Discrete-Time Immersion and Invariance Adaptive Control of a Slider-crank Mechanism

Transcription

1 Pepints of the 9th Wold Congess he Intenational Fedeation of Automati Contol Cape own South Afia August -9 0 Disete-ime Immesion and Invaiane Adaptive Contol of a Slide-an Mehanism Yapa Yalçın Dept of Contol Engineeing Istanbul ehnial Univesity Masla 69 Istanbul uey; yaliny@ituedut Abstat: A disete-time adaptive nonlinea ontol poedue is developed based on immesion and invaiane ontol and ug ba-stepping fo the egulation of slide-an system A onvenient paameti fomulation of the system dynamis is ablished A novel paamete imato design is pesented and the ba-stepping ontolle is onstuted onsideing the etainty equivalene piniple INRODUCION he slide-an mehanism has a vey wide usage in mahine design Some of the appliations ae found in intenal ombustion engines in eletial swith geas paaging and tetile engineeing (Lin et al 997 Ha et al 006 It is mainly used to onvet otay motion to a eipoating motion o vie vesa Seveal ontol tehniques have been pesented fo the system in the liteatue by sliding mode ontol (Fung and Chang 009 enegy ontol (Komaita and Fuuta 008 genealized minimum vaiane ontol (Saitoh and Fuuta 007 and omputed toque tehnique (Lin et al 998 Most of the mehanisms in the engineeing appliations onsist of motos geas and nonlinea omponents he ontol pefomane of these systems is usually dominated by the effets of nonlineaities and unetainties heefoe the nonlinea adaptive design poedue fo the ase study onsideed in this pape may be appliable fo a wide ange of nonlinea eleto-mehanial systems On the othe hand to develop modelling and ontol tehniques fo disete-time nonlinea systems have gained moe impotane as in the engineeing patie omputeontolled systems ae pefeed beause of the simpliity and fleibility of thei implementation Immesion and Invaiane (I&I ontol is a eently pesented new methodology in the liteatue poposed by (Astolfi et al 007 Astolfi et al 008 Kaagiannis and Astolfi 008 fo designing nonlinea and adaptive ontolles fo nonlinea systems whih elies upon the notion of system immesion and manifold invaiane In I&I ontol appoah the behaviou that satisfies the ontol objetive is fomulated as a taget system and the ontolle is onstuted suh that the plant dynamis immesed into the taget dynamis In (Yalin and Astolfi 0 0 a disete-time adaptive ontolle design via state-feedba fo the adaptive egulation of linealy paameteised disete-time systems in stit-feedba fom is pesented In this pape the adaptive egulation of the slide-an mehanism is onsideed and a disete time adaptive nonlinea ontol poedue is poposed based on immesion and invaiane ontol appoah and ug ba-stepping Fo the pupose fistly a paameti model fo the system dynamis is obtained he system dynamis ae not dietly linea in paametes but it is epessed in a fom suh that the intended ontol poedue an be developed utilizing it hen a paamete imato is designed Aftewads the ba-stepping ontolle is onstuted onsideing etainty equivalene piniple Fig Slide-an systems DERIVAION OF SYSEM DYNAMIC EQUAIONS In this setion the dynami equations fo the onsideed an-slide mehanism shown in Figue ae deived by the Eule-Lagange tehnique and these equations ae epessed in a fom in the net setion suh that an immesion and invaiane adaptive ontol algoithm an be developed and a ba-stepping ontolle an be onstuted In Figue and ae length of the ams m m and m masses of the ams and the slide espetively he slidean system is a -dof unde-atuated mehanial system he atuation is pefomed though the base joint he angula motion of base joint is tansfomed to a linea motion in the slide so that the slide moves aodingly Let s wite the ineti enegy funtion of the system K I I m m ( Copyight 0 IFAC 5

2 9th IFAC Wold Congess Cape own South Afia August -9 0 whee I and I ae the inetias of ams is the linea veloity of the seond am and is linea veloity of the mass m Besides onside the potential enegy of the system as P mg mg ( Note that the following elation is valid fo the angles ( heefoe os os os namely os ( Now onside os os and y togethe with ( then y whee i j (5 (6 y os theefoe y os (7 (8 ( os Hee note that ug the following epession fo is obtained os os and elations (- equivalently whih is utilized in the sequel os hus an be witten as ( os os os [ ( os os ] whee as in (Ha et al 006 (9 (0 Finally e os os and y 0 then ( os and when (9 is onsideed ( os os ( ( Moeove the potential enegy of the system an be ewitten as P m g m g ( m m g ( hus the Lagangian fo the system is obtained as follows os L K P I I m [ ( os os ] ( m ( os ( m m g Hene onsideing the inetias as I I m and m to aptue the vaious am-joint stutues and unetainties and Fv sgn( fv sgn( vm g the Lagange Equations of the system is obtained as given below d L L dt ( P ( os P ( [( os P ( P ( os 5 P ( ] 0 os F v (5 Hee is the toque applied to base joint Fv is the visous fition is the visous fition oeffiient ae v positive salas and P ( P ( 5 os (6 P ( os ( os os (os 6 7 v os os (os 8 9 F f with sgn( v sgn( ( os 6

3 9th IFAC Wold Congess Cape own South Afia August -9 0 m 5 5m ( m m m 8 f m g v v 9 m m 6 ( m m 7 ( m m m 0 ( m m g (7 Note that the equations (5-(7 an be witten in the fom of A( B( C ( (8 whee A( P ( os P ( B( ( os P ( P ( os 5 P ( sgn( ( os C( 0 os Finally onsideing (8 fo equations of the system an be witten as B( C( v A( A( whee A( v CONROLLER DESIGN the dynami (9 In this setion a disete-time adaptive nonlinea ontol poedue is developed he nonlinea system ( an be onsideed as a time vaying linea system If all the paametes ae nown an adaptive linea ontolle an be designed In this ase A ( B ( and C ( an be alulated at eah sample and the linea ontolle to be designed an be aodingly adapted If thee is unetainties in the paametes of the system a nonlinea adaptive ontolle is needed o pefeed to design e it may not be possible to dietly alulate the values of A B and C polynomials Hee it is assumed that and ae nown and m m m I and I (namely and ae unnowns and a disete-time adaptive nonlinea ontolle based on immesion and invaiane ontol appoah is designed heefoe let s ewite system dynamis (9 in the following fom with BC ( ( A v (0 whee ( 5 5 A( [ os ( os ( os os 0 0] and ( BC ( [0 os ( 5 ( ( os ( os ( os ( os ( Note that m and os ( os v g sgn( ( os os ( os g os ] m m m m m 5 5 ( 5 ( 6 7 ( 8 9 g 0 5 vg he disete-time dynamis of the system is equied to be able to pefom a disete-time design poedue Fowad Eule appoimation to deivatives is used to obtain disetetime dynamis of the system as given below ( ( BC v A he following notation yields ( 7

4 9th IFAC Wold Congess Cape own South Afia August -9 0 BC ( v ( and when A A A u v BC BC (5 and ae onsideed the epession fo the disetetime dynamis ae obtained in the following fom ( BC ( A u (6 In the sequel an indiet adaptive ontolle onstution is deived ug immesion and invaiane ontol appoah Fo the imation of the paametes following imation eo is onsideed z ( ( (7 A whee 0 and is suh that ( ( A( namely ( I ( ( eists Note that and fo A I z (8 ( ( A( ( ( z ( ( ( I A it is ewitten as hene paamete imations ae obtained as (9 Also note that z ( ( (0 heefoe A z z ( ( ( ( ( A ( ( A A ( A( ( ( ( ( A BC A A( ( ( ( ( ( A ( ( u BC A A( ( ( u A( ( z ( ( ( z BC u ( ( z A A( ( z ( ( ( z is obtained Let s selet the update law as ( A( ( u ( BC A A( ( ( hen the imation eo dynamis beomes A z z ( z BC ( u A( A ( ( ( z ( z Ug the following notation BC A ( ( A( ( ( u ( ( A ( ( an be ewitten as z ( I ( ( ( z (5 Let s selet and ( ( I M M K K M ( ( whee (7 M ( A (8 K M M ( K I K K (0 0 (9 p K ( M I p ( M I p (0 Note that whee ( ( R M and ( R M R I M M K K ( L I K K ( 8

5 9th IFAC Wold Congess Cape own South Afia August -9 0 hene z ( I ( R M L R R M L R z ( ( Poposition: Conside the system ( in whih and satisfy (7 he zeo equilibium of the system is asymptotially stable namely lim z ( 0 Poof: Conside the andidate Lyapunov funtion V and V z z z z whee In ( z [ I] z I ( R M L R R M L R ( (( R M L R I ( R M L R ( I ( I I onside and satisfy (7 then ( z z ( (5 R M L R Let I R M L R and ewite ( as I ( I ( I I ( whee 0 ae suh that 0 hus ( and this maes V 0 heefoe lim z ( 0 Rema: In the imation of the paametes the loo-ahead values of the state vaiable is equied hese values ae i alulated utilizing the system equations (6 with paamete values obtained in the pevious imations he ba-stepping onstution of the ontolle is pefomed in the sequel Conside system equations (6 and denote then and assume as vitual input note that selet and note that thus ( u BC A( aing the ontol signal u as u yields ( BC ( A and ( ( BC BC ( ( A A his an be ewitten as ( ( ( ( BC A BC A ( ( A A ( ( z ( BC A ( ( A A ( ( ( z ( ( BC A A A ( ( z ( z ( BC A BC A ( ( A A hene the losed loop system dynamis ae obtained as Note that when z 0 these dynamis beome BC A A BC A( A( 0 hus onsideing the Poposition it an be laimed that if the update law ( togethe with (6 is used the ontol objetive is ahieved Note that A( A( z A ( ( ( z ( ( z z 0 SIMULAION RESULS In this pape basially the egulation poblem is onsideed theefoe the simulation esults fo 0 ae pesented in the sequel In the simulations system paametes ae taen as g 98gm s 0m m m 05g m 5 g m 05g whih yields [ ] and 00 is used to obtain disete-time dynamis Estimato paametes ae seleted as and 0 he simulations ae un with the initial onditions [ 05] and (0 [ ] Figue illustates the time histoies of the paamete imation eos z [ z z z z z 5] whih onvege to zeo while Figue Figue and Figue 5 illustate the time histoy of the state angula position of fist joint angula veloity of fist joint and the edge position fo both the adaptive ontol ase and the nown paamete ontol ase (Dasheddot lines oesponds to adaptive ontol solid lines oesponds to nown paamete ontol Note that the edge position is alulated ug the elation os 9

6 9th IFAC Wold Congess Cape own South Afia August -9 0 Fig ime histoy of the paamete imation eo z Fig 5 ime histoy of the edge position Fig ime histoy of the vaiable (angula position Fig ime histoy of the vaiable (angula veloity REFERENCES Astolfi A D Kaagiannis and R Otega (008 Nonlinea and Adaptive Contol with Appliations Spinge Astolfi A D Kaagiannis and R Otega (007 Stabilization of unetain nonlinea systems via immesion and invaiane Euopean Jounal of Contol vol pp 0-0 Fung RF and CF Chang (009 Foe/Motion Slidingmode Contol Of hee ypial mehanisms Asian Jounal of Contol vol no pp 96-0 Ha J R Fung KY Chen and SC Hsien (006 Dynami modeling and identifiation of a slide-an mehanism Jounal of Sound and Vibation vol 89 pp 09 0 Kaagiannis D and A Astolfi (008 Nonlinea adaptive ontol of systems in feedba fom:an altenative to adaptive bastepping Systems and Contol Lettes vol 57 pp Kao CC CW Chuang and RF Fung (006 he self-tuning PID ontol in a slide an mehanism system by applying patile swam optimization appoah Mehatonis vol 6 no 8 pp 5 5 Komaita Y and K Fuuta (008 Enegy ontol of slide-an mehanism SICE Annual Confeene pp 99 0 Lin FJ YS Lin and SL Chiu (998 Slide-an mehanism ontol ug adaptive omputed toque tehnique Contol heoy and Appliations IEE Poeedings - vol 5 pp 6-76 Lin FJ RF Fung HH Lin and CM Hong (00 A supevisoy fuzzy neual netwo ontolle fo slide-an mehanism Mehatonis vol pp 7 50 Saitoh M and K Fuuta (007 Genealized minimum vaiane ontol of slide-an mehanism SICE Annual Confeene pp Wai RJ and FJ Lin (998 A fuzzy neual netwo ontolle with adaptive leaning ates fo nonlinea slide-an mehanism Neuoomputing vol 0 pp95 0 Yalin Y and A Astolfi (0 Immesion and invaiane adaptive ontol fo disete time systems in stit feedba fom Systems and Contol Lettes vol 6 no pp 7 Yalin Y and A Astolfi (0 Disete time immesion and invaiane adaptive ontol fo systems in stit feedba fom Poeedings of the 50th IEEE Confeene on Deision and Contol and Euopean Contol Confeene (CDC-ECC 0 Olando FL USA pp 7 0