Direct Torque Control of a Single-Sided Linear Induction Motor Based on Sliding Mode
DIRECT TORQUE CONTROL OF A SINGLE-SIDED LINEAR INDUCTION MOTOR BASED ON SLIDING MODE
Bo Yang, Horst Grotstollen EIM-E/250 Institute of Electrical Drives and Power Electronics University of Paderborn Warburger Str. 100, 33098 Paderborn, Germany Tel: +49-(0)5251-605482 Fax: +49-(0)5251-605483
Abstract - A single-sided linear induction motor with gle-sided short stator linear induction motor. This requires a reaction plates is introduced into a doubly fed long stator proper control strategy for drive control, especially in the linear drive railway system in order to solve the switch transition area between long and short stator drive. problem and can be also applied in little-used areas of the railway. Due to the irregularity and incontinuity of the reaction plate in the switch area, a robust drive con- troller is required. As a result, a direct torque control based on sliding mode is applied, since it has shown supe- Primary Reaction Plate rior dynamic performance and can react to critical envi- ronmental conditions fast and flexibly.
Key words - Direct torque control, Single-sided linear induction motor, Sliding mode, Doubly fed linear motor Fig 1. Sketch of a Switch 1. INTRODUCTION As well known, the induction motor seems to be the most Since 1997, a novel mechatronic railway system, which complicated in terms of controllability. Furthermore, the integrates linear drives into the conventional railway system switch crossing even puts forward a bigger challenge based on a modular design, has been developed within the because of irregularity and incontinuity of reaction plates research project NBP-„Neue Bahntechnik Paderborn“[1]. (Fig.1). Consequently a direct torque control (DTC) based on sliding mode is taken into consideration ([2],[3]), since This new system is based on the operation of shuttles DTC provides fast and precise torque response. Sliding instead of trains, which are guided by wheels and rails and mode control theory is one of the prospective control driven via a doubly fed long stator linear motor. The primary approaches for electrical machines, due to its order reduc- (stator) is installed between the rails, and the secondary tion, disturbance rejection, strong robustness and simple (rotor) is fixed below the undercarriage. The primaries are implementation [2]. divided into segments supplied by different power supply substations. Both the primary and the secondary are fitted with three phase windings, so that they can generate the respective magnetic field completely independent from each 2. CONTROL DESIGN other. Due to double feeding, energy can be transmitted from the primary to the on-board supply system. Hence, neither 2.1 Sliding Mode Direct Torque Control overhead wires nor contact rails are required in this railway system. In addition, a relative movement between two shut- The induction machine model, with the stator currents tles on the same primary segment becomes possible [4]. (secondary) and rotor flux (reaction plate flux) as state vari- ables, in the stationary reference frame is described by: For long stator linear drive railway system, a problem arises with a switch. It is very difficult to install normal pri- · β i α = – γ i ++------ψ βpω ψ +γ u maries through the switch, a special design on the primary S 1 Sα T Rα m Rβ 2 Sα would be required. For example, a bending switch is applied R β to long stator maglev railway systems: Transrapid in Ger- · γ ψ β ω ψ γ iSβ = – 1iSβ ++------Rβ– p m Rα 2uSβ many and Yamanashi in Japan. Unfortunately, this solution TR cannot be practicable for NBP railway system due to full (1) ψ· M 1 ψ ω ψ traffic of the shuttles and flexible convoy operation. Rα = ------iSα – ------Rα – p m Rβ TR TR In order to generate the thrust force successively, a simple ψ· M 1 ψ ω ψ reaction plate composed of copper and iron is applied in the Rβ = ------i β – ------β + p α T S T R m R switch area. Certainly, the reaction plate can be also utilized R R in little-used areas of the railway to reduce the cost of the where track. Then, the secondary and the reaction plate form a sin- 2 γ ψ γ σ ⁄ () ⁄ 2 Rβ 2MiSβ = 1 – M LSLR , TR = LR RR d = ()γγ –k ψ + ------2– γ pω ψ – ------00 1 1 2 Rβ T 2 m Rα T R R (9) 2 γ ψ γ M 1 M 2 Rα 2MiSα β = ------γ = ------γ = γ R + ------d = ()γk – γ ψ α –2------– γ pω ψ β + ------σ , 2 σ , 1 2 S 01 1 1 2 R T 2 m R T LSLR LS LRTR R R It will be shown by using the Lyapunov function, if the ω TR is the rotor time constant, m is the rotor mechanical sliding mode occurs on the surface s = 0 in a finite time, in speed, which is proportional to the linear speed of the car- other words, if the sliding surface s is attractive. ,,, riage vm . MLR RR LS and RS represent mutual induct- 1 T ance, rotor inductance, rotor resistance, stator inductance V = ---s s (10) and stator resistance respectively. 2 T The thrust force between the stator and the rotor, i.e., And its time derivative is V· = s s· with between the secondary and the reaction plate is given by · ·· · ()·· ·· · ()(), s1 ==eT + k1eT T – Tref + k1eT = b1 + Du 11 3πp M ()ψ ψ ·· (11) Fx = ------τ------iSβ Rα – iSα Rβ .(2)s· ==e·· + k e· φ + k e· =b + ()Du ()21, 2 LR 2 φ 2 φ ref 2 φ 2
As far as linear induction motor is concerned, DTC means γ – k βpω γ2 γ 1 1 2 ω2 β ω γ β ω m φ b1 = 1 – k1 1 ++------p m Tk+ – 1 p m + 1 p m – ------to control the thrust force by setting the stator voltage. TR TR Hence, the thrust force F , or rather the active part of the ω ω x p m p m Mmi (12) + ------+ 2γ pω – k pω φ + γ ψ u· – γ ψ u· – ------force, is a control variable. Clearly, the flux of the reaction 1 m 1 m d 2 Rα Sβ 2 Rβ Sα TR TR plate plays an important role to the thrust force and must be · – βpω φ – T·· – k T· controlled. In view of simplification, the flux square of the m ref 1 ref ψ2 ψ2 ψ rotor Rα + Rβ , instead of the flux R is set as another M M 1 β b = 2------m – ------+ γ φ ++------φ pω T control variable 2 T T i T 1 d T m R R R R (13) ψ ψ 2 · · ·· Ti= β α – i α β + k – ------φ – k φref – φref S R S R 2 T 2 R 2 2 .(3) φψ= + ψ Rα Rβ with Then the errors are equal to φ ψ ψ d = iSα Rα + iSβ Rβ ,φφ (14) eT =TT– ref eφ = – ref .(4) 2 2 mi = iSα + iSβ φ where Tref and ref are set values of the active thrust force and the square of the rotor flux. The time derivative of the sliding surface functions should take the parameter variation (∆b ,∆D ) into account. An integral and a differential sliding surface functions are In view of parameter uncertainty of the linear motor, the applied to a rotary motor in [3]. In this paper two differential time derivative of the sliding functions are modified as sliding surface functions are adopted for the single-sided linear induction motor, namely s· ==()b + ∆b + ()D + ∆D u bDuz++ (15)
e· + k e ∆ ∆ s1 T 1 T z = b + Du (16) s == (5) s e· + k e 2 φ 2 φ µ () · T Tsign s , > V ==s s· s bb– – k s – c1 1 + z where k1 k2 0 . c µ () c2sign s2 (17) The stator voltage u can be chosen as in [3]: T µ µ ≤ T < = – kcs s – c1 s1 – c2 s2 ++s1z1 s2z2 –kcs s 0 uSα –1() u ==– D bk+ cs + ui ,(6) u Clearly, the time derivative is definitively negative as Sβ µ µ long as c1 and c2 are assigned as the upper limits of z1 and z . where kc is a positive control gain and 2 To moderate the chattering problem, a smoothing factor is –1 µ sign() s introduced in the sliding surface functions. u = –D c1 1 (7) i µ () c2sign s2 s () i Sat si = ------λ- (18) si + i d d 00 01 with λ > 0 . D = (8) i γ M ψ γ M ψ 2 2------Rα 2 2------Rβ When the control error trajectories reach the sliding sur- TR TR · · face i.e. s1 ====s2 s1 s2 0 , it is obvious that the comprising two auxiliary variables: actual thrust force and rotor flux will converge to the set val- ues. 2.2 Sliding Mode Flux Observer Consider the sliding surface, if
The above mentioned control strategy can be imple- β ρ > ()βη ψ˜ ω ()η ψ˜ mented under the assumption that all the states are measured 1 ------1 + Rα + p m 2 + Rβ TR and known. However, the rotor (reaction plate) flux is not measurable easily, especially for the linear motor. Conse- β , (23) ρ > ------()βη + ψ˜ β + p ω ()η + ψ˜ α quently, a sliding mode observer is developed to estimate the 2 T 2 R m 1 R R rotor flux. ρ , ρ > 3 4 0 The differential equations of the observer are proposed as η η ψ · β where 1 and 2 describe the upper limits of Rα and ˜ α γ ψ˜ α β ω ψ˜ β γ Λ iS = – 1iSα ++------R p m R ++2uSα 1 ψ β , then the sliding surface for the current and the flux is TR R attractive. · β ˜ β γ ψ˜ β ω ψ˜ γ Λ iS = – 1iSβ +++------Rβ– p m Rα 2uSβ 2 In order to prove the stability of the sliding surface, the TR (19) Lyapunov function is utilized: · Lm 1 ψ˜ α ψ˜ ω ψ˜ Λ · T T R = ------iSα – ------Rα – p m Rβ + 3 V ==s s· []e e []e· e· T T i S iS 1 2 1 2 R R β ()β()η ψ ()ω ω ()η ψ · L < e ------e – + ˜ Rα + p e – + ˜ Rβ + m 1 1T 3 1 m 4 m 2 ψ˜ Rβ = ------i – ------ψ˜ β ++pω ψ˜ α Λ R T Sβ T R m R 4 R R β e ------()βe – ()η + ψ˜ β + p()ω e – ω ()η + ψ˜ α < 0 2 T 4 2 R m 3 m 1 R ψ˜ ψ˜ R ˜iSα , ˜iSβ , Rα and Rβ are the estimated values of iSα , ψ ψ Λ Λ iSβ , Rα , Rβ . 1 to 4 represent the inputs of the · T T V ==sψ s·ψ []e e []e· e· observer, which should be determined yet. R R 3 4 3 4 2 2 The estimate errors are equal to e e (24) 3 ρ 4 ρ < = – ------– 3 e3 – ------– 4 e4 0 TR TR ˜iSα – iSα e1 Clearly, the time derivative of the sliding surface for cur- e ˜iSβ – iSβ e ==2 (20) rents and flux are clearly negative. e ψ˜ α – ψ α 3 R R A smoothing factor is also applied to the sliding surface e ψ˜ ψ of the observer in order to reduce the chattering. 4 Rβ – Rβ s and their time derivatives are () j Sat sj = ------λ- (25) sj + j β β ω Λ ------e3 ++p me4 1 with λ >0. TR j e· β 1 β ω Λ ------e4 – p me3 + 2 e· TR e· ==2 . (21) 3. EXPERIMENTAL RESULTS e· 1 ω Λ 3 – ------e3 – p me4 + 3 · TR The proposed control scheme was implemented at the test e4 1 bed in the above mentioned Electrical Drives and Power – ------e ++pω e Λ T 4 m 3 4 Electronics Laboratory. An 8-m test bed was built for a lin- R ear drive module with 8 primaries and 2 secondaries. For the The input variables of the flux observer can be chosen as: purpose of testing the proposed drive control a 2.4-m bilayer reaction plate is mounted at the test bed. Λ ρ () 1 1sign e1 All the controls for the linear drive module are executed Λ ρ () 2 2sign e2 on a DSP (Motorola PowerPC 750, 480MHz) platform in = – (22) Λ ρ () this system. 3 3sign e3 Λ ρ () 4 4sign e4 Obviously, not only the rotor flux but also the stator cur- rents are estimated. Because the stator currents are measura- ble, the estimate errors of stator currents can be obtained in Secondary an easy way. If the sliding surface for the stator currents are attractive, the current convergence of the estimated currents Reaction Plates is guaranteed, regardless of the estimate errors of rotor flux Primary e3 and e4 . Then, the estimate error of rotor flux can be fixed in terms of Λ and Λ . 1 2 Fig 2. Test Bed and Reaction Plates with Gap The sliding surface of the flux observer are defined as T T T T s ==s s e e , sψ ==s s e e . iS 1 2 1 2 R 3 4 3 4 100 50 uSalpha iSalpha uSbeta iSalphae 0 0 Current-alpha(A)
Stator Voltages(V) 100 -50 0 5 10 15 20 0 5 10 15 20 50 0.02 iSbeta Phiref iSbetae Phi 0 0.01
-50 0 Current-beta(A) 0 5 10 15 20 0 5 10 15 20 Square of Rotor Flux 2 0.2 Tref psiLalpha T psiLbeta 0 0 Rotor Fluxes active force (N) -2 -0.2 0 5 10 15 20 0 5 10 15 20 0.5 5 vm xm vref 4.5 xref 0 4 Position (m) Position Speed(m/s) -0.5 3.5 0 5 10 15 20 0 5 10 15 20
Fig 3. Experimental results of sliding mode DTC
Speed control can be achieved through the thrust force 4. CONCLUSION control. As a result, a trapezoidal speed reference command is applied to the linear drive module. Fig. 3 shows the exper- In this paper, a new direct torque control based on sliding imental results with both the sliding mode controller and mode and a sliding mode flux observer are designed for a sliding mode observer working. The active thrust force, or single-sided linear induction motor in a doubly fed long sta- rather the speed, and the square of the rotor flux are control- tor drive system. The proposed algorithm is verified by led to the reference values. experiments. In order to simulate the critical condition in the switch area, a reaction plate with a gap is specially introduced into ACKNOWLEDGMENT the experimental environment. This work was partly developed in the course of the Col- laborative Research Centre 614 - Self-Optimizing Concepts and Structures in Mechanical Engineering - University of Rotor-Flux-Oriented Control DTC with Sliding Mode Paderborn, and was published on its behalf and funded by vref vref vm vm the Deutsche Forschungsgemeinschaft.
REFERENCES Speed (m/s) [1]B. Yang, M. Henke, H. Grotstollen: Pitch Analysis and Control error error Design for the Linear Motor of a Railway Carriage, IEEE Ind. Application Society Annual Meeting (IAS 2001), Chicago, pp.2360- 2365.
error (m/s) error [2]V.I. Utkin: Sliding Mode Control Design Principles and Applica- tions to Electric Drives, IEEE Trans. on Ind. Electronics, 1993. [3]Shir-Kuan-Lin, Chin-Hsing-Fang: Sliding-mode direct torque control of an Induction Motor. IEEE Ind. Electr. Conf. 2001 Fig 4. Comparison of field oriented control with sliding mode DTC (IECON’01), pp.2171-7. 2001. It is shown from the experimental results, DTC with Slid- [4]B. Yang, H. Grotstollen: Application, Calculation and Analysis ing Mode is faster and more robust to the parameter uncer- of the Doubly Fed Longstator Linear Motor for the Wheel-on-Rail tainties than the conventional rotor-flux-oriented control NBP Test Track, 10th International Power Electronics & Motion under the same conditions. Control Conference (EPE-PEMC 2002), Dubrovnik, Croatia, 2002.