CONTROL BY INTERCONNECTION OF THE TIMOSHENKO BEAM. DEIS, University of Bologna, viale Risorgimento 2, Bologna (Italy)

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1 CONTROL BY INTERCONNECTION OF THE TIMOSHENKO BEAM Alessandro Macchelli Claudio Melchiorri EIS, Universiy of Bologna, viale Risorgimeno 2, 4036 Bologna Ialy {amacchelli, Asrac: In his paper, he dynamical conrol of a mixed finie and infinie dimensional mechanical sysem is approached wihin he framework of por Hamilonian sysems. As an applicaive example of he presened mehodology, a flexile eam, modeled according o he Timoshenko heory, wih a mass under graviy field conneced o a free end, is considered. Afer he disriued por Hamilonian dph model of he eam is inroduced, he conrol prolem is discussed. The concep of srucural invarian Casimir funcion is generalized o he infinie dimensional case and he so-called conrol y inerconnecion conrol echnique is exended o he infinie dimensional case. In his way, finie dimensional passive conrollers can sailize disriued parameer sysems y shaping heir oal energy, i.e. y assigning a new minimum in he desired equilirium configuraion ha can e reached if a dissipaion effec is inroduced. Keywords: disriued por Hamilonian sysems, energy shaping, Timoshenko eam, conrol. INTROUCTION Flexile eams are generally modeled according o he classical Euler-Bernoulli heory: his formulaion provides a good descripion of he dynamical ehavior of he sysem if he eam s cross secional dimension are small in comparison of is lengh. In his case, he effecs of he roary ineria of he eam are no considered. A more accurae eam model is provided y he Timoshenko heory, according o which he roary ineria and also he deformaion due o shear are considered. The resuling Timoshenko model of he eam is generally more accurae in predicing he eam s response han he Euler-Bernoulli one, u, on he oher hand, i is more difficul o uilize for conrol purposes ecause of is complexiy. The dph van der Schaf and Maschke, 2002 formulaion of he Timoshenko eam Macchelli and Melchiorri, 2003 provides a eer undersanding of he model from a physical poin of view, oviously wihou reducing he complexiy of he model iself. I makes clear how he kineic and he poenial elasic energy domains inerac and how he sysem can exchange power wih he environmen hrough is order and/or a disriued por. Moreover, from he conrol poin of view, he dph represenaion of he sysem allows o easily exend well-esalished passive conrol sraegies, ha were originally developed for finie dimensional por Hamilonian sysems, o deal wih disriued sysems. This paper is organized as follows: in Sec. 2, he dph model of he Timoshenko eam is presened y inroducing he underlying irac srucure. Then, he mixed finie and infinie dimensional por Hamilonian model of he plan is presened in Sec. 3 and, in Sec. 4, some consideraions on he srucural invarians Casimir funcions of he closed-loop sysem are developed. This is he

2 saring poin in he definiion of he conroller, ha is widely discussed in Sec. 5. Finally, some concluding remarks and some suggesions for fuure work are illusraed in Sec PH MOEL OF THE TIMOSHENKO BEAM 2. Background. The classical formulaion According o he Timoshenko heory, he moion of a eam can e descried y he following sysem of PE: ρ 2 w 2 I ρ 2 φ 2 EI 2 φ x 2 K K 2 w x 2 K φ x = 0 φ w = 0 x where is he ime and x 0, L] is he spaial coordinae along he eam in is equilirium posiion. Then, wx, is he deflecion of he eam from he equilirium configuraion, denoed y w 0, and φx, is he roaion of he eam s cross secion due o ending. Assume ha he moion akes place in he wx-plane. The coefficiens ρ, I ρ, E and I are he mass per uni lengh, he mass momen of ineria of he cross secion, Young s modulus and he momen of ineria of he cross secion, respecively. The parameer K is equal o kga, where G is he modulus of elasiciy in shear, A is he cross secional area and k is a consan depending on he shape of he cross secion. The oal energy is given y he following relaion kineic energy {}}{ L 2 ] 2 w φ H := ρ I ρ dx 2 0 L K φ w 2 ] 2 φ EI dx 2 0 x x }{{} poenial elasic energy 2 ha poins ou ha, as every infinie dimensional sysem, he eam is characerized y a spaial domain := 0, L], wih order = {0, L}, and y he presence of wo ineracive energy domains, he kineic and he poenial elasic. 2.2 Timoshenko eam in dph form The saring poin in he definiion of a por Hamilonian sysem oh finie and infinie dimensional is he idenificaion of a suiale space of power or energy variales, sricly relaed o he geomery of he sysem, and he definiion of a irac srucure on his space of power variales, in order o descrie he inernal and exernal inerconnecion of he sysem. The poenial elasic energy in 2 is a funcion of he shear and of he ending, given y he following -forms: ɛ, x = ] w, x φ, x dx x ɛ r, x = φ, xdx x 3 The associaed co-energy variales are he shear force and he ending momenum, given y he 0-forms funcions σ, x = K ɛ, x and σ r, x = EI ɛ r, x, where is he Hodge sar operaor, Araham and Marsden, 978. Besides, he kineic energy is funcion of he ranslaional and roaional momena, i.e. of he following -form: p, x = ρ w, xdx φ p r, x = I ρ, xdx 4 and he associaed co-energy variales are he ranslaional and roaional momena, given y he 0-forms funcions v, x = ρ p, x and v r, x = I ρ p r, x. If N is an n-dimensional Riemannian manifold, he space of k-forms on N, i.e. he space of k- linear alernaing funcions, is given y Ω k N. So, we have ha p, p r, ɛ, ɛ r Ω and ha w, φ Ω 0. The oal energy 2 ecomes he following quadraic funcional: Hp, p r, ɛ, ɛ r = Hp, p r, ɛ, ɛ r 5 wih H : Ω Ω Ω, Hp, p r, ɛ, ɛ r = 2 ρ p p I ρ p r p r K ɛ ɛ EI ɛ r ɛ r he energy densiy. Consider a ime funcion p, p r, ɛ, ɛ r Ω Ω wih R, and evaluae he energy H along his rajecory. A any ime, he variaion of inernal energy, ha is he power exchanged wih he environmen, is given y dh d = δ p H p δ p r H p r δ ɛ H ɛ δ ɛ r H ɛ r 6 The differenial forms p, pr, ɛ and ɛr are he ime derivaives of he energy variales p, p r, ɛ and ɛ r, and represen he generalized velociies flows, while δ p H, δ pr H, δ ɛ H, δ ɛr H are he variaional derivaive of he oal energy H : Ω Ω R. They are relaed o he

3 rae of change of he sored energy and represen he generalized forces effors. The dph formulaion of he Timoshenko eam can e oained or y expressing in erms of p, p r, ɛ and ɛ r 3 and 4, or, in a more rigorous way, y revealing he underlying irac srucure of he model. Firs of all, i is necessary o define he space of power variales. The space of flows is given y F := Ω Ω Ω Ω Ω 0 Ω 0, and he space of effor E, he dual of F, is given y E := Ω 0 Ω 0 Ω 0 Ω 0 Ω 0 Ω 0, since he concep of dualiy over he space of forms can e given y he following: Proposiion. Assume ha N is an n-dimensional manifold. Then, he dual space Ω k N of Ω k N can e idenified wih Ω n k N and he dualiy produc eween Ω k N and Ω k N y β, α := β α 7 wih α Ω k N and β Ω n k N. The same resul holds for Ω k N. Proof. See van der Schaf and Maschke, Given he dualiy produc defined in 7, i is possile o inroduce he pairing operaor on he space of power variales y means of he following definiion, ha can e easily specialized for he Timoshenko eam case. efiniion 2. Suppose ha α i, β i Ω k N Ω n k N, i =, 2. Then α, β, α 2, β 2 := β 2, α β, α 2 = β 2 α β α 2 N Since he space of power variales and he pairing operaor are specified, i is possile o give he following: efiniion 3. Suppose ha F and E are linear spaces wih a dual pairing,. A irac srucure is a linear suspace F E such ha =, wih denoing he orhogonal complemen wih respec o he pairing,. The previous definiion is quie general and can e reformulaed as follows. Given f i, e i F E, i =, 2, hen f, e, f 2, e 2 = 0. I is immediae ha, if f, e, hen N 0 = f, e, f, e = 2 e, f 8 In oher words, if f, e F E is a couple of power conjugaed variales, he fac ha hey elong o he irac srucure implies power conservaion, i.e. he dual produc is equal o 0. The irac srucure is he geomerical ool y means of which i is possile o deal wih power conserving inerconnecion in physical sysems. Once a proper inerconnecion srucure is defined, he por Hamilonian model of a physical sysem, oh finie and infinie dimensional, follows auomaically. Wih he following proposiion, he main resul of his secion is presened. Proposiion 4. Consider he space of power variales F E and he ilinear form pairing operaor, inroduced y ef. 2; hen, define he following linear suspace of F E: = { f p,..., f r, e p,..., e r F E f p 0 0 d 0 e p f pr f ɛ = 0 0 d e pr d 0 0 e ɛ, f ɛr 0 d 0 0 e ɛr f f r e e r = e p e pr e ɛ e ɛr } 9 where denoes he resricion on he order of he spaial domain. Then =, or, equivalenly, is a irac srucure. Proof. As in van der Schaf and Maschke, 2002, he proof can e divided in wo seps. In he firs one, i is verified ha, while, in. Suppose ha ω i, ei p,..., e r,i F E, wih i =, 2; hen, if ω, ω 2, i happens ha ω, ω 2 = 0. From he definiion 9 of he irac srucure, we have he second one, ha = fp i,..., f r,i ω, ω 2 = de 2 p e ɛ e 2 p de ɛ de p e 2 ɛ e p de 2 ɛ de 2 p r e ɛ r e 2 p r de ɛ r de p r e 2 ɛ r e p r de 2 ɛ r f, e,2 f,2 e, f r, e r,2 f r,2 e r, Since de e 2 = de e 2 e de 2, and, for he Sokes heorem, if α Ω 0, hen dα = α, we deduce ha ω, ω 2 = 0 and. In order o prove ha, consider ω 2. From he definiion of irac srucure we have ha ω, ω, ω 2 = 0. Since ω, from 9 we have:

4 0 = ω, ω 2 = de ɛ e 2 p fp 2 e p e ɛ de ɛ r e 2 p r fp 2 r e ] p r de p e p r e 2 ɛ fɛ 2 e ] ɛ de p r e 2 ɛ r fɛ 2 r e ɛ r e p e,2 f,2 e ɛ e p r e r,2 f r,2 e ɛ r From he Sokes Theorem and he properies of he exerior derivaive, is is possile o oain ha 0 = e ɛ de 2 p e 2 p r fɛ 2 ] e p fp 2 de 2 ɛ e ɛ r de 2 p r fɛ 2 r ] e pr fp 2 r e 2 ɛ de 2 ɛ r ] ] e ɛ e 2 p f,2 ] e ɛ r e 2 p r f r,2 ] e p e 2 ɛ e,2 ] e p r e 2 ɛ r e r,2 for every ω. Since he exerior produc of forms is non degeneraive, we deduce ha he previous relaion holds if and only if ω 2. So, and his complees he proof. Consider he oal energy 5 as he Hamilonian of he sysem, i.e. a quadraic funcional of he energy variales p, p r, ɛ and ɛ r ounded from elow. The rae of change of hese energy variale generalized velociies can e conneced o he irac srucure 9 y seing f p = p, f ɛ = ɛ, f p r = pr and f ɛr = ɛr, where he minus sign is necessary in order o have a consisen energy flow descripion. Moreover, he rae of change of he Hamilonian wih respec o he energy variales, ha is is variaional derivaives, can e relaed o he irac srucure y seing e p = δ p H, e ɛ = δ ɛ H, e pr = δ pr H and e ɛr = δ ɛr H. Then, he disriued Hamilonian formulaion wih oundary energy flow of he Timoshenko eam can e given y he following: efiniion 5. The dph model of he Timoshenko eam wih irac srucure 9 and Hamilonian H 5 is given y p 0 0 d 0 δ p H p r ɛ = 0 0 d δ pr H d 0 0 δ ɛ H ɛ r 0 d 0 0 δ ɛr H f δ p H f r e = δ pr H δ ɛ H e r δ ɛr H 0 Noe ha 0 coincides, afer some manipulaions, wih. The power conserving propery 8 implies ha for every f p,..., e r e p f p e pr f pr e ɛ f ɛ e ɛr f ɛr e f e r f r = 0 Then, from 6, he following proposiion can e easily proved Proposiion 6. Consider he dph model of he Timoshenko eam 0. Then dh d = e f e r f r = e, xf, x e r, xf r, x ] x=l x=0 or, in oher words, he increase of energy kineic/poenial energy of he eam is equal o he power supplied hrough he order. I is imporan o noe ha he Timoshenko eam has een already reaed wihin he framework of dph sysems also in Golo e al., This approach differs from he one presened in his paper in he mahemaical descripion of he space of energy/power variales L 2 funcions and differenial forms and in he definiion of dualiy over he space of power variales, Golo, Moreover, in he proposed approach, he Sokes heorem is he key poin in undersanding he relaion eween power variales on he order and evoluion of he energy variales inside he domain. 3. MOEL OF THE PLANT Consider he mechanical sysem of Fig., in which a flexile eam is conneced o a rigid ody wih mass m and ineria momenum J in x = L and o a conroller in x = 0. The conroller acs on he sysem wih a force f c and a orque τ c. Since he Timoshenko model of he eam is valid only for small deformaions, i is possile o assume ha he moion of he mass is he cominaion of a roaional and of a ranslaional moion along x = L. The por Hamilonian model of he mass

5 PSfrag replacemens { f L f r f L ] T = e f c τ c 0 L q 2 m, J Fig.. Flexile link wih mass in x = L. Plan overview. is given y: q = p H ṗ = q H p H f e = p H q 2 where q = q, q 2 ] T Q are he generalized coordinaes, wih q he disance from he equilirium configuraion and q 2 he roaion angle, p T Q are he generalized momena and Hp, q := 2 pt M p mgq = p 2 2 m p2 2 J mgq is he oal energy Hamilonian, wih g he graviy acceleraion. Moreover, he por variales are given y f, e R 2. As regard he conroller, we assume ha i can e modeled y means of he following finie dimensional por Hamilonian sysems q c = pc H c ṗ c = qc H c c pc H c G c f c 3 e c = G T c pc H c where q c Q c are he generalized coordinaes, wih dimq c = 2, p c T Q c are he generalized momena and f c, e c R 2 are he power conjugaed por variales. Moreover, H c q c, p c is he Hamilonian and i will e specified in he remaining par of his secion in order o drive he whole sysem in a desired equilirium configuraion The por causaliy of oh he mass and he conroller is assumed o e wih flows as inpu and effors as oupus. As poined ou in Sramigioli, 200, i is possile o inerconnec wo por Hamilonian sysems only if a por dualizaion is applied on one of he sysem. In his way, a sysem can have an effor as inpu and a flow as oupu. The ond graph represenaion of he closed loop sysem made of he Timoshenko eam, he mass in x = L and he finie dimensional rag replacemens por Hamilonian conroller acing in x = 0 is given in Fig. 2, where SGY is he symplecic gyraor ha implemens he por dualizaion. Then, he inerconnecions consrains eween H c SGY H SGY Fig. 2. Bond graph represenaion of he closedloop sysem. he por variales of he susysems are given y he following power-preserving relaions: H e L e r L ] T = f { f 0 ff r 0 ] 4 T = ec e 0 e r 0 ] T = fc From 0, 2, 3 and 4 i is possile o oain he mixed finie and infinie dimensional por Hamilonian represenaion of he closedloop sysem. The oal energy H cl is defined in he exended space χ := q, p, q c, p c, p, p r, ɛ r, ɛ ] T and i is given y he sum of he energy funcions of he susysems, ha is H cl := H H c H 5 Moreover, i is easy o verify ha he energy rae is equal o dh cl T H cl = d p H cl p T H cl H cl c p c p c where c and H c have o e designed in order o drive he sysem o he desired equilirium posiion, in general differen from he rivial one, for which only some damping injecion, i.e. c > 0, is enough. The asic idea is o shape he oal energy H cl y properly choosing he conroller Hamilonian H c in order o have a new minimum of energy in he desired configuraion ha can e reached if a dissipaive effec is inroduced. 4. STRUCTURAL INVARIANTS IN THE CLOSE-LOOP SYSTEM In general, he energy variales of he conroller are no relaed o he energy variales of he plan and, as a consequence, i is no clear how he oal energy can e shaped and he desired minimum assigned. The firs sep is o relae he energy variales of he conroller o he energy variales of he plan independenly from he Hamilonian of he conroller and, if i is possile, from he Hamilonian of he plan. These invarians are called, in he finie dimensional framework, Casimir funcions, van der Schaf, 999; Orega e al., 999. The Casimir funcions are srucural invarians in he sense ha hey are invarian independenly from he Hamilonian of he sysem. In he case of mixed finie and infinie dimensional sysem, i is possile o give he following: efiniion 7. Consider a mixed finie and infinie dimensional por Hamilonian sysem wih sae space given y X X, where X is he sae space of he finie dimensional par and X he sae space of he infinie dimensional one, and Hamilonian H : X X R. A funcional C : X X R is a Casimir funcional for he sysem if and only if dc d = 0, H Hamilonian of he sysem

6 ha is C is consan independenly of he Hamilonian of he sysem. For he sysem of Fig., we have ha X = Q T Q Q c T Q c and X = Ω Ω Ω Ω, wih Hamilonian given y 5. Given χ := q, p, q c, p c, p, p r, ɛ r, ɛ ] T X X and a funcional C : X X R, we have ha dc C d = T q q T C p ṗ T C q c T C ṗ c q c p c δ p C p δ p r C p r δ ɛ C ɛ δ ɛ r C ɛ r ha has o e equal o zero for every Hamilonian H, H c and H. I is possile o prove, ha his is rue if and only if dδ ɛ C = 0 dδ p C δ pr C = 0 dδ ɛr C δ ɛ C = 0 dδ pr C = 0 C p = C ] ] δp C = L δp C = 0 = 0 p c δ pr C L δ pr C 0 ] ] C q = δɛ C L C δɛ C = G 0 δ ɛr C c L q c δ ɛr C 0 In oher words, he following proposiion holds. 6 Proposiion 8. Consider he mixed finie and infinie dimensional por Hamilonian sysem of Fig. 2, ha is he resul of he power conserving inerconnecion 4 of he susysems 0, 2 and 3. If X X is he exended sae space of he sysem, hen a funcional C : X X R is a Casimir for he closed-loop sysem if and only if relaions 6 hold. In order o conrol he flexile eam wih he finie dimensional conroller 3, he firs sep is o find Casimir funcionals for he closed-loop sysem ha can relae he sae variales of he conroller q c o he sae variales ha descrie he configuraion of he flexile eam and of he mass conneced o is exremiy. In paricular, we are looking for some funcionals C i, i =, 2, such ha C i q, p, q c, p c, p, p r, ɛ, ɛ r := q c,i C i q, p, p c, p, p r, ɛ, ɛ r, wih i =, 2, are Casimir funcionals for he closed loop sysem, ha is hey saisfies he condiions of Prop. 8. Firs of all, from 6, i is immediae o noe ha every Casimir funcional canno depend on p and p c. Moreover, since i is necessary ha dδ ɛ C i = 0 and dδ pr C i = 0, we deduce ha δ ɛ C i and δ pr C i have o e consan as funcion on x on and heir value will e deermined y he oundary condiions on C i. Since, from 6, δ pr C i = 0, we deduce ha δ pr C i = 0 on. Bu dδ p C i = δ pr C i = 0, hen, from he oundary condiions, we deduce ha also δ p C i = 0 on. As a consequence, all he admissile Casimir funcionals are also independen from p and p r, ha is C i q, q c, ɛ, ɛ r := q c,i C i q, ɛ, ɛ r, wih i =, 2. Assuming G c = I, we have ha ] ] C δɛ C = = x=0 q c 0 δ ɛr C x=0 7 and, consequenly, δ ɛ C = on. From 6, we have ha dδ ɛr C = δ ɛ C = = dx; hen, δ ɛr C = x c, where c is deermined y he oundary condiions. Since, from 7, δ ɛr C x=0 = 0, hen c = 0; moreover, we deduce ha δ ɛr C x=l = L, relaion ha inroduces a new oundary condiion in x = L. A consequence is ha ] ] C q = δɛ C x=l = δ ɛr C x=l L The firs conclusion is ha C q,..., ɛ r = q c, Lq 2 q xɛ r ɛ 8 is a Casimir for he closed loop sysem. Following he same procedure, i is possile o calculae C 2. From 6, we have ha C 2 q c = 0 ] ] δɛ C = 2 x=0 δ ɛr C 2 x=0 9 and hen δ ɛ C 2 = 0 on ; moreover, dδ ɛr C 2 = 0 and, consequenly, δ ɛr C 2 = on since 9 holds. Again from 6, we deduce ha ] ] δɛ C 2 x=l C 2 q = δ ɛr C 2 x=l So we can sae ha C 2 q, q c, ɛ, ɛ r = q c,2 q 2 = 0 ɛ r 20 is anoher Casimir funcional for he closed loop sysem. In conclusion, he following proposiion has een proved. Proposiion 9. Consider he mixed finie and infinie dimensional por Hamilonian sysem of Fig. 2, ha is he resul of he power conserving inerconnecion 4 of he susysems 0, 2 and 3. Then 8 and 20 are Casimir funcionals for his sysem. Since C i, i =, 2, are Casimir funcionals, hey are invarian for he sysem of Fig. 2. Then, for every energy funcion H c of he conroller, we have ha q c, = Lq 2 q xɛ r ɛ C 2 q c,2 = q 2 ɛ r C 2 where C and C 2 depend on he iniial condiions. If he iniial configuraion of he sysem is known,

7 hen i is possile o assume hese consans equal o zero. Since H c is an arirary funcion of q c, i is possile o shape he oal energy funcion of he closed-loop sysem in order o have a minimum of energy in a desired configuraion: if some dissipaion effec is presen, he new equilirium configuraion will e reached. 5. CONTROL BY ENERGY SHAPING In his secion, he sailizaion of he mechanical sysem of Fig. y energy shaping is presened. The desired equilirium configuraion expressed in erms of energy variales can e calculaed as he soluion of 0 wih f,r p = p r = ɛ = ɛ r = 0 0,L = 0, e r 0,L = e 0 = 0, e L = mg ha leads o he following equilirium soluion: p, x = 0 p r, x = 0 ɛ, x = mg ɛ r, x = mg K EI x L 22 If α and q are he desired roaional angle and posiion of he mass, ha is φ, L = α and w, L = q, he desired configuraion 22 of he eam can e easily expressed in erms of is verical displacemen and cross secion roaion: w, x = mg x L3 6EI α mg x L q 23 K φ, x = mg 2EI x L2 α The energy funcion H c of he conroller 3 will e developed in order o regulae he closed-loop sysem in he configuraion χ = q, p, p c, p, p r, ɛ, ɛ r In he remaining par of his secion i will proved ha, y choosing he conroller energy as H c q c,p c := 2 pt c Mc p c 2 K c,q c, q c, 2 2 K c,2q c,2 q c,2 2 Ψ q c, q c, Ψ 2 q c,2 q c,2 24 wih M c = Mc T > 0, K c,, K c,2 > 0, qc, and qc,2 given y 2 evaluaed in he equilirium configuraion, Ψ and Ψ 2 o e properly chosen, he configuraion χ previously inroduced is gloally asympoically sale. As in he case of finie dimensional Hamilonian sysem, he sailiy of a mixed finie and infinie dimensional Hamilonian sysem can e proved if i can e shown ha he equilirium is a sric exremum of he oal energy of he closed-loop sysem. The only difference is ha, in order o prove he sailiy for he infinie dimensional par, i is necessary o fix a norm: i is imporan o noe ha he sailiy wih respec o his given norm will no assure he sailiy wih respec o a differen one. The sailiy definiion in he sense of Lyapunov for mixed finie and infinie dimensional sysem can e given y he following: efiniion 0. The equilirium configuraion χ for a mixed finie and infinie dimensional sysem is said o e sale in he sense of Lyapunov wih respec o he norm if for every ɛ > 0 here exiss δ ɛ > 0 such ha χ0 χ < δ ɛ χ χ < ɛ for all > 0, where χ0 is he iniial configuraion of he sysem, Swaers, As proposed in Rodriguez e al., 200; Swaers, 2000, in order o verify he sailiy of χ, i is necessary o show ha χ is an exremum of he closed-loop energy funcion H cl inroduced in 22, wih H c given y 24, ha is he condiion H cl χ = 0 has o hold. Moreover, if χ is he displacemen from he equilirium configuraion χ, inroduce he non linear funcional N χ := H cl χ χ H cl χ 25 ha is proporional o he second variaion of H cl. Then, he configuraion χ is sale if i is possile o find γ, Γ > 0 such ha Swaers, 2000: 0 N χ Γ χ γ 26 In his case, he norm can e convenienly chosen as he Euclidean norm on he exended space of he closed-loop sysem. I easy o prove ha H cl χ = 0 if and only if Ψ = mg and Ψ 2 = mgl in 24. Moreover, since H cl χ = 0, he linear funcional inroduced in 25 is given y N χ = 2 pt M p 2 pt c Mc p c H p, p r, ɛ, ɛ r 2 K c, L q 2 q 2 2 K c,2 q 2 ɛ r 2 x ɛ r ɛ Since N χ > 0 if χ 0, he sailiy proof is compleed if proper consans γ, Γ saisfying 26 are found. If Γ = 2 max{ M, Mc, K c,, K c,2, 2 ρ, 2 I ρ, 2K, 2EI}, hen

8 N χ Γ p 2 p c 2 χ 2] Γ L q 2 q x ɛ r ɛ 2 Γ q 2 ɛ r 2 If γ = 2 and Γ = Γ max{2, 2L 2 2L }, hen condiion 26 is saisfied and he configuraion χ is sale. Summarizing, he following proposiion has een proved. Proposiion. Consider he mixed finie and infinie dimensional por Hamilonian sysem of Fig. 2, ha is he resul of he power conserving inerconnecion 4 of he susysems 0, 2 and 3. If in 3 i is assumed ha G c = I and H c is chosen according o 24, hen he configuraion descried in 22, 23 is sale in he sense of Lyapunov. 6. CONCLUSIONS AN FUTURE WORK Saring from he dph formulaion of he Timoshenko eam, i has een shown ha i is possile o exend he conrol y inerconnecion and energy shaping o rea mixed finie and infinie dimensional sysems. In paricular, he conrol of a mechanical sysem made of a flexile eam wih a mass conneced a one of is exremiy under graviy field has een presened. The finie dimensional conroller, acing on he sysem hrough he oher exremiy, is developed y properly exending he concep of Casimir funcions o he infinie dimensional case. Fuure work will deal wih he exension of hese conceps o he modeling and conrol of simple kinemaic chains wih flexile links. Acknowledgmen. This work has een suppored y he European projec GeoPlex, IST Furher informaions a hp:// Orega, R., A.J. van der Schaf, B. Maschke and G. Escoar 999. Energy-shaping of porconrolled hamilonian sysems y inerconnecion. In: Proc. of he 38h CC Conference. Rodriguez, H., A.J. van der Schaf and R. Orega 200. On sailizaion of non-linear disriued parameer por-conrolled hamilonian sysems via energy shaping. In: Proc. of he 40h CC Conference. Sramigioli, S Modeling and IPC conrol of ineracive mechanical sysems: a coordinae free approach. Springer-Verlag. London, UK. Swaers, G.E Inroducion o Hamilonian fluid dynamics and sailiy heory. Chapman & Hall / CRC. van der Schaf, A.J L 2 -gain and Passiviy Techniques in Nonlinear Conrol. Springer- Verlag. London, UK. van der Schaf, A.J. and B.M. Maschke Hamilonian formulaion of disriued parameer sysems wih oundary energy flow. J. of Geom. and Ph. 7. REFERENCES Araham, R. and J.E. Marsden 978. Foundaion of Mechanics. Benjamin/Cummings Pulishing Company. Golo, G Inerconnecion srucures in por-ased modeling: ools for analysis and simulaion. Ph hesis. Universiy of Twene. Golo, G., V. Talasila and A.J. van der Schaf A hamilonian formulaion of he imoshenko eam model. In: Proc. of Mecharonics Macchelli, A. and C. Melchiorri isriued por hamilonian formulaion of he imoshenko eam: Modeling and conrol. In: Proc. of 4h MATHMO, Vienna.