Poceedings of Intentionl cientific Confeence of FME ession 4: Automtion Contol nd Applied Infomtics Ppe 9 Discete Model Pmetition NOKIEVIČ, Pet Doc,Ing,Cc Deptment of Contol ystems nd Instumenttion, Fculty of Mechnicl Engineeing, VŠB-Technicl Univesity of Ostv petnoskievic@vsc Astct: The ppe dels with the methods of the identifiction nd discete model pmetition of the systems opeting in the closed loop The systems e descied using the discete time models Box Jenkins, AMAX, AX Thee e shown the methods of the diect nd indiect identifiction with nd without the test joint signl The methods e pesented with the esults clculted using the pogm MATLAB imulink Keywods: identifiction, discete time models, model pmetition, closed loop systems Intoduction A discete time model descies the eltion etween inputs nd outputs t the discete time points tht e euidistnt nd the time etween two points is the time unit the smpling peiod The Box-Jenkins model stuctue of the discete time model B( ) C( ) ε() A( ) D( ) seems to e uite genel But the AMAX model B( ) C( ) ε() A( ) A( ) cn descie lot of line systems with sttioney distunces Mny systems wok unde feedck contol A lot of technicl systems hs n integto on the output nd the open loop system is unstle o the system is so low dmped tht it is not possile to elise the identifiction expeiment in the open loop In this cse it is necessy to use the methods fo the identifiction of the systems opeting in the closed loop They e thee min ppoches to identifying system woking in the close loop: Diect identifiction Indiect identifiction Joint input - output identifiction The sic pinciples of the identifiction of the systems woking unde feedck e explined in the next chptes of this ppe Indiect identifiction The indiect identifiction is possile to elise with the joint test signl nd without the extenl joint test signl Let the closed loop system hs the stuctue shown in Fig nd the given vlue w () ()
w(t) W() Contolle ystem (t) Z() (t) Z() The sttiony nd mesule test signl u () k is joint to the output of the contolle The test signl nd the distunce () k e uncoelted Afte then we cn conside the signl u () k s independent input of the closed loop system We cn eliminte the system input the ction vlue u () k ( u () k nd () k coelte in the closed loop system) nd expess the output of the system y Y U Z() (3) The closed loop system is descied using the identifile tnsfe function Y (4) U The pmetes of the clculted tnsfe function Ĝ e detemined fom the estimted tnsfe function ˆ nd fom the known tnsfe function of the contolle indiect using the expession ˆ Fig Closed loop system nd the coesponding open loop stuctue ˆ ˆ (5) The tnsfe function Ĝ() cn e pmeteised using the stndd identifiction methods o ecusive methods The descied ppoch convets the polem of the identifiction in the closed loop to the identifiction in the open loop The system is identified using the identifiction of the model highe ode (of the closed loop system) nd the clcultion could e moe complicted, the numeicl polems could occu nd the clcultion could tke moe time
3 Indiect identifiction without joint input Conside closed loop system s shown in Fig with the mesule output nd the given vlue w (t) We suppose tht the system is without distunce nd tht the given vlue w(t) contols the system so much tht the chnges of the output e osevle This condition is vey impotnt to otin the line independent vectos of the vlues of the output [Isemnn 99, ödestöm 989, Noskievič 999] w(t) W() Fig Indiect identifiction without joint test signl The tnsfe function of the contolle is given y Q( ) (, (6) ) ( ) nd the tnsfe function of the identified system is given y B( ) ( ) A( ) m m The tnsfe function of the closed loop system could e expessed like d (7) d d () () Q( )B( ) k () () ( )A( ) Q( )B( ) k (8) The pmetes of the tnsfe function i, i,,,, i, i,,, k e the functions of the pmetes of the tnsfe function of the contolle nd of the identified system Afte identifiction of the closed loop system we hve the estimted pmetes ˆ,, ˆ nd ˆ,, ˆ k fom witch is possile to clculte the pmetes i nd i of the identified system using the eution
m m (9) In genel cse is nue of the eutions fo the clcultion of the pmetes i i, igge thn the nue of the pmetes i i, nd we detemine the pmetes fom the eution [ ] * ˆ Θ T T, () whee [ ] T m â â ˆ ˆ ˆ Θ () is the vecto of the identified pmetes, is the mtix ( ) *,,,, p p is vecto of the estimted pmetes of the tnsfe function of the closed loop system The ccucy of this ppoch is vey good if the system distunce is eo nd the influence of the noise is vey pue Let the closed loop system is given y the lock scheme in Fig 3 The estimted tnsfe function of the closed loop system is #$%#&'()*,- #$%#&'()*, /345$$$6/7875$ 9-97 -9-345$$$6/345$ 9-97 :(;,< :< -89-85$$$6/75$ 9-97 -95$ 9- #>;&#??,& @*>A9BC<C;,A %DC;,$E#C(, Fig3 imultion model of the closed loop system indiect identifiction without joint signl
4 3 4 3 58 664 596 858 58 963 45 55 ) ( () Using the eution 4 3 4 3 (3) nd pseudo invese of the mtix we cn clculte the estimted pmetes of the identified system 3679 3679 64 3679, (4) o expess the tnsfe function 3679 3679 64 3679 ) ( (5) The good ccucy is ovious fte comping the pmetes of the given nd estimted tnsfe function of the system in Fig3 4 Diect identifiction without joint test input This identifiction ppoch is sed on the mesuement of the output nd input We suppose the known stuctue of the contolle nd model stuctue the ode of the AMAX model If the odes of the contolle nd of the system comply with the given conditions fo the odes [Noskievič 999, Isemnn 99] the distunce (t) nd the input signl doesn t coelte despite of pesence of the feedck By the identifiction the system is diven only y the distunce (t), the given vlue w(t) is eo, Fig4 (t) Z() w(t) W() P Fig4 Diect identifiction in the closed loop without joint signl
The tnsfe function of the distunce could e expess y Y ( ) Z( ) P D( )( ), (6) A( )( ) B( )Q( ) nd fte nging A( - )( - )B( - )Q( - ) D( - )( - )Z() (7) The eo e(t) is eul - fo the given vlue w nd the tnsfe function of the contolle could e expess y U ( ) Y ( ) Q( ( ) ) Q( - ) - ( - ) (8) Afte sustitution of the lst expession in the eution (7) we otin A( - )( - )-B( - )( - ) D( - )( - )Z() (9) Afte nging the eution we otin the clssicl expession of the AMAX model fo the open loop system A( - ) B( - ) D( - )V() () The conclusion is, tht it is possile to estimte the pmetes of the mthemticl model of the contolled system fom the dt set of the input nd output of the closed loop system fo the given vlue w(t) nd fo the distunce (t) tht enough dive the system The ccucy of this identifiction ppoch depends on the stuctue of the contolle nd on the filling of the conditions fo the ode of the model nd contolle [Noskievič 999, Isemnn 99] Fig shows the lock scheme of the identifiction with out joint test signl in the closed loop only y woking of the distunce on the output The pmetes of the AX model wee clculted fom the dt vectos of the system input nd output using the #$%#&'()*, @*>A9BC<C;,A %DC;,$E#C(,- )#&D*$>*$HI (;) / #>(;*>; :< -6/-75$$$6//5$ 9-97 -9/75$$$9/85$$$6/7-5$$$6/-5$ 9-97 9 98 #>;&#??,& /--5$$$6/F75$ 9-97 -6/5$$$6//5$ 9-97 :(;,< :<7 #$%#&'()*,- Fig5 Diect identifiction without joint test signl MATLAB pocedues The esult is shown in Fig6
Fig6 Comping of the outputs of the system nd identified model Fig7 Output of the closed loop system nd input of the system
5 Diect identifiction with the joint test input The diect identifiction with the joint test input use the system input nd the system output y the woking of the joint test input signl u s (t), Fig8 w(t) W() (t) Z() P Fig8 Diect identifiction - closed loop system with the joint test input It is possile to deive the identicl eutions fo the sought mthemticl model of the system like fo the diect identifiction without joint test input signl Only one diffeence is hee tht the input signl doesn t depend only on the output of the contolle ut depends lso on the joint input test signl u s (t) This elimintes the coeltion etween the distunce (t) nd the input signl of the system Fo the known ode of the system it is possile fom the dt vectos nd clculte the pmetes of the model in the fom AMAX The joint signl is time vying ndom o pseudo ndom signl This method hs ette esults in the compison with the method without joint signl Fig9 shows the simultion model of the diect identifiction with the joint signl nd with the distunce on the output of the system The AMAX model 45 579 3638 3486 () ws detemined fom the dt set of the input nd output dt vectos in Fig Fig9 imultion model of the diect identifiction with the joint test signl nd distunce on the output
Fig Output nd input signls of the systems used fo the diect identifiction 6 Conclusions Identifiction of the systems opeting in the closed loop ws the topic of this ppe These methods e vey impotnt, ecuse lot of systems wok unde feedck contol This is typicl in the pocess industy, mny othe systems lso in non technicl pplictions iologicl, economicl systems e woking only with the feedck in the closed loop systems It is vey impotnt to know how to identify the open loop systems when it must opete unde feedck contol duing the identifiction expeiment It ws shown tht the diect nd indiect identifiction methods with nd without joint test signl llow to identify the contolled system with the vey good ccucy Fom the numeicl point of view the diect ppoch is the simplest one Diffeent ppoches wee shown on the exmples using the pogm MATLAB imulink In the pctice the optiml expeiment nd ccucy of the identified model cn e otined in diffeent wys 7 efeences NOKIEVIČ, P 999 Modelování identifikce systémů vyd Ostv : MONTANEX, s, 999 76 s IBN 8-75-3- NOKIEVIČ, P & POPÍŠEK, 999 Expeimentální identifikce elektohydulického sevopohonu In : MATLAB 99 s, s IBN 8-75-3- IEMANN, 99 Identifiktion dynmische ysteme pinge - Velg : Belin Heideleg, 99 IBN 3-54-5494- ÖDETÖM,T & TOICA,P 989 ystem identifiction Pentice Hll Intentionl (/UK) Ltd,IBN -3-8836-5