Interntionl Crpthin Control Conference ICCC 00 ALENOVICE, CZEC REPUBLIC y 7-30, 00 COOSING TE NUBER OF ODELS OF TE REFERENCE ODEL USING ULTIPLE ODELS ADAPTIVE CONTROL SYSTE rin BICĂ, Victor-Vleriu PATRICIU nd Vile PODARU Deprtment of Computer Engineering, ilitry Technicl Acdemy, Buchret, Romni, bic.mrin@mt.ro Abtrct: The n topic of thi pper i to preent computer ided method to find the necery number of nonl model N of n dptive reference model control ytem uing multiple model o tht the ytem hve deired qulity of dpttion. Firt, the uthor define the qulity of dpttion function of ytem performnce chrcteritic vector. To find n nlyticl generle rule nd mthemticl reltion for N i very difficult or impoible from now. Thi i the reon in thi pper i chooed the ce of econd order ytem with only one unknown prmeter. Even in thi ce finding n nlyticl method without uing computer i verry difficult. Key word: Adptive Control Sytem, Qulity, odel, Sytem Performnce Chrcteritic. 1 Introduction Adptive control ytem theory give u olution for growing control qulity, but not for the evlution of the improving degree of the qulity. To elinte thi difficulty the evlution of the dpttion qulity the rtio of two volume of ytem performnce chrcteritic pce i poible olution [Bică 1000, Bică 000]. The method of dpttion of the dptive reference control ytem uing multiple model conit of identifiction the rel model of the ytem, the election of the nonl model nd then the election of the reference model mot pproprite to the functioning condition of the ytem [ărgărit 1998]. It i necery certin number of nonl model N for obtining the precribed qulity of the ytem. The don of the prmeter pce i bounded for the dptive reference model control ytem uing multiple model. Thi llow u to prtitioner it in ub-don, every ub-don hving model in hi center. The qulity of dpttion will be evlute for ytem with contnt prmeter 137
coreponding to the center of the ub-don. Nrendr [Nrendr 1997] hown tht the qulity of the ytem depend on the number of the model, being better if the number of model N incree. But, from now, nobody give n nlitycl method to find the number of the model o tht the dptive control ytem hve deired qulity. Evluting qulity of dptive control ytem Becue the utomtic control ytem without dptive controller re mny time untble, firt we mut define tbiliztion index the rtio Q N N (1 where N i the number of ll nlyed ce nd N i the number of tble ce. For the dptive ytem (which i tble in ll the ce we firt define n dpttion qulity index the rtio of two volume of ytem performnce chrcteritic pce (don Q V Vn, ( where V i the volume of non-dptive control ytem obtined for wort-ce nlyi n nd V i the me volume for dptive control ytem, computed only for the N ce in which the non dptive ytem i tble. Q i necery to elinte from nlyi the ce tht led to inignificnt vlue Q n for the index (for exmple if the ytem i untble V grow to infinity nd the Q index become zero. For compring the dpttion qulity of the me proce with different dptive controller the Q index i ued. The V n volume i given by p n ( i mx i n i 1 V γ γ, (3 where: γ ( γ 1, γ,..., γ p i the ytem performnce chrcteritic vector; γ i mx nd γ i n re the mximum, repectively the nimum, vlue of the i-th prmeter of the ytem performnce chrcteritic vector; The obtining of the V volume for every type of dptive control ytem differ, nd depend on two poible itution: both the compented nd uncompented ytem cn be chrcterized by the me ytem performnce chrcteritic vector; the compented ytem nd uncompented ytem cn not be chrcterized by the me ytem performnce chrcteritic vector (e.g. the uncompented ytem h overhoot nd compented ytem h not. Ce 1. Both the compented nd non-compented ytem h the me ytem performnce chrcteritic vector. 138
The V volume depend on ued dptive method. For dptive control ytem with explicit reference model the V volume depend on the preciion of the model reliztion: ( 1 p p ( i mx i n i 1 V γ γ, (4 where: γ γ, γ,..., γ i the ytem performnce chrcteritic vector of the model; γ i mx nd i n γ re the mximum, repectively the nimum, vlue of the i-th prmeter of the ytem performnce chrcteritic vector of the model; Ce. The compented nd uncompented ytem re not chrcterized by the me ytem performnce chrcteritic vector. It i better tht the V volume to be etimted with n integrl error criterion [Dutrche 1993] nd the Q index i obtined : Q I In, (5 where I n i the integrl error performnce index of non-dptive control ytem nd I i the me integrl error performnce index for dptive control ytem. We will chooe the criterion tht i mot pproprite for the functionl requet of the ytem. 3 Chooing the number of model of the reference model uing multiple model dptive control ytem Given imple econd order ytem with the trnfer function of the uncompented cloed loop ytem, repectivelly uncompented open loop ytem: ( d + ξ+ ( + ξ with only one uncertin prmeter vrying between two knew lit [, ] compented with erie dptive compentor it i hown in figure 1. n mx (6 (7 nd Figure 1. Serie compented dptive control ytem 139
The rnge of vrition of i divided into N intervl. Figure. The digrm of nturl rdin frequency of nonl model For every, i 1, N correpond nonl model with the trnfer function of the cloed loop ytem ( nd the trnfer function of the open loop ytem ( : d ( d + ξ + ( + ξ (8 (9 For every nonl model with the trnfer function ( it i obtined the trnfer function of the compentor ci (. Tking into ccount the trnfer function of the reference model of the ytem of the cloed loop ytem, repectively open loop ytem, i: ( D + ξ + ( the trnfer function of the compentor will be: ci ( ( ( + ξ ( + ξ ( + ξ D d The mximum error i obtined if the nturle rdin frequency i t equl ditnce from nd + 1. The rel trnfer function of the uncompented ytem cn be model or + ( model. pproximted with ( 1 If the trnfer function of the ytem i pproximted with ( compentor ci ( Di ( : (10 (11 (1 will be chooed the nd the rel trnfer function of the compented open loop ytem i ( + ( + ξ ξ Di ( ci ( d ( + ξ 140 (13
If the trnfer function of the ytem i pproximted with 1 ( compentor ci 1 ( i ( Di+ 1 : + will be chooed the + nd the rel trnfer function of the compented open loop ytem ( + + 1 1 ( + ξ ξ ( ( ( + ξ Di+ 1 ci + 1 d + Generlly, to relie deired qulity for the compented ytem we cn ue the dptive qulity index ( Q for the reference model of the ytem nd Q for the reference model uing multiple model dptive control ytem, or the ytem performnce chrcteritic of the ytem (the reference model performnce chrcteritic re the mximum overhoot σ, the ettling time t t, nd the ytem performnce chrcteritic re σ nd t. Thi ce it i recommended to ue the ytem performnce chrcteritic. t It i necery to chooe the vlue of the nturl rdin frequency o tht the difference between the ytem performnce chrcteritic of the reference model nd tht of the compented ytem re mller thn n impoed error ε i. The ytem performnce chrcteritic re function of the number of nonl model N nd the vlue of nturl rdin frequency of the nonl model, i 1,,...N. Tking into ccount the reltion bove preented it i difficult to find n nlyticl correpondence between N nd deired qulity of the ytem. Thi i the reon we ue the computer to imulte the reference model uing multiple model dptive control ytem, to find out the ytem performnce chrcteritic, to compre with the reference model chrcteritic nd uing the method tril nd error to etblih the nturl rdin frequency of the nonl model. Becue the complexity of the dptive compentor depend on the number of nonl model N, the method i uefull in helping help u to obtin the nimum number of nonl model tht ure the deired qulity of the ytem. 4 Exmple We will preent uing n exmple the econd order ytem with trnfer function given by (8 nd (9. The nturl rdin frequency i vrying on the intervl [ 0.1; 0.9 ], nd the dmping fctor ξ i contnt ξ 0.15. The reference model h the trnfer function [ărgărit 1998]: ( 1 ( 5+ 1( 3+ 1 (14. (15 For the non-dptive ytem with compentor with the nturl rdin frequency n 0.55 we will obtin σ mx 5.05%, σ n 0%, t t mx 113.684, t t n 7.556, nd the volume V will be: n 141
V ( σ σ ( t t (5.05-0(113.684-7.556 434.7545. (16 n mx n t mx t n To obtin the deired qulity we find tht we need 10 nonl model with the nturl rdin frequency, i 1,10 ditributed follow: 0.0, 0.40, 0.6, 0.87, 0.315, 0.35, 0.40, 0.470, 0.580, 0.775 rdin. For the dptive ytem σ mx 0.315%. σ n 0%, t t mx nd the volume V will be: V ( σ σ ( t 30., tt n 7.0375 mx n tmx t t n (0.315-0(30.-7.037 1.0033. (17 The dpttion qulity index i: Q V Vn 1.0033 434.7545 0.00308, (18 howing tht for 10 reference model optiml ditributed V 1.0033, Q 0.00308 nd the dptive ytem dinihe more thn 433 time the volume in ytem performnce chrcteritic pce. For comprion, by tking 10 nonl model uniform ditributed (the intervl of vrition of i divided in 10 equl intervl, t t mx 44.750, t t n 7.056, the volume V 19.1449 nd the dpttion qulity index i Q V Vn 0.044, the dptive ytem dinihe more thn time the volume in ytem performnce chrcteritic pce (compring with 433 in the ce of nonl model optiml ditributed. 5 Concluion The method llow, now only by trying, to find the number nd prmeter of the nonl model of model reference uing multiple model dptive control ytem. We obtin high improving of the qulity of the ytem by n optiml choice of the prmeter of nonl model without ny chnge in the number of nonl model. Reference 1. BICĂ,. 1999. Prmetric Relibility of Automtic Control Sytem, PhD Thei, ilitry Technicl Acdemy, Buchrtet.. BICĂ,. 000. Automtic Control Sytem. Prmetric Relibility. ilitry Publihing oue, Buchret, ISBN 973-3-0600-8. 3. DUITRACE, I., DUITRU, S., IU, I., UNTEANU, F., USCĂ, G., CALCEV, C. 1993. Automtizări electronice, Editur Didctică şi Pedgogică, Bucureşti, 1993. 4. ĂRGĂRIT, L. 1998. Contribuţii l intez itemelor dptive multimodel, Teză de doctort, Bucureşti. 5. NARENDRA K.S., BALAKRISNAN J.1997. Adptive Control Uing ultiple odel, IEEE Trnction on Automtic Control, Vol. 4, No., Februry. 14