Application of Differential Geometry in Magnetic Levitation Control

Transcription

1 Appl. Math. In. Sci. 8, No. 5, (2014) 2437 Applied Matheatics & Inoration Sciences An International Journal Application o Dierential Geoetry in Magnetic Levitation Control Hengkun Liu, Yungang Li and Hu Cheng Research center o Maglev, National University o Deense Technology, Changsha, Hunan, China Received: 11 Sep. 2013, Revised: 9 Dec. 2013, Accepted: 10 Dec Published online: 1 Sep Abstract: To enhance the reliability o levitation, an especial installation called joint-structure is applied to the aglev vehicle. Due to the joint-structure, each suspension point o the aglev vehicle is regulated by two independent controllers. When one controller o the suspension point breaks down, the syste can still be stably suspended by the other controller. However, there is strong orce-coupling between the two controllers or the joint-structure, which akes the controller designed based on totally separated suspension point unsuccessul in application. To realize stable suspension or joint-structure, dienrential geoetry ethod is introduced to obtain a globally decoupling and linearized odel or the syste. Then the control paraeters are designed by pole assignent. Experiental and siulation results validate the eectiveness o the proposed control algorith. Keywords: Maglev vehicles, Nonlinear decoupling control, Dierential geoetry 1 INTRODUCTION Maglev technology has been widely utilized to dierent applications[1], aong which aglev vehicles rapidly develops since its presentation. To enhance the reliability o aglev vehicles, joint-structure is introduced, and each joint-structure is regulated by two controllers. I one controller breaks down, the other controller can still guarantee the stability o the syste, and the stability o the vehicle accordingly. As the two controllers or the sae joint-structure is strongly coupled together, decoupling control is the key to the stability o the joint-structure. There are quite a lot o decoupling control strategies, such as inversion atrix ethod [2, 3], relative apliication coeicient atching ethod[4], diagonal doinance approach[5], state eedback control[6],adaptive decoupling control[7, 8], intelligent decoupling control[9, 10], nonlinear and robust decoupling control[11, 12], and so on. Based on the characteristics o the aglev syste and the experiental setup, this article uses dierential geoetry ethod to realize the decoupling and the stability o the joint-structure. 2 MODELING Fig. 1: Sketch o joint-structure Ignoring the elasticity o the guideway and the eect o the air-spring on the syste perorance, the sketch o the joint-structure can be presented by Fig. 1 [13]. The deinition o the sybols in Fig. 1 and the sybols which will be used in the ollowing are presented as below. s 01 = s 02 = s 0 is the expected suspension gap, s 1 ands 2 is the easured suspension gap, 1 = 2 = is ass o the suspension object, g is the Gravity acceleration, F 1 and F 2 is the electroagnetic orce, u 1 and u 2 is the control voltage, i 1 and i 2 is the current in the coil, N 1 = N 2 = Nis the turns o the coil, R 1 = R 2 = R is the resistance o the coil, A 1 = A 2 = A is the pole area o Corresponding author e-ail: liberry@sina.co

2 2438 H. Liu et. al. : Application o Dierential Geoetry in Magnetic... a single side o the electroagnet, L 1 and L 2 is the inductance o the coil, µ 0 is the pereability o vacuu, k r is the rigid coeicient between two sides. Suppose the agnetootive orce o the agnetic ield concentrates in the air gap, the atheatical odel o the syste shown in Fig. 1 can be obtained as[13], u 1 = Ri 1 + µ 0N 2 A di 1 2s 1 dt µ 0N 2 Ai 1 ds 1 2s 2 1 dt s 1 = F 1 (i 1,s 1 )+g k r (s 1 s 2 ) u 2 = Ri 2 + µ 0N 2 A di 2 2s 2 dt µ 0N 2 Ai 2 ds 2 2s 2 2 dt s 2 = F 2 (i 2,s 2 )+g+k r (s 1 s 2 ) F i (i i,s i )= ( µ 0 N 2 A4 ) (i i s i ) 2 In (1), the irst and the second equations are separately the electricity equation and echanical equation or the suspension syste on the let, while the third and the ourth equations are separately the electricity equation and echanical equation or the suspension syste on the right. 3 ESSENTIAL THEORY OF DIFFERENTIAL GEOMETRY For the convenience o proble analysis, the decoupling control theory used in this paper is listed as ollowing[14]. Deinition 1: Let open sets, U R n, x U, and given a sooth scalar unction λ(x) and a n-diention vector ield (x) in U. Then a new scalar unction noted L λ(x) is deined as ollowing. L λ(x)= n i=1 (1) λ i (x 1,x 2,...,x n ) (2) x i This new unction L λ(x) is called the lie-algebra o λ(x) along (x). And this unction can be calculated iteratively. For exaple, lie-algebra o λ(x) sequentially along (x) and g(x) is, L g L λ(x)= (L λ) g(x) (3) x Or lie-algebra o λ(x) along (x) or k ties is, L k λ(x)=d(l k 1 λ) (x) (4) Deinition 2: For the ollowing ultiple inputs and ultiple outputs syste, ẋ= (x)+ i=1 y j = h j (x), j g i (x)u i Its relative order r i (x 0 ) is the lie algebra, which satisies, (5) L g j L r i 1 h i (x 0 ) 0( j= 1,...,) And or k<r i 1, satisies, L g j L k h i(x 0 )=0( j= 1,...,) On the basis o the two deinitions, two theores or decoupling o nonlinear systes are introduced. First, introduce the decoupling atrix nonsingular theore. Theore 1: I a ultiple variables nonlinear syste has a relative order r i or x 0 to all i, its decoupling atrix A(x) shown in (6) is nonsingular at x 0. L g1 L r 1 1 h 1 (x) L g L r 1 1 h 1 (x) L g1 L r 2 1 h 2 (x) L g L r 2 1 h 2 (x).. L g1 L r 1 h (x) L g L r 1 h (x) Then introduce theore 2, which is utilized to explain the conditions or decoupling o the syste, and how to realize decoupling or nonlinear coupled syste. Theore 2: I nonlinear syste has relative order at x 0, or saying decoupling atrix A(x) is nonsingular at x 0, then the input-output decoupling proble at x 0 can be solved by a static state eedback, and one solution is the eedback deined by the ollowing atrix, where, u(x)=α(x)+β(x)v α(x)= A 1 (x)b(x) β(x)=a 1 (x) b(x)= [ L r 1 h 1 (x) L r h (x) ] T I the conditions described in theore 2 are satisied, decoupling or nonlinear syste can be realized by equations (7) and (8). The proposed nonlinear decoupling ethod or aglev syste can be obtained based on the deinitions and theores introduced in this part. 4 DECOUPLING AND LINEARIZATION OF THE SYSTEM For the convenience o description, introduce the ollowing variables transor, (6) (7) (8) x 1 = s 1 x 2 = y 1 = i 1 x 4 = s 2 x 5 = y 2 x 6 = i 2 (9) k=µ 0 N 2 A 4 (10)

3 Appl. Math. In. Sci. 8, No. 5, (2014) By the transor in (9), the odel (1) can be changed as ollowing, ẋ 1 = x 2 ẋ 2 = g kx2 3 x1 2 k r(x 1 x 4 ) ẋ 3 = x 2 Rx 1 x 1 2k ẋ 4 = x 5 + x 1 2k u 1 ẋ 5 = g kx2 6 x4 2 + k r(x 1 x 4 ) ẋ 6 = x 5x 6 Rx 4x 6 x 4 2k Y = [ (11) + x 4 2k u 2 ] X (12) Copared with the standard or in (5), it can be known: (x)= x 2 g kx 2 3 x 2 1 k r (x 1 x 4 ) x 2 x1 Rx 1 2k x 5 g kx 2 6 x k r (x 1 x 4 ) x 5 x 6 x4 Rx 4 x 6 2k (13) g 1 (x)= [ 0 0 x 1 2k ] T (14) g 2 (x)= [ x 4 2k ] T (15) y 1 = h 1 (x)=x 1 (16) y 2 = h 2 (x)=x 4 (17) First, the decoupling atrix o the syste can be calculated according to (6) as, [ ] x3 x1 0 A(x)= (18) 0 x 6 x4 And the other expressions to calculated the control variables can also be obtained as, b(x)= [ ] T b 1 b 2 = [ ] Rx 2 (19) 3 x 1 k r (x 2 x 5 ) Rx2 6 x 4 + k r (x 2 x 5 ) [ Rx3 k α(x)= r x 1 (x 2 x 5 ) ] Rx 6 + k r x 4 (x 2 x 5 ) x 6 [ ] x1 x3 0 β(x)= 0 x 4 x6 (20) (21) By equation (7), the decoupling control law can be calculated as, [ ] [ u1 Rx3 k = r x 1 (x 2 x 5 ) ] u 2 Rx 6 + k r x 4 (x 2 x 5 ) x 6 [ ][ ] (22) x1 x3 0 + v1 0 x 4 x6 v 2 That s to say, u 1 = R k rx 1 (x 2 x 5 ) x 1 v 1 (23) u 2 = Rx 6 + k rx 4 x 6 (x 2 x 5 ) x 4 x 6 v 2 (24) Up to now, the syste has been decoupled. To realize linearization o the decoupled syste, the ollowing variable transor is introduced, z 1 = x 1,z 2 = x 2, (25) z 3 = ż 2 = g kx2 3 x1 2 k r(x 1 x 4 ) (26) z 4 = x 4,z 5 = x 5, (27) z 6 = ż 5 = g kx2 6 x4 2 + k r(x 1 x 4 ) (28) It can be obtained by calculation that, ż 3 = v 1, ż 6 = v 2 Now, the odel or joint-structure has been decoupled and linearized, and the state space odel ater decoupling is, where, ż=az+bv (29) y= Cz (30) z= [ z 1 z 2 z 6 ] T (31) And the state atrix or the syste is, A= (32) [ ] T B= (33) [ ] C= (34)

4 2440 H. Liu et. al. : Application o Dierential Geoetry in Magnetic... Fro state space atrix (32) (34), the state space odel or two separated subsystes can be obtained, and the state space or the irst subsyste is, A 1 = (35) B 1 = [ ] T (36) C 1 = [ ] (37) It can be seen ro (32) (34) that, the state atrix o the second subsyste is the sae to the irst one. Thus only the controller design or the irst subsyste is conducted. 5 CONTROL ALGORITHM DESIGN In this section, control algorith design reers to the controller design or subsyste (35) (37). For the convenience o ipleentation, state eedback control is utilized. Suppose the eedback control law is[15], v 1 = k 1 (z 1 z 01 )+k 2 z 2 + k 3 z 3 (38) And set the control goals o the syste ater eedback as, overshoot is less than 5%, settling tie is 0.1s. Then the doinant poles or the syste is calculated as, s 1 = j (39) s 2 = j (40) Suppose the third pole o the syste is, s 3 = 200 (41) Then the characteristic equation or the syste is, s s s (42) Substitute eedback control law (38) into the irst subsyste ater linearization (35) (37), and the characteristic equation or the closed loop syste can be obtained as, s 3 k 3 s 2 k 2 s k 1 = 0 (43) Coparing the characteristic equation ater eedback control (43) with the expected characteristic equation (42), the control paraeters can be chosen as, k 1 = (44) k 2 = (45) k 3 = 280 (46) Substitute (25) and (26) into (38), the control variable or the linear syste can be calculated as, v 1 = k 1 (x 1 s 0 )+k 2 x 2 + ( k 3 g kx2 3 x1 2 k ) r(x 1 x 4 ) (47) Substitute control law or linear syste (47) into the expression o control variable (23), the control variable o the irst subsyste or the joint-structure can be obtained as, u 1 = R k rx 1 (x 2 x 5 ) x 3 x k 1 (x 1 s 0 )+k 2 x ( k 3 g kx2 3 x1 2 k ) r(x 1 x 4 ) (48) Siilarly, the control variable or the second subsyste can be calculated as, u 2 = Rx 6 + k rx 4 (x 2 x 5 ) x 6 x k 1 (x 4 s 0 )+k 2 x ( x 6 k 3 g kx2 6 x4 2 + k ) r(x 1 x 4 ) (49) Equations (48) and (49) are the control law applied to the physical controllers. 6 SIMULATION AND EXPERIMENTS For one application syste, the paraeters o the two subsystes described in Fig. 1 are totally the sae. k= , =725kg s 0 = 0.012, R=4.44Ω g=9.8nkg, k r = N When the power o one controller o the joint-structure is suddenly shut down, the siulation results o the other controller are shown in Fig. 2. When the power o one controller o the joint-structure is suddenly shut down, the experiental results o the other controller are shown in Fig. 3. Fro siulation results shown in Fig. 2, it can be seen that when one controller breaks down, the current or the other controller increase rapidly ro 18A to 24A, and the axiu variety o the suspension gap is 0.5. Fro experiental results shown in Fig. 3, it can be seen that when one controller breaks down, the current or the other controller increase rapidly ro 17A to 25A, and the axiu variety o the suspension gap is 1.6. And the suspension gap inally reaches the expected value because o the eect o integrator.

5 Appl. Math. In. Sci. 8, No. 5, (2014) The authors are grateul to the anonyous reeree or a careul checking o the details and or helpul coents that iproved this paper. Reerences Fig. 2: Siulation result o air-gap and current Fig. 3: Experient results o air-gap variety and current 7 CONCLUSIONS The nonlinear odel or joint-structure o aglev vehicles are built in this paper. And the obtained odel is a ultiple-inputs ultiple-outputs coupling odel. Nonlinear state eedback technique is introduced to decouple and linearize this odel. Then pole assignent is utilized to design the controller. The proposed control algorith is siple, and convenient to ipleent. The control strategy proposed in this paper is validated by both siulation and experient. Siulation and experiental results show that, both the two controllers or a single joint-structure can realize stable suspension, which eans the proposed ethod is eective or decoupling control. And it can be also shown that, when one controller breaks down, the other controller can realize the stability o the whole joint-structure syste by itsel. Joint-structure unction is eectively realized by the proposed ethod, which can iprove the saety o aglev vehicles. [1] S. J. Joo and J. H. Seo. Design and analysis o the nonlinear eedback linearizing control or an electroagnetic suspension syste [J]. IEEE Trans. Contr. Syst. Technol, 5, (1997). [2] R.J.kavanagh.Multivarible control syste synthesis [J]. AIEE.Trans Apr Ind., 77, (1988). [3] K Weischedel, T J Mcavoy. Feasiblity o decoupling in conventionally controlled distillation coluns[j]. Ind Eng Chen, 19, (1990). [4] F G Shiskoy.The stability o interacting control loops with and without Decoupling [J]. Proc IFAC ultivariable Technological Syste Cont 4th International Syposiu, (1977). [5] H H Rosebrocok. Design o Multivariable Control Syste using Inverse Nyquist Array [J]. Pro IEEE, (1969). [6] Eler G Gilbert. The Decoupling o Multivariable Systes by State Feedback [J]. SIAM.J Control, (1989). [7] Narendra K and Parthasarathy K. Identiication and control o dynaic systes using neural networks IEEE Trans [J]. On Neural Networks, 1, 4-27 (1990). [8] Henriquest J and Dourado A. A Multivariable adaptive control using a recurrent neural network Proceedings o Eann98[J]. Engineering Applications o Neural Networks, Gibraltar, 9, (1998). [9] Morari,M. Advances in Model Predictive Control Conerence [M]. Oxord University Press, (1994). [10] Shaw A and Doyle F. Multivariable non-linear control applications or a high purity distillation colun using a recurrent dynaic neuron odel [J]. Journal o Process Control, 7, (1997). [11] R Y Chiang, M G Saonov, K Haiges, K Madden and J Tekawy. A ixed H controller or a super aneuverable ighter peroring the Herbst aneuver [J]. Autoatica, 29, (1993). [12] W Chan, R J Anderson, P Beae, D H Jones, D Notkin and W E Warner. Decoupling Synchronization ro Local Control or Eicient Sybolic Model Checking o State Charts [J]. In Proceedings o the 1999 International Conerence on Sotware Engineering (ICSE99), (1999). [13] HengKun Liu. Research on Suspension Control Probles o EMS High-Speed Maglev Train Double Bogies Join- Structure [R]. Doctor dissertation o National University o Deense Technology, (2005). [14] XiaoHua X., Weibing G. Nonlinear control and decoupling [M], Science Publisher, (1993). [15] Ling W. Autoatic control principle [M]. Tinghua University Publisher, (1990).

6 2442 H. Liu et. al. : Application o Dierential Geoetry in Magnetic... Hengkun Liu was born in Jangjin, Chongqing, China, in He received the PH.D degree in 2005works or National University o Deense Technology as a teacher. His research interests include control theory and its application, and aglev train levitation control technique. Yungang Li was born in Tianen county, Hubei Province, China, in He received the PH.D degree in 1996works or National University o Deense Technology as a Proessor. His research interests include control theory and its application, and aglev train levitation control technique. Hu Cheng was born in Xiangan county, Hubei Province, China, in He received the PH.D degree in 2011works or National University o Deense Technology as a teacher. His research interests include control theory and its application, and aglev train levitation control technique.