngle nput, sngle output system s consdeed wth the tansfe functon as gven below T n s N n (s) D (s) n n1 0 n 0 y s x s (1) whee y s and x s ae numeato and denomnato polynomal coeffcents of hghe ode system espectvely. Now, the poblem s to fnd the th ( < n) ode ROM wth the genealzed wth the tansfe functon consdeed as T s N (s) D (s) s b a s () whee and ae denomnato and numeato polynomal coeffcents of the ROM espectvely. Fgue 1: Classfcaton of optmzaton algothms Intenatonal Jounal of Contol Theoy and Applcatons 70

3 Ode Reducton of Contnuous LTI Systems usng Hamony Seach Optmzaton wth Retenton of Domnant Poles 3. PROPOSED METHOD In ths secton, the poposed method s dscussed whch utlzes the domnant pole etenton and Hamony seach optmzaton algothm fo educton of hghe ode system nto lowe one Domnant pole etenton [4] In the poposed appoach, the poles of ROM ae obtaned by the domnant pole etenton method. In ths method, the numbe of poles equed fo educed model s the domnant poles of hghe ode model. Let the n th ode ognal hghe ode system have n poles located at -p 1, -p,..,- p n, then the domnant poles of th ode ( < n) educed ode ae p 1, -p,.,-p. Hence, the denomnato of educed ode model s D ( s) ( s p )( s p )...( s p ) (3) Hamony Seach Algothm [5] Once the denomnato polynomal s obtaned, the numeato polynomal coeffcents ae obtaned by Hamony seach (HS) optmzaton algothm. Hamony seach, one of the meta-heustc optmzaton algothms s poposed by Geem [5] n 001 whch s based on the pefect hamony seach n muscal pocess. The mpovsaton pocess of a sklled muscan s ntelaced to develop a qualtatve optmzaton pocess called hamony seach algothm. A muscan follows thee choces fo mpovsaton (1) Play a memozed musc (hamony memoy) () play a famous pece wth adjustable ptch (ptch adjustng) o (3) compose somethng new (andomzaton). Smlaly HS follows the above muscal ule: 1. Selecton of a possble soluton s value fom Hamony Memoy (HM);. Selecton of adjacent soluton fom hamony memoy; 3. Selecton of a andom soluton value fom the value of possble ange. Ptch adjustng ate (PAR) and hamony memoy consdeng ate (HMCR) ae two man paametes to mplement the above ules on HS algothm. The Flow chat of HS mplementaton s shown n Fgue. whch s explaned by the fou steps gven below: Step 1: (Intalzaton): Intalze Hamony memoy sze (HMS) that s the numbe of possble soluton vectos along wth hamony memoy consdeaton ate (HMCR), ptch adjustment ate (PAR), bandwdth (BW) and numbe of teatons. Afte that, defne the ftness functon whch should be mnmzed. The ntal soluton vectos n the HS s the andomly geneated n dmensonal vectos wth uppe and lowe bound value and gven by k x x and(0,1) *( x x ) (4) j jl ju jl whee, x ju and x jl ae lowe and uppe bound of soluton vectos, j = 1,,.., N and k = 1,,.., HMS. Step : (Impovsng of a new soluton): Impovsaton s the pocess of New Hamony geneaton. Ths new soluton s selected on the bass of HMCR. The tem HMCR s the pobablty of selectng a component fom the hamony memoy. HMCR s selected fom 0.7 to 0.9, snce convegence speed of algothm and selecton of best hamoncs ae hghly dependent upon ts value. If HMCR value s vey less, then only few best hamoncs ae selected and convegence speed s slow and f t s vey hgh, then almost all vectos ae selected and the best soluton vecto cannot be obtaned..in ths pocess, a andom paamete [0, 1], s geneated, f ths paamete s geate than HMCR, then new vecto s geneated by andom selecton othewse, t s geneated by consdeaton ate. Theeafte, vecto geneated by consdeaton ate s mutated by pocess of ptch adjustment wth pobablty of PAR. Ths pocess s gven as 71 Intenatonal Jounal of Contol Theoy and Applcatons

4 Akhlesh K. Gupta, Deepak Kuma and Paulson Samuel next next x x and(0,1) * BW (5) j j Step 3: (Replacement of soluton vectos): The HM s updated by new soluton f t the ftness value of the new soluton s bette than the that of the wost membe n the HM, othewse t s elmnated. Step 4: The step and step 3 ae contnuously epeated untl maxmum numbe of teaton o some temnaton cteon s met. 4. NUMERICAL EXAMPLES Example 1: Consde the HOS wth tansfe functon [18-1] as gven below G s 3 s 7s 4s 4 s 10s 35s 50s The poles of the above system ae as follows: p 1.00 p.00 p 3.00 p Hee, the objectve s to educe ths system n a second ode model. The poles of the ROM ae obtaned by the domnant pole etenton appoach as dscussed n secton 3.1 whch state that the fast dynamcs poles can be elmnated snce these have less effect on system pefomance wheeas the domnant poles should be etaned n educed model. Theefoe, the poles of educed ode model ae as follows: p 1.00 p.00 1 Now, the numeato polynomal of educed ode model s obtaned by Hamony seach optmzaton algothm by mnmzaton of G G ; whee G s educed ode tansfe functon. The paametes of HS algothm ae as HMS = 100, HMCR = 0.9, PAR = 0.3 and BW = 0.. Theeafte, steady state eo s suppessed by DC matchng and fnally, the tansfe functon of the educed ode model s as follows G s 0.789s s 3s Fgue 3 demonstates the smlaty between the step esponses of educed ode models obtaned by poposed method and exstng methods [4, 19, 1]. Fgue. 4 s plotted fo the convegence of squaed eo wth espect to tme whch shows that the poposed method conveges vey apdly as compaed to the methods [4, 19, 1]. A qualtatve compason of poposed method and method avalable n lteatue s shown n table 1 n tems of tansent behavo wheeas Table shows the pefomance ndces compason fo dffeent eo ndces. Table 1 Qualtatve compason of tansent paametes fo Example 1 Reducton Method Settlng tme Rse Tme Oveshoot (%) Ognal system Poposed Method Nawal and Pasad [1] Boby and Pal [18] Desa and Pasad [19] Desa and Pasad [0] Pama et al. [4] Pal [6] Intenatonal Jounal of Contol Theoy and Applcatons 7

5 Ode Reducton of Contnuous LTI Systems usng Hamony Seach Optmzaton wth Retenton of Domnant Poles Fgue : Flow chat of hamony seach algothms Fgue 3: Step esponses of ROM fo Example 1 73 Intenatonal Jounal of Contol Theoy and Applcatons

6 Akhlesh K. Gupta, Deepak Kuma and Paulson Samuel Example : The state space model of tape-dve functon [7] wth ffth ode dynamcs wth sngle nput and thee outputs s shown as below: A= ; B= 0; C= Hee, the am s to educe the above 5 th ode system nto nd ode lowe model of a close appoxmaton wth the ognal system. The poles of the ognal system ae as follows: p 0.055, p , p 1.899, p 3.970, p Accodng to secton 3.1, the poles nea to ogn ae slow decayng poles wheeas the poles fa away the ogn wll decay vey fast. Hence, the poles of the educed ode model ae p 1 and p. Once the denomnato polynomal s obtaned the numeato polynomal of educed model s obtaned by the Hamony seach by mnmzng the ftness functon. The hamony seach algothm s appled to fnd the nd ode educed model. Hence, the state space matx of educed model s gven by: A = ; B = ; C = The step esponse of tape dve ognal system and nd ode ROM obtaned by the poposed algothm s shown n Fgue. 5. It s obseved that the educed model obtaned by poposed method gves close appoxmaton to ognal system. Fgue 4: Convegence of squaed eo obtaned by dffeent method fo Example 1 Intenatonal Jounal of Contol Theoy and Applcatons 74

7 Ode Reducton of Contnuous LTI Systems usng Hamony Seach Optmzaton wth Retenton of Domnant Poles Fgue 5: Step esponses of ROM fo Example Table Pefomance ndex compason of vaous educed ode appoach fo Example 1 Reducton Method Reduced Model ISE ITSE IAE ITAE Poposed Method 0.789s s 3s Nawal and Pasad [1] s s s Desa and Pasad [19] s s 1.65s Boby and Pal [18] s s.7561s Desa and Pasad [0] 0.8s s 1.47s Pama et al. [4] 0.744s s 1.458s Pal [6] 16s 4 30s 4s x x x x x x x x x x x x x x x x x x x x x x x x x x x x Intenatonal Jounal of Contol Theoy and Applcatons

8 5. CONCLUSION Akhlesh K. Gupta, Deepak Kuma and Paulson Samuel Hamony seach and domnant pole etenton based mxed method s pesented to develop the educed ode model fo lnea tme nvaant contnuous tme systems. The stablty guaanteed domnant pole etenton method s used to fnd the poles of educed model and afte that, the numeato polynomal s obtaned by the metaheustc hamony seach optmzaton method by mnmzaton of ftness functon. Numecal examples of SISO and SITO system s consdeed to valdate the supeoty of poposed method ove exstng methods and the compaatve analyss s ncluded n tems of step esponse, pefomance ndces, tansent paametes and convegence of squaed eo. Futhe, the poposed method can easly be extended fo dscete tme systems. REFERENCES [1] C F. Chen and L. S. Shch, A novel appoach to Lnea model smplfcaton, Intenatonal jounal of Contol, 1968, vol. 8, no. 6, pp [] S. C. Chuang, Applcaton of contnued-facton method fo modelng tansfe functon to gve moe accuate tansent esponse, Electoncs Lettes, 1970, vol. 6, no. 6, pp [3] Y. Shamash, Contnued facton methods fo educton of constant lnea multvaable systems, Int. J. Syst. Sc., 1976, vol. 7, no.7, pp [4] V. Zakan, Smplfcaton of lnea tme-nvaant systems by moment appoxmants, Int. J. Cont., vol. 18, no. 3, pp [5] Y. Shamash, Stable educed-ode models usng Padé type appoxmatons, IEEE Tansactons on Automatc contol, 1973, vol. 19, no.5, pp , [6] T.C. Chen, C.Y. Chang, and K.W. Han, Stable educed-ode Padè appoxmants usng stablty equaton method, Electoncs Lettes, 1980, vol. 16, no. 9, pp [7] R. K. Appah, Lnea model educton usng Huwtz polynomal appoxmaton, Intenatonal Jounal of Contol, 1978, vol. 8, no. 3, pp [8] R. K. Appah, Padé methods of Huwtz polynomal wth applcaton to lnea system educton, Intenatonal Jounal of Contol, 1979, vol. 9, no. 1, pp [9] M. Hutton, B. Fedland, Routh appoxmatons fo educng ode of lnea, tme-nvaant systems, IEEE Tansactons on Automatc contol, 1975, vol. AC-0, no. 3, pp [10] J. Pal, System Reducton by a Mxed Method, IEEE Tansactons on Automatc Contol, 1980, vol. AC-5, no. 5, pp [11] R. Pasad, S. P. Shama, A. K. Mttal, Lnea model educton usng the advantages of Mhalov cteon and facto dvson, Jounal of Insttuton of Engnees IE (I), 003, Jounal-EL 84, pp [1] G. Pama, R. Pasad, S. Mukhejee, A Mxed Method fo Lage-Scale Systems Modellng usng Egen Spectum Analyss and Caue Second Fom, IETE Jounal of Reseach, 007, vol. 53, no., pp [13] R. V. Rao, G. G. Waghmae, A new optmzaton algothm fo solvng complex constaned desgn optmzaton poblems. Engneeng Optmzaton, 016, pp. 1-4, do: /030515X [14] Sataksh, S. Mukhejee, R.C. Mttal, Ode educton of lnea dscete systems usng a genetc algothm, Appled Mathematcal Modellng, 005, vol. 9, no. 6, pp [15] G. Pama, R. Pasad, & S. Mukhejee, Ode educton of lnea dynamc systems usng stablty equaton method and GA, IJECEECE, Wold Academy of Scence, Engneeng and Technology, 007, vol. 1, no., pp [16] S.N. Svanandam, S.N. Deepa, A compaatve study usng genetc algothm and patcle swam optmzaton fo lowe ode system modellng, Int.J.Comput. Intenet Manage. 009, vol. 17, no. 3, pp [17] Z. Gallehda, M. Kaa, and O.P. Malk, Model Ode Reducton usng PSO Algothm And It s Applcaton to Powe Systems, IEEE Intenatonal Confeence on Electc Powe and Enegy Conveson Systems, EPECS 09, 009, Shajah. Intenatonal Jounal of Contol Theoy and Applcatons 76

9 Ode Reducton of Contnuous LTI Systems usng Hamony Seach Optmzaton wth Retenton of Domnant Poles [18] B. Phlp and J. Pal, An evolutonay computaton based appoach fo educed ode modellng of lnea systems, IEEE Intenatonal Confeence on Computatonal Intellgence and Computng Reseach (ICCIC), 010, Combatoe, Inda. [19] S.R. Desa, R. Pasad, A novel ode dmnuton of LTI systems usng Bg Bang Bg Cunch optmzaton and Routh Appoxmaton, Appled Mathematcal Modellng, 013, vol. 37, no , pp [0] S.R. Desa, R. Pasad, A new appoach to ode educton usng stablty equaton and bg bang bg cunch optmzaton, Systems Scence & Contol Engneeng, 013, vol. 1, no. 1, pp [1] Amt Nawal & B. Rajenda Pasad, A Novel Ode Reducton Appoach fo LTI Systems Usng Cuckoo Seach Optmzaton and Stablty Equaton, IETE Jounal of Reseach, 016, vol. 6, no., pp [] J. S. Yadav, N. P. Patda, J. Sngha and C. Adl, A Combned Conventonal and Dffeental Evoluton Method fo Model Ode Reducton, IJECEECE, Wold Academy of Scence, Engneeng and Technology, 011, vol. 5, no. 9, pp [3] H. Nas Soloklo, M. Maghfoo asang, Chebyshev atonal functons appoxmaton fo model ode educton usng hamony seach, Scenta Ianca, 013, vol. 0, no. 3, pp [4] Y. Shamash, Lnea system educton usng Padè appoxmaton to allow etenton of domnant modes, Int. J. Contol, 1975, vol. 1, no., pp [5] Z. W. Geem, J. H. Km, G. V. Loganathan, A new heustc optmzaton algothm: hamony seach, Tans. Soc. Model Smul. Int., 001, vol. 76, no., pp [6] J. Pal, Stable educed ode Pade appoxmants usng the Routh Huwtz aay, Electonc Lettes, 1979, vol. 15, no. 8, pp [7] G. F. Fankln, J. D. Powell and A. Emam-Naen, Feedback Contol of Dynamc Systems (Addson-Wesley, 1994). 77 Intenatonal Jounal of Contol Theoy and Applcatons