Passivity based control of magnetic levitation systems: theory and experiments Λ

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1 Passivity based control of agnetic levitation systes: teory and experients Λ Hugo Rodriguez a, Roeo Ortega ay and Houria Siguerdidjane b alaboratoire des Signaux et Systées bservice d Autoatique Supelec Supelec 99, Gif-sur-Yvette 99, Gif-sur-Yvette FRANCE FRANCE fcortes, rortegag@lss.supelec.fr, Siguerdidjane@supelec.fr Keywords: control of agnetic levitation systes, Hailtonian systes, nonlinear control, energy saping, passivity. Abstract Te present paper is concerned wit te application of a new passivity based controller (PBC), called interconnection and daping assignent (IDA) PBC, to te proble of stabilization of agnetic levitation systes. We tae as case in point te ubiquitous two degrees of freedo levitated ball. Te IDA PBC etodology involves te solution of a linear partial dierential equation, wic ay be diicult to find in soe applications. We sow ow for agnetic levitation systes it is possible to incorporate te pysical insigt to overcoe tis proble, and actually generate a faily of soot stabilizing controllers. We present experiental results wic illustrate te perforance of te proposed scees as copared to standard (e.g., PID) linear designs. Proble forulation Te odel of te two degree of freedo levitation syste depicted in Fig. is given by _ i + RIi = ui; i =; ; Ẍ = f f g (.) Ÿ = f f were X; Y are te rotor positions in te orizontal and vertical directions, respectively, R are te coils resistances, is te ass of te rotor, g is te acceleration of gravity, and i;i i;f i;u i i = ; ; denote te total agnetic flux, te current in te coil, te force, and te control voltage associated to te i t actuator, respectively. Te following assuptions on te agnetic device are ade: (A) Te agnetic forces of te vertical and orizontal otions are decoupled (see, e.g., []). (A) Te total flux, rotor position and coil current are related as I i = (c +( )i Y ) i; i =; I i = (c +( )i X) i; i =; were c; are soe positive constants. Λ Tis wor as been partially supported by CONACyT of Mexico y Autor to wo all correspondence sould be addressed Figure : Two degree of freedo levitation syste. (A) Te forces produced by te actuators satisfy f i = i ; i =; ; (.) Te control objective is to place te rotor at a desired constant position (X Λ;Y Λ). In te next sections we design an IDA PBC tat acieves tis objective. A brief description of te IDA PBC etodology is given in Appendix A. Controller design for te vertical dynaics Te vertical dynaics can be described by a port controlled Hailtonian (PCH) odel as R» _y = R + u u z } z } J R g were we ave defined te state vector y = te systes total energy is (.) i >, ; ;Y; Y _ and H = (c y ) y + (c + y ) y + gy + y Let us rear tat, for a given constant desired position Y Λ = y Λ, te equilibriu points of (.) can be paraeterized in ters of te total flux of one actuator, say y Λ,as y Λ =»qg + y Λ ;yλ;yλ; > (.)

2 Proposition Consider te vertical dynaics odel (.) of te levitation syste (.) in closed loop wit te nonlinear static state feedbac controller u = R R (c y)y (y yλ) ( R + Ra) Λ ~y +~y + Ray y u = R (c + y)y + R fi (y y Λ) fir a ~y +~y + Ray Λ fi y (.) were ~y i = yi y iλ, i =; ;, and te controller paraeters are cosen as ; ;R a >, fi<. Ten, te equilibriu y Λ (.) is asyptotically stable. Proof First we prove tat te original interconnection atrix is not adequate to stabilize (.) via IDA PBC. (Tis serves as a justification for te assignent of a new interconnection atrix, wic is done in te sequel.) As discussed in Appendix A te controller design procedure requires te solution of te linear PDE (A.). If we do not odify te interconnection and daping structures, tat is, if we tae J a = Ra =, tis equation reduces to (J R) a =gu wic, upon replaceent of (.), yields R a =u ; R a =u ; a = a = Tis eans tat te function H a can only depend on y and y, tat is, only te electrical coordinates can be saped. Tus, te resulting Lyapunov function would be of te for H d = (c y ) y+ (c + y ) y+gy + y +H a(y;y ) Even toug, wit a suitable selection of H a(y;y ), H d could ave an extreu at y Λ, te H d = (c y ) H d H a (c+y ) H a y y y is sign indefinite for all H a(y;y ). Consequently tis extreu is a saddle point and te equilibriu will not be stable. A careful observation of te Hessian reveals tat te proble is te lac of an eective coupling between te electrical and te ecanical subsystes. Indeed, te interconnection atrix J only couples position wit velocity. To overcoe tis proble we propose to enforce a coupling between te fluxes y, y and te velocity y, tus, we propose to add an interconnection atrix J a = fi fi were ; fi are paraeters to be defined. Besides te added interconnection we will find convenient fro considerations pertaining to te proof of asyptotic stability to odify te daping atrix, and we propose R a = diagf; ;R a; g wit R a >. Wit tese coices of J a; R a te PDE (A.) becoes R a a = +u R a fi a = +u R a a + a =R a a +fi a a = fi (.) Te last two equations define a linear PDE (for H a) tat can be easily solved using sybolic prograing. Once H a is obtained te first and second equations define te control signals. A solution of te PDE is te following H a = (c y)y (c + y)y + y fi y +R agy +Φ(z ;z ) were Φ(z ;z ) is an arbitrary continuously dierentiable function of z = fi y y; z = y + y + Ray As explained in Appendix A tis function as to be cosen to satisfy te equilibriu assignent and Lyapunov stability conditions. Te new Lyapunov function is given by H d = y fi y +R agy +gy + y +Φ(z ;z ) tis function will ave an extreu at y Λ if d (yλ) = y Λ fi y Λ g R ag A suitable coice of Φ(z ;z ) is given by wit Φ(z ;z )= fi x Λ ~z g~z + ~z fi ~z = ~y ~y (.) ~z = ~y +~y + Ray We write for future reference te gradient of H d d = (y y Λ g) + ~z fi (y y Λ) ~z R a ~z + y (.) Now, to assure tat y Λ is a iniu of H d we loo at its Hessian (evaluated at y H d (yλ) = + ) R a y Λ fi Ra R a ( y Λ R a + R a Tis atrix will be positive definite if ; ;R a > and fi <, fro wic we conclude tat H d is a bona fide Lyapunov function.

3 It is easy to verify tat te controller (.) is obtained fro te first two equations of (.). Evaluating _ H d along te closed loop dynaics (.), (.), wic for future reference we write explicitly as we get _y = R d d _y = R d fi d _y = R a d + d _y = d +fi d d (.) d d d _H d = R R R a (.) We ave tus establised tat, under te conditions of Proposition, y Λ is a stable equilibriu point. Invoing La Salle s invariance principle we will now prove tat te stability is actually asyptotic. Fro (.), and setting te rigt and side of (.) equal to zero, we obtain te residual dynaics _y = d ; _y = fi d ; _y = d ; _y = (.9) fro wic we conclude tat, in te residual set, y = μy a constant. Now, fro te tird ter of (.) we ave d =) ~z =, and fro te fourt one we obtain ~z =) d = μy. Now, to sow tat te syste is in an equilibriu we ave to prove tat μy =. To tis end, we calculate and dierentiating again d d = y y μy _y = dt d dt d = _y μy = μy = Tis proves tat μy =, and consequently d =. We ten ave tat te full gradient d, tus te syste is in an equilibriu μy. To prove tat μy = y Λ, fro (.) we ave te following two iplications: d = ) y = y Λ, and d = ) y = y Λ. Tis, togeter wit ~z =, wic iplies μy = y Λ, copletes te proof. ΛΛΛ Rear It is interesting to analyze te ters tat appear in te control law (.). Te first rigt and ters are siply RI i;i =;, ence tey are feedbac linearization ters tat cancel te voltage drop in te coils, see (.). Te second rigt and ters are proportional to force errors (see (.)). Te reaining linear ters are proportional to te errors in all coordinates. Controller design for te orizontal dynaics In view of te decoupling assuption A of Section te control of te orizontal coordinate can be treated independently of te vertical position, and odulo te dierence in te gravity eect te controller design follows exactly te procedure described in te previous section. Terefore, witout any additional detail, we only present te odel and te controller. Te orizontal dynaics can also be described by a PCH odel as _x = R R J R i (.) ; ;X; X _, wit te Hailto- were we ave defined x = (x) + g» u u H(x) = (c x ) x + (c + x ) x + x Notice te absence of te gravity induced ter. Te equilibria (denoted by x Λ ) can again be paraetrized in ters of te total flux of one actuator as x Λ =[xλ;xλ;x Λ; ] > (.) We ave te following proposition wose proof is straigtforward fro te proof of Proposition, wit te sae desired interconnection and daping atrices. Proposition Consider te orizontal dynaics odel (.) of te levitation syste (.) in closed loop wit te static state feedbac controller u (x) =R (c x ) x R ( x x Λ ) ( R + Ra) ~x + fi ~x +~x + Rax i x u (x) =R (c+x ) x R fi ( x + x Λ ) ( R fi + fira) ~x + fi ~x +~x + Rax i fi x (.) were ~x i = xi x iλ, i =; ;, and te controller paraeters are cosen as ; ;R a >, fi<. Ten, te equilibriu x Λ (.) is asyptotically stable. Experiental results We ave carried out experiental studies for te sipler levitation syste sown in Fig., wose PCH odel is given by _x = J R R wit te Hailtonian function defined as C (x) + H(x) = (c x)x + x + gx g u (.) and equilibriu points, for given a desired rotor position x Λ,given by x Λ = p g; xλ; Λ >. To evaluate te eect of te design paraeters we ave tried tree dierent IDA PBCs:

4 Figure : Magnetic suspension. l Modified interconnection: We leave R untouced and odify only te interconnection as J a = wit a tuning paraeter. Te IDA PBC procedure leads to u(x) = R (c x)x R (x g) K p( ~x+~x) Kvx (.) were K p = R ;Kv =, and stability is ensured for ; > l Modified interconnection and daping: In tis case we coose J a as above and also select R a = R R a Notice tat we ave reoved te original electrical resistance. Tis as been otivated by te fact tat, in tis way, we eliinate te quadratic nonlinear ters in x of (.). Indeed, te IDA PBC is now given as u(x) = R (c x)x Kp( ~x +~x) Kvx (.) Ra wit K p = ;Kv =, and te tuning paraeters range in ; ;R a >. l Outer loop integration: In order to reduce te steady state error we observed in our experients tat it is necessary to add an integral ter to tese controllers. As discussed in [], tis well nown odification can also be accoodated by our teory. Indeed, we now tat te closed loop dynaics of (.) wit eiter of te nonlinear controllers (.) or (.) is again a PCH syste of te for (A.), wose passive output is Te experiental rig is coposed by a agnetic suspension, consisting of a rotor and an actuator, connected to a personal coputer wit te interface DT. Te rotor axis is in line wit te acceleration due to te gravity and is subject to te force exerted by te actuator tus causing vertical displaceent. Te actuator, consists of a coil and a voltage controlled current source. Te rotor position is easured wit a dierential transforer. It as an infinite resolution, oreover it is robust wit respect to electrical disturbances. Concerning te coputing unit, te interface DT, contains a bits A/D converter (wit ultiplexed inputs) and two bits D/A outputs. Te sapling period can be adjusted between μs and s. Te paraeters of te rotor and actuator are: = :Kg; = : N=A; R = :Ω; c = :; g =9:=s. To ipleent te nonlinear controllers we ave used a linear Luenberger observer to estiate te agnetic flux. We suppose tat te dynaics of te observer are fast wit respect to te suspension dynaics. Using a siple linearized odel analysis, and invoing te tie scale separation assuption, we deterined te observer gains as L = [; 9; ] >. Te initial condition for te observer was cosen equal to [x Λ; ; ] >. After soe relatively straigtforward coissioning stage we fixed te gains of te IDA PBC (.) to K p = :; = :;Ki =, and K p = :;=:;Ki =for (.). We also tested a classical linear PID controller wit te best set of gains being K = ;T d = :;T i = :. It sould be pointed out tat te linear controller was ipleented by analogue circuits, wile te IDA PBCs were approxiated nuerically. Tis explains te relatively iger level of noise on te responses of te IDA PBCs presented below. First, we carried out a sall step cange in te rotor position fro to. In Figs., and we sow te reference step and te actual position as well as te control signal for te PID, and te IDA PBCs (wit integral action) (.) and (.), respectively. In tis sall range of operation te PID beaves satisfactorily, and responds faster tan te IDA PBCs. However, notice te large oversoot, bot in te regulated output and in te control action, wic actually its te saturation level. On te oter and, te transient of te IDA PBC is sooter wit no saturation proble. Wit respect to te steady state error, we can see tat te PID and te controller (.) ave an acceptable perforance.. x.. y d = g > (x) d (x) Hence, we add an integral ter to te control as. Displaceent x and x Λ. _ = y d v = u(x) K i were K i >. Tis leads to te new augented PCH syste»» _x Jd R d g d = _ g > (x; ) were H d = H d (x) + K i. It is easy to prove tat asyptotic stability is preserved. Figure : Syste response wit te PID controller. Te second test was a larger step (fro : to :), were te linearized odel is not valid anyore. Te transient beaviours for te tree controllers is depicted in Figs., and. We can see fro Fig. tat te PID as again te input saturation proble, furterore, te rotor as actually touced te lower pysical liit. Te IDA PBCs, on te contrary, still beave sootly witout retuning and aintain te rotor witin te pysical bounds wit-

5 . x x Displaceent x and x Λ. Displaceent x and x Λ. Figure : (.). Syste response wit te nonlinear controller Figure : (.). Syste response wit te nonlinear controller. x References... Displaceent x and x Λ. Figure : (.). Syste response wit te nonlinear controller out saturating te actuator. As expected, in bot cases te controller (.) tat reoves te undesirable nonlinear ter as te best perforance. It is wort pointing out tat tis eect was obtained by applying a far fro obvious positive daping injection, wic is naturally suggested witin te fraewor of our IDA PBC teory x.... x Displaceent x and x Λ. Figure : Syste response wit te PID controller. A [] M. de Queiroz and D. Dawson. Nonlinear Control of Active Magnetic Bearings: A Bacstepping Approac, Tran. Control Systes Tecnology, Vol., No.,, Septeber 99. [] R. Ortega, A. van der Scaft, B. Masce and G. Escobar, Stabilization of port controlled Hailtonian systes: Energy balancing and passivation, Autoatica, (to appear). See also: IEEE Conf. Dec. and Control, Poenix, AZ, USA, Dec. 999, pp.. [] A. J., van der Scaft, L Gain and Passivity Tecniques in Nonlinear Control, Springer Verlag, Berlin, 999. IDA PBC design etodology We briefly review ere te IDA PBC design etodology wic is used to derive te controller, te interested reader is referred to [] (see also []) for furter details. Te procedure taes o fro a port controlled Hailtonian description of te syste of te for ρ _x = [J(x) R(x)] ± (x) +g(x)u w = g > (x) (x) were x R n is te state vector, u R and w R are te power variables, wic are conjugated in te sense tat teir product as units of power, for instance currents and voltages in electrical circuits, or forces and velocities in ecanical systes. H(x) is te energy function (wic by definition is bounded fro below), te sew syetric atrix J(x) = J > (x) captures te interconnection structure, and R(x) =R > (x) is te dissipation atrix due to resistances and frictions. Te control objective is to stabilize via a state feedbac u = fi(x) an equilibriu x Λ. Te controller design proceeds in tree steps: ) (Interconnection and daping assignent) Fro pysical considerations we fix te desired interconnection and daping atrices as. Displaceent x and x Λ. Figure : (.). Syste response wit te nonlinear controller J d (x) = J(x) +J a(x) = J > d (x) R d (x) = (A.) R(x) +R a(x) =R > d (x) ) (Integrability conditions) We copute a faily of energy functions and corresponding controls fi(x) tat can be assigned preserving te PCH structure. Tat is, we want our closed loop dynaics to be of te for _x =[J d (x) R d (x)] (x) (A.)

6 wit te closed loop energy function H d (x) = H(x) +H a(x) (A.) It is easy to see tat tis is acieved if and only if te PDE [J d (x) R d (x)] a (x) = (x)+g(x)fi(x) (A.) is solvable for H a(x). A necessary and suicient condition for its solution is» > a (x) (x) wic given J a(x); Ra(x) directly defines a PDE for fi(x). ) (Energy Saping) Fro (A.) we ave» > d _H d (x) (x) R d (x) (x)» (A.) Hence, H d (x) is a Lyapunov function if it as a iniu at te desired equilibriu x Λ. Furterore, to ensure asyptotic stability, te following iplication sould be also true _H d (x) =) x = x Λ (A.) In te tird step of te procedure we coose, aong te faily of controls obtained in step ) above, one tat ensures asyptotic stability.