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1 DECETRALIZED ODEL REFERECE ADAPTIVE PRECIZE COTROL OF COPLEX OBJECTS S.D. Zemyakov, V.Yu. Rutkovsky, V.. Gumov ad V.. Sukhaov Isttute of Cotro Scece by V.A. Trapezkov Russa Academy of Scece 65 Profsoyuzaya, 7997, oscow, Russa Te: 007 (950) ; e-ma: Abstract: Ths paper deas wth the cotro probem of compex obects o the base of the mode referece adaptve prcpe. For decomposto ad precse cotro ew agorthms are dscovered. A space robotc modue s cosdered as the exampe of a compex obect. Computer smuato demostrates good resuts. Key words: Compex Obect, Decomposto, ode Referece Adaptve Cotro, Lyapuov Fucto, Smuato. ITRODUCTIO As a compex we cosder a obect wth some tercoected subsystems (Voroov, 985; Šak, 99). A mathematca mode () of such a obect s usuay oear ad ostatoary oe. Sythess of cotro agorthms for such a obect s ot a smpe probem. Usua f ot a sge method for ths goa s decomposto ad aggregato (Voroov, 985; Šak, 99). The decomposto coud be reazed o physca or mathematca prcpes (Šak, 99). For every subsystem a oca good cotro ( ay sese) s dscovered provded that tercoectos are mssg. The t s ecessary to prove that the good behavor of the system o the whoe takes pace (Voroov, 965; Šak, 99). I ths paper we use other a approach: for every subsystem a compoet of tercoectos s seected ad compesated o the base of adaptve cotro. ore correcty the approach cudes the use of computer aded cotro (Zemyakov ad Rutkovsky, 004); of a programmed adaptve cotro; of a mode referece adaptve cotro (Zemyakov ad Rutkovsky, 966; Pertrov, et a., 980). A oca adaptve cotro system for a obect subsystem coud be sytheszed o the base of tradtoa agorthms for a mode referece adaptve approach. I resut the adaptve system w be oear ad ostatoary. It s kow that for such a system s dffcut to suppy good dyamcs ot speakg about precse cotro (Zemyakov ad Rutkovsky, 966; Pertrov, et a., 980). So ths paper we derve ew otradtoa agorthms for a system wth referece mode that are capabe to guaratee the desered precse cotro.. PROBLE STATEET Cosder a compex obect (Zemyakov ad Rutkovsky, 004) descrbed by the dffereta equato T A( qq ) + qd s( qq ) es Sq ( ) () s q R, s D ( q) ( d ( q)) s d, t R, Aq ( ) ( a ( q)) 0 a ( q) a ( q) a ( q) + qt q qs (, ts,, ). s s st t t >, Durg the obect s operatg matrces A(q), D s(q), S(q) (s, ) are kow (Zemyakov ad Rutkovsky, 004); vectors q q(), t q q () t are measurabe. For every q (, ) there exsts a fucto q 0 () t ad a equato

2 + kq kq 0 () t, () the fucto q 0 () t ad the umbers k > 0, d > 0 are prescrbed advace. The probem: It s ecessary to dscover cotro agorthms tqq (,, ) that guaratees the moto (). 3. DECOPOSITIO OF A OBJECT ATHEATICAL ODEL Let a obect wth () s decomposed to subsystems accordg to the physca prcpe (Šak, 99) A ( qq ) + ( qq, ) S ( q ) (, ), A ( q ) > 0, q, vectors q, T q ( q, q,, q ), T (,,, ). are compoets of Dmesos of the vectors q, (, ) are equa ad. The for ay subsystem coud be wrtte the A ( qq ) S ( q ) + F( tqqq,,, ) (, ), A ( q) R, A ( q ) > 0, F () S () q A ( q) ( q, q ) R, (3). 4. ATHEATICAL ODEL OF A SUBSYSTE WITH ACTUATORS Dfferet subsystems coud have actuators of dfferet ature. I ths paper we cosder oy dc motors as actuators. The of dc motor s we kow (Krutko, 99) J, d r ( r) k k k τ + d d m m ω u rq R R (, ;, ), (4) s the umber of subsystems wth dc motor actuators. d I (4) s a movg momet of a motor; s a momet for a oad rotato; r s a reducto coeffcet; J, R, τ, km, k ω are motor costructve parameters; u are cotro agorthms to be dscovered. Let the of a obect subsystem wth dc motor actuators has the (3). To the equato (3) t s ecessary to add the equatos for actuators (4) preseted matrx d A R( ), d d τ + ρu β q (5) (, ), A dag ( J ( r ), J ( r ),..., J ( r ) ), R dag r r r (,,..., ), d T d d d ( ) (,,..., ), T ( ) (,,..., ), dag τ ( τ, τ,..., τ ), k m k k m m ρ dag(,,..., ), R R R k mk k k ω kmkω m ω dag(,,..., ) R R R β. From (3) ad (5) the of a subsystem movemet together wth actuators accepts the

3 [ A ( q) + S ( q) A ] d S ( qr ) + F( tqqq,,, ), d d τ + ρu β q, (, ). (6) For cotro agorthms sythess t s possbe (Krutko, 99) to take the codto From the equato (0) we see that f t s vad the equaty [ f ( t, q,, q) + S ] 0 () the the movemet of a subsystem wth umber s decomposed o separate subsubsystems wth τ ad to smpfy the system (6) to the [ A ( q) + S ( q) A ] S ( q) R ρ U S ( qr ) β q + F( tqqq,,, ). 0 (7) + k q k q 0 () t ( ),. () From () ad () we see that the cotro agorthm (9) gves the probem souto f the equaty () takes pace. 5. ADAPTIVE PROGRAED COTROL OF A OBJECT SUBSYSTE The of a obect subsystem wth dc motor actuators (7) coud be preseted the [ A ( q) + S ( q) A ] S ( q) R ρ U + + f ( tqqq,,, ), f tqqq A q S qa [ F( tqqq,,, ) S ( qr ) β q ]. (,,, ) [ ( ) + ( ) ] The codto det( A ( q) + S ( q) A ) 0 (8) 6. ODEL REFERECE ADAPTIVE COTROL Beow we w try to hod up the equaty () o the base of mode referece adaptve cotro (Pertrov, et a., 980). Adaptve agorthms w ot be tradtoa as (Zemyakov ad Rutkovsky, 966; Pertrov, et a., 980) but more costructve for precse cotro. If the equaty () does ot take pace tha the equato () takes the + k q k q 0 () t + f () t + S ( ),, f ( t) f ( tqqq,,, ). Let us choose a referece mode the (3) s usuay rght. Let us take a cotro agorthm the U ( S ( q) R ρ ) [ A ( q) + S ( q) A ] + 0 [ K ( q q ) D q S ], ( q ) ( q ( t), q ( t),..., q ( t)), 0 T (,,..., ), (,,..., ), T. K dag k k k D dag d d d ( S ) ( S, S,..., S ) Equates (8) ad (9) together gve a equato + D q + K q K q 0 () t + + [ f ( tqqq,,, ) + S]. (9) (0) + k q k q 0 () t m m m (, ) ad wth the equaty ε q q m (4) we receve from (3) ad (4) the equato wth respect to the error ε the ε + d ε + k ε [ f ( t) + S ] Wth otato ( ),. (5) 3

4 ε x, ε x f () t + S y, f () t µ (), t ψ,, S equato (5) coud be wrtte the matrx x Ax + β, y ψ + µ (), t ( ) T (, T x x x), ( β ) (0 y ), 0 A. k d (6) We w try to choose a agorthm for S purposefu varato from the codto of the asymptotca covergece of the system (6) movemet wth respect to the movemet x 0, y 0. For ths goa we take a Lyapuov fucto the V ( x, y ) κ ( x ) P ( x ) ( y ) T + (7) P s a postve defte matrx, κ cost > 0. The dervatve of V (, x y ) wth respect to tme aog a souto of the system (6) s the equaty T V ( x, y ) κ ( x ) Q ( x ) + + y [ σ + µ ( t) + ψ ], (8) Q s the prescrbed egatve defte matrx, σ ( ), p x + p x prk are eemets of the matrx P ( p ) ( r, k,). rk For a aaytca resut we suppose that the sg of the coordate y s kow. Reay t s possbe oy to approach to ths assumpto by dfferet way. I the paper we do ot cocetrate o ths questo. Oe smpe possbe way w be accepted for smuato. The we choose the desred agorthm the ψ σ ksgy ( ), (9) k > 0. We suppose that the equaty k > µ () t takes pace. The we have the resut V ( x, y ) > 0, V ( x, y ) < 0 whch esure the souto of the probem. 7. SIULATIO RESULTS For smuato we cosder a space robotc modue (SR) (Zemyakov ad Rutkovsky, 004) as a compex obect wth (). As a mechaca obect SR has a bg umber of degrees of freedom. of a SR s oear mutcoected ostatoary oe. Automatc cotro by such a obect s ot a smpe probem. At the same tme SR operatg demads precse dyamc accuracy to provde for ts safety, for the safety of obects wth whch t teracts. O the base of physca prcpe (Šak, 99) the of SR coud be dvded o two tercoected subsystems: subsystem for the carryg body ad subsystem for mapuators (Zemyakov ad Rutkovsky, 004). Cotro agorthms for the carryg body subsystem coud be sytheszed o the base of rey cotro, Potryag maxmum prcpe ad the drect Lyapuov method (Zemyakov ad Rutkovsky, 004). Here we w try to sythesze cotro agorthms for the mapuators subsystem o the base of adaptve cotro (Zemyakov ad Rutkovsky, 966; Pertrov, et a., 980). We assume that the decomposto probem s soved ad ow t s ecessary to show that agorthms (9) reay provde precse cotro of q () t wth respect to q 0 () t (). Let k 0.5 ad d 0.7 (3), Q (8). The P (7) ad hece σ (0.x x ) (8). 4

5 Let q 0 ( t) s(0.5 t). I fg..a uder the umber we see kq t ad uder the umber 0 () 0 kq () t + f () t. The term f () t dsturbs 0 () kq t sgfcaty. I fg..b uder umbers ad respectvey we see q () t ad qm () t uder codto S () t 0. Of course wthout adaptato fg..b the dfferece ε () () t q t qm() t s sgfcat. I fg..c S () t 0. We see that q () t practcay cocdes wth q () t. Fg. shows the same resut for aother fucto q 0 () t. a m Compex Cotro Systems. auka, oscow, (Russa). Šak D., (99). Decetrazed Cotro of Compex Systems. Academc Press, Ic., ew York,. Zemyakov S.D. ad Rutkovsky V.Yu. (004). Computer Aded odeg ad Aaytca Sythess of Cotro Agorthms for a Spacecraft wth Dscretey chagg Structure. I: Proc. of the 6-th IFAC Symposum o Aut. Cotro Space. State Uversty of Aerospace Istrumetato. Sat-Petersburg. Russa. Zemyakov S.D. ad Rutkovsky V.Yu. (966). Sythess of ode Referece Adaptve Cotro System. I: Automatka ad Teemechaka. 3. P auka, oscow (Russa). Petrov B.., Rutkovsky V.Yu. ad Zemyakov S.D. (980). Adaptve Coordate-Parametrc Cotro of ostatoary Obects. auka, oscow (Russa). Krutko P.D. (99). Cotro of Robot Actuators. auka, oscow (Russa). b c Fg.. a b c Fg.. ACKOWLEDGEET The work reported ths paper s a cotrbuto to proect fuded by Russa Foudato for Basc research for whch authors are gratefu. REFERECES Voroov A.A., (985).Itroducto to Dyamcs of 5