Partial stabilization of a class of hydraulic mechanical systems with Casimir functions

Transcription

1 Proceedings of the 7th World Congress he International Federation of Atomatic Control Seol, Korea, Jly 6-, 28 Partial stabilization of a class of hydralic mechanical systems with Casimir fnctions Sator Sakai Chiba University, Deartment of Mechanical Engineering, Yayoi-cho, Inage-k, Chiba, Jaan (el: ; satorsakaiatfaclty.chiba-.j Abstract: his aer gives a new modeling and assivity based control of a class of flid mechanical systems with Casimir fnctions. First, we roose two stabilization methods, a new dynamic asymtotic stabilization method and a new artial stabilization method. Second, we give a new model of a class of hydralic mechanical systems sing Casimir fnctions. Frthermore, the roosed two stabilization methods are alied to the derived model. Finally, the validity of or methods are confirmed by nmerical simlation.. INRODUCION his aer gives a new modeling and assivity based control of a class of hydralic mechanical systems sch as hydralic robot arms, nematic robot arms and so on. It is well known that modeling and control of this class of hydralic mechanical systems are mch difficlt than those of mechatoronical systems. his is de to that the driving system is comlex and consists of comressible flid systems, which are nonlinear dynamical systems with nknown (or hard-to-be identified) arameters. o solve these roblems, this aer gives fndamental reslts abot modeling and control of this class of hydralic mechanical systems, esecially of the driving system, by focsing and develoing ort-hamiltonian systems and control theory. Port-Hamiltonian systems (van der Schaft (2)) are generalization of Hamiltonian systems in classical mechanics and can model many systems sch as electromechanical systems, mechanical systems with nonholonomic constraints (Maschke and van der Schaft (994)), distribted systems and their mixed systems (Macchelli and Melchiorri (25)) as well as classical mechanical systems. Passivity is the most imortant roerty of ort-hamiltonian systems and some assivity based control methods, originally from akegaki and Arimoto (98), were develoed, sch as, Energy-Casimir methods (van der Schaft (2)), the generalized canonical transformations (Fjimoto and Sgie (2)), IDA-PBC (Ortega and Garcia-Canseco (24)) and IPC aroach (Stramigioli et al. (998)) and so on. hese methods can give nonlinear robst controllers, for examle, the generalized canonical transformations give nonlinear robst dynamic ott feedback stabilizers (Sakai and Fjimoto (25)). Modeling and control of several flid systems are already discssed in ort-hamiltonian form. For examle, Ramkrishna and van. der. Schaft (26) discss infinite dimensional canal systems in three dimensional sace and Johanson (26) discss the for-tank systems based on IDA-PBC. hese flid systems have free-srface and are incomressible. Riccardo et al. (26) discss the modeling of hydralic arms and show some exerimental reslts. Gernot and Schlacher (25) discss the control of the hydralic arms in ort-hamiltonian form. In these aroaches, the modeling is based on the standard rocedre in ort-hamiltonian framework. Aart from these aroaches, we discss the modeling and control of the class of flid mechanical systems, sch as hydralic robot arms, nematic robot arms, based on a new strctral roerty, (natral) Casimir fnctions. Advantages of ort-hamiltonian systems are from their strctral roerties, sch as assivity, which do not exist in general nonlinear systems.inthisaer,wefocs on new strctral roerties of secial ort-hamiltonian systems, that is, ort-hamiltonian systems with Casimir fnctions which do not exist in general ort-hamiltonian systems. In a word, a key oint of this aer is that the class of hydralic mechanical systems are fond to be secial ort-hamiltonian systems, that is, ort-hamiltonian systems with Casimir fnctions. he Casimir fnctions are sed in the modeling and control as a new strctral roerty of ort-hamiltonian systems in addition to the standard strctral roerties, sch as assivity. his aer is organized as follows. In Section 2, we refer ort-hamiltonian systems and their roerties, esecially Casimir fnctions. In Section 3, we roose two stabilization methods, a new dynamic asymtotic stabilization method and a new artial stabilization method. In Section 4, we give a new model of the class of hydralic mechanical systems sing Casimir fnctions. At the same time, a very fndamental coordinate of the class of hydralic mechanical systems is discovered. In Section 5, the roosed two stabilization method are alied to the new model and a new assivity based control are roosed. In Section 6, the validity of or methods are confirmed by nmerical simlation and finally we conclde this aer in Section 7. In this aer, I n is n n identity matrix, R m n is the real sace of m rows and n colmns matrix /8/$2. 28 IFAC / KR-.74

2 7th IFAC World Congress (IFAC'8) Seol, Korea, Jly 6-, POR-HAMILONIAN SYSEMS 2. Port-Hamiltonian systems A ort-hamiltonian system with a Hamiltonian H(x) R is a system described by ẋ = J(x) H(x) y = g(x) H(x) + g(x) () with,y R m, x R n and a skew symmetric matrix J(x), i.e. J(x)=J(x) holds. he following roerty is known. Lemma (van der Schaft (2)) Consider the ort- Hamiltonian system (). Sose the Hamiltonian H(x) satisfies H(x). hen the int-ott maing y of the system is assive with resect to the storage fnction H, and the feedback Σ sd : = D(x) y (2) withamatrixd(x) εi > R m m renders (,y). Frthermore if H(x) is ositive definite and if the system is zero-state detectable, then the feedback (2) renders the origin asymtotically stable. he zero-state detectability and the ositive definiteness of the Hamiltonian assmed in Lemma do not always hold for general ort-hamiltoniansystems. In scha case, the generalized canonical transformation is sefl. 2.2 Casimir fnctions One of the roerties of ort-hamiltonian systems are the existence of Casimir fnctions. Casimir fnctions (with resect to J) are defined as the soltions of the following PDE, C(x) J(x). (3) Casimir fnctions are the secial first integrals, that is, Ċ (4) for any Hamiltonian H(x) at =. Casimir fnctions are not bonded from below nor er in general. Casimir fnctions do not alway exist for ort-hamiltonian systems in general. Note that in this aer we do not treat Casimir fnctions of the closed-loo systems bt treat only Casimir fnctions of controlled systems (lants). he former Casimir fnctions are discssed in the Energy-Casimir method and are artificially designed fnctions. From this oint of view, we call the latter natral Casimir fnctions in this aer. J was relaced by a negative semidefinite matrix in order to describe dissiative elements van der Schaft (2). 3. DYNAMIC AND PARIAL SABILIZAION FOR POR-HAMILONIAN SYSEMS 3. Dynamic stabilization for ort-hamiltonian systems In this sbsection, we give a dynamic stabilization method for ort-hamiltonian systems. In the revios section, ort-hamiltonian systems can be stabilized by the static stabilizers. However,the dynamic stabilizers are also sefl and will be alied later. heorem Consider the following (mechanical) ort- Hamiltonian systems ] ] H m ] I = ṗ I H m + G Σ hm : (5) y = G H m where x =(,) R n are the osition and the momentm, G is nonsinglar matrix, the Hamiltonian is H m = (/2)( M )+U() and M = M >, U() U() =. hen, the following dynamic feedback ṙ = P(x) G H m D H r Σ dd : (6) = P(x) H r makes Ω = {(,) y = =} asymtotically stable, where r R l, H r =(/2)r Rr, P is tall matrix, D = D > and R = R >. Proof of heorem he closed-loo system is Σ cl : ] ṗ = I I GP(x) ṙ P(x) G D(x) (H m + H r ) (H m + H r ) (H m + H r ) (7) and (may be non-canonical) Hamiltonian systems with a new Hamiltonian H m +H r and dissiation. Since the time derivative of H m + H r is Ḣ m + Ḣr = H r D H r (8) and H m + H r is bonded from below, H r (9) as t and the { =} is asymtotically stable. his imlies r, that is, ṙ becase H r belongs to class K with resect to r. From Eation (7) and the tall matrix P(x),wehave P(x) (H m + H r ) = H =. () 2792

3 7th IFAC World Congress (IFAC'8) Seol, Korea, Jly 6-, 28 his means that the set {y =} is asymtotically stable. In all, the set {y = =} is asymtotically stable. (Q.E.D.) Note that the roosed dynamic asymtotic stabilization is a generalization of the reslt of or revios reslt (Sakai and Fjimoto (25)). It is easy to extend Σ hm to more general ort-hamiltonian systems bt Σ hm will be alied directly later. 3.2 Partial stabilization for ort-hamiltonian systems with Casimir fnction Advantages of ort-hamiltonian systems are from their strctral roerties, sch as assivity, which do not exist in general nonlinear systems. In this sbsection, we focs on strctral roerties of secial ort-hamiltonian systems, that is, ort-hamiltonian systems with Casimir fnctions which do not exist in general ort-hamiltonian systems. he following theorem gives a new artial stabilization method for ort-hamiltonian systems with Casimir fnctions. heorem 2 Consider the following ort-hamiltonian systems with Casimir fnctions C(x) H ] =J(,) ṗ] H + G Σ hc : () y = G H where x =(, ), G is nonsinglar matrix. Sose there exist a coordinate transformation x φ(x) =(x r,c) sch that H(φ)=H r (x r )+H c (C). (2) where x r R r, r =rank(j) and H r is bonded from below. hen the feedback = D(x) y r (3) makes the set {y r =} asymtotically stable where y r =G ] φ I r ] Hr r. (4) Proofofheorem2 From the existence of Casimir fnctions C(x), there exist a coordinate transformation which convert the system () into ] ] H(x r,c) xṙ Jr (x = r,c) r C H(x r,c) + φ G] C (5) y =G ] φ H. C where J r = Jr R r r. he time derivative of the fnction H r is given as Ḣ r = H r ẋ r + H r r C Ċ = H r H(x r,c) J r +I r ] φ r r = H r H r J r +I r ] φ ] r r G] = H r I r ] φ r ] G] G] (6) de to the secial Hamiltonian strctre (2). he system with int and ott y r is assive (lossless) with resect to the storage fnction H r,thatis, Ḣ r = y r. (7) and the controller (3) makes the set {y r =} asymtotically stable since Ḣ r = y r Dy r (8) and H r is bonded from below. (Q.E.D.) Note that the above ott y r is different from the sal ott y of ort-hamiltonian fnction and a new ott based on the strctral roerties of secial ort- Hamiltonian systems, that is, ort-hamiltonian systems with Casimir fnctions. Frthermore, not all states, bt only the artial state x r can be stabilized in heorem MODELING USING CASIMIR FUNCIONS In this section, modeling of a class of hydralic mechanical system (sch as Fig. and Fig.2) is discssed with Casimir fnctions. In this section, this class of hydralic mechanical systems are discssed in ractical way, that is, we start not from infinite dimensional model bt from finite dimensional model and we take int not as tore bt as sool dislacement. his finite dimensional model with sool dislacement int is well-known in ractical control sitations (Jelali and Kroll (22)). In addition, it is also known that the mlti-degree of freedom case can be searated into this one degree of freedom cases comletely. Sool B S A Pielines Hydralic Cylinder Fig.. Hydralic robot arms (examle) Robot Arm 2793

4 7th IFAC World Congress (IFAC'8) Seol, Korea, Jly 6-, 28 A m 2 Fig. 2. he driving system f 4. Generalized continity eation αa A continity eation for incomressible flid is ṗ i = E V (Qin i Q ot i ) (9) where i is the ressre of chamber i, Q in is flow to chamber i, Q ot is flow from chamber i, E is the blk modls and V is flid volme, For examle, see Jelali and Kroll (22) for this modeling assmtions in detail. As the ort-hamiltonian systems are generalization of classical Hamiltonian system (with no ints and otts), we give the following generalization of continity eation where Σ f : ẋf ẋ f2 ] = y f ] A g αa o + ] g f 2 = A αa] H f f (2) H f = 2V E(x2 f + x 2 f2) (2) x f =(x f,x f2 ), x fi =(V/E) i, g i = g i (x fi )arefrom Bernolli s eation (omitted from its nieness), f is the sool int dislacement, A and αa (α (, )) are the area in the chambers. Note that the system Σ f is NO ort-hamiltonian systems and that the state is not ressre and different from that of the model by Gernot and Schlacher (25). 4.2 Interconnection with generalized continity eation Now the mechanical system Σ hm and the flid system Σ f are interconnected by the following { o = y = y f. hen the interconnected system is (22) Σ fm ṗ ẋ f ẋ f2 I I GA αga = AG αag y fm =G f f f f2 + ] G f f (23) where H fm = H f + H m and G f =g g 2 ] and is easily confirmed to be ort-hamiltonian systems since J matrix art is again skew symmetric. Note that the interconnected system of the flid system Σ f and the mechanical system Σ hm is again a ort- Hamiltonian system althogh the flid system Σ f is NO ort-hamiltonian system. his sitation is different from that in classicalmechanics where we shold take only energy conservation in Navier- Stokes eations and, aart from this, take only mass conservation in continity eation. hanks to the formlation of the generalized continity eation and the existing ort-hamiltonian framework, we can take not only energy conservation bt also mass conservation into accont simltaneosly. 4.3 Modeling based on Casimir fnctions In this section, we give the most imortant reslt abot the modeling of the class ofhydralicmechanicalsystems. Lemma 2 Consider the flid-mechanical Hamiltonian systems Σ fm. hen there exist a Casimir fnction C f = αx f + x f2. (24) Proof of Lemma 2. It is confirmed that C f satisfies the PDE (3) by a direct calclation, that is, Ċ f holds for any Hamiltonian H fm with zero-int. (Q.E.D.) heorem 3 Consider the flid-mechanical Hamiltonian systems Σ fm. hen there exist a coordinate transformation φ sch that the transformed systems satisfy the condition (2). Proof of heorem 3. Consider the following coordinate transformation, x fr C f =φ(,,x f,x f2 ) I 2n α = +α 2 +α 2 α + x f x f2. (25) It is calclated that the flid-mechanical systems are transformed to a new ort-hamiltonian systems which satisfies 2794

5 7th IFAC World Congress (IFAC'8) Seol, Korea, Jly 6-, 28 the conditions (2) as H fm = H r +(/2)Cf C f where H r is given in the following artial dynamics corresonding to that by J r art in Eation (5) H r ] I ṗ = I +α2 A x fr +α 2 A H r H r fr + gr] f (26) where g r is omitted becase of its nieness and H r = H + 2V Ex2 fr (27) is bonded from below. (Q.E.D.) 5. CONROL USING CASIMIR FUNCIONS In this section, we give a new stabilization method for the class of hydralic mechanical systems based on the revios reslts in this aer. he roosed controller in this section can stabilize only mechanical art, even if the arameters of flid systems, the blk modles E, is nknown and the mechanical art has few daming effect. Lemma 3 Consider the flid-mechanical systems Σ fm and sose that U() is the ositive definite fnction. hen the feedback f = Dgr H r fr makes the set {(,)=} asymtotically stable. (28) ProofofLemma3. First, from heorem 2 and heorem 3, the set {y r = gr H r =} (29) fr is asymtotically stable. Second, since the closed-loo system of (26) and (28) is eivalent to that of heorem, the feedback makes the set {y = H =} (3) asymtotically stable. his imlies that the set {(,) = } is asymtotically stablized de to the zero-state detectability. (Q.E.D.) Note that the roosed controller stabilizes for any arameter E. It is very imortant that x fr is a new (natral) coordinate for flid-mechanical system and never reorted in existing reslts (Merrit (967), Jelali and Kroll (22)). 6. NUMERICAL SIMULAION In this section, we confirm the validity of or methods by nmerical simlation. All arameters of lants are normalized as. Fig.3 shows the time resonse of a standard linear mechanical-siring SISO system with the stabilizer in heorem. All states converts to the origin smoothly and the validity of heorem is confirmed Fig. 3. ime resonse of the state of the linear system hen we show the hydralic arm in Fig.2 with the stabilizer in Lemma 3. Initial states are (,,, ). Fig.4 shows the time resonses of the state at D =/2. Only the state of mechanical systems (,) converts to the origin smoothly. his imlies that the validity of the artial stabilization methods in heorem 2 is confirmed. In Fig.4 the settling time is abot 2s. Fig.5 shows the time resonses of the state at D =/5. In this case, only the state of mechanical systems (,) also converts to the origin smoothly bt the settling time is abot 2s even when the gain D is smaller than that in Fig.4. his is the same tendency as that by Sakai and Fjimoto (25), where the athors gave a effective gaintning gideline based on the analysis of the linearized system. Fig.6 and Fig.7 show the time resonse of the discovered coordinate x fr in the case of D =/2and/5, resectively. he states convert to the origin. In all, the validity of or methods are confirmed Fig. 4. ime resonse of the all state (D =(/2)) x x 2 x

6 7th IFAC World Congress (IFAC'8) Seol, Korea, Jly 6-, Fig. 5. ime resonse of the all state (D =(/5)) Fig. 6. ime resonse of x fr (D =(/2)) Fig. 7. ime resonse of x fr (D =(/5)) 2 7. CONCLUSIONS First, we roose two stabilization methods, a new dynamic asymtotic stabilization method and a new artial stabilization method. Second, we give a new model of a class of hydralic mechanical systems sing Casimir fnctions. At the same time, a very fndamental coordinate (x fr ) of the class of hydralic mechanical systems is discovered. hird, the roosed two stabilization methods are alied to the new model of the class of hydralic mechanical systems and a new assivity based control are roosed. Finally, the validity of or methods are confirmed by nmerical simlation. REFERENCES K. Fjimoto and. Sgie. Canonical transformation and stabilization of generalized hamiltonian systems. Systems & Control Letters, 42(3):27 227, 2. G. Gernot and K Schlacher. Energy-based nonlinear control of hydralically actated mechanical systems. In Proc. of the 44th Conference on Decision and Control, ages , 25. M.Jelali and A.Kroll.Hydralic Servo Systems. Sringer, 22. K.H. Johanson. Interconnection and daming assigment assivity-based control of a for-tank system. In Proc. of 3rd IFAC worksho on Lagrangian and Hamiltonian systems and control, ages 69 74, 26. A. Macchelli and C. Melchiorri. Control by interconnection of mixed ort hamiltonian systems. IEEE rans. Atomatic Control, 5(): , 25. B. M. J. Maschke and A. J. van der Schaft. A Hamiltonian aroach to stabilization of nonholonomic mechanical systems. In Proc. 33rd IEEE Conf. on Decision and Control, ages , 994. H. Merrit. Hydralic Control Systems. John Willey & Sons, 967. R. Ortega and Eloisa Garcia-Canseco. Interconection and daming assignment assivity-based control: A srvey. Eroean Jornal of Control, ages 27, 24. P. Ramkrishna and Arjan van. der. Schaft. A orthamiltonian aroach to modeling and interconnections of canal systems. In Network modeling and control of hysical systems, age WeA8, 26. M. Riccardo, Z. Roberto, and F. Paolo. Dynamic model of an electro-hydralic three oint hitch. In Proc.ofthe 26 Amecican Control Conference, ages , 26. S. Sakai and K. Fjimoto. Dynamic ott feedback stabilization of a class of nonholonomic hamiltonian systems. In Proc. IFAC World Congress 25, ages , 25. S. Stramigioli, B. M. J. Maschke, and A. J. van der Schaft. Passive ott feedback and ort interconnection. In Proc. 4th IFAC Sym. Nonlinear Control Systems, ages 63 68, 998. M. akegaki and S. Arimoto. A new feedback method for dynamic control of manilators. rans. ASME, J. Dyn. Syst., Meas., Control, 3:9 25, 98. A. J. van der Schaft. L 2 -Gain and Passivity echnies in Nonlinear Control. Sringer-Verlag, London,