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Attou. Sliding mode Control - MSAP


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Attou. Sliding mode Control - MSAP

  1. 1. SLIDING MODE CONTROL OF A PERMANENT MAGNETS SYNCHRONOUS MACHINE A. Attou A. Massoum E. Chiali ICEPS, University of Djilali Liabes Sidi Bel-Abbes, Algeria. ICEPS, University of Djilali Liabes Sidi Bel-Abbes, Algeria. ICEPS, University of Djilali Liabes Sidi Bel-Abbes, Algeria. Abstract— This paper presents a speed sliding mode controller for a vector controlled (FOC) of the permanent magnet synchronous machine (PMSM) fed by a pulse width modulation voltage source inverter. The sliding mode control (SMC) is used to achieve robust performance against parameter variations and external disturbances. The problem with this conventional controller is that it has large chattering on the torque and the drive is very noisy. In order to reduce a torque ripple, the sign function is used. The proposed algorithm was simulated by Matlab/Simulink software and simulation results show that the performance of the control scheme is robust and the chattering problem is solved. Keywords— PMSM, field oriented control, sliding mode control. I. INTRODUCTION Permanent magnet synchronous motor (PMSM), which possesses such advantages as high efficiency, high torque ratio and the absence of rotor losses . But, if we compare the PMSM with DC motors, PMSM are more difficult in speed control and not suitable for high dynamic performance applications because of their complex inherent nonlinear dynamics and coupling of the system. So PMSM commonly run at essentially constant speed, whereas dc motors are preferred for variable-speed drives [8],[14]. Robustness as a desirable property of the automatic control systems is defined as the ability of the control system to yield a specified dynamic response to its reference inputs despite uncertainties in the plant mathematical model and unknown external disturbances. In its basic form, the sliding mode control (SMC) is a kind of nonlinear robust control using a systematic scheme based on a sliding mode surface and Lyapunov stability theorem. It features disturbance rejection, strong robustness and fast response [9],[8],[12]. The design of this control law consists of two phases: - Depending on the initial conditions of a trajectory is precomputed. This trajectory is used to modify the sliding surface so that the trajectories of the change on the surface for all t 0. - A discontinuous control is designed to ensure that the system evolves on the sliding surface, despite the presence of a certain class of uncertainties and disturbances.[6] The main advantages of this strategy are: A prior knowledge of the convergence time and setting of the control law independent of that time. - Establishment of the sliding mode at the initial time, which gives the control law robust behavior throughout the system response. - The control strategy is applicable regardless of the order of sliding mode (equal to or higher relative degree of the system). - The generation of the path enabling the convergence in finite time [5],[6]. In this work is composed of PMSM modeling in the Park frame and an overview of the sliding mode control of the PMSM supplied with PWM inverter. In the last step, a comment on the results obtained in simulation and a conclusion where we emphasize the interest of this method of control. - II. MACHINE EQUATIONS With the simplifying assumptions, the model of a 3-phase permanent magnet synchronous motor can be expressed in the so-called dq frame by application of the Park transformation, in the form of state is written [10][15]. x = F(x) + m i=1 g i (x)U i (1) With x1 Id x = x2 = Iq (2) x3 U U Ui = 1 = d U2 Uq (3)
  2. 2. high frequency which is generally triangular shaped hence the name triangular-sinusoidal [5]. 1 0 Ld g1 = 0 ; g2 = 1 Lq 0 0 f1 x F x = f2 x = f3 x (4) − Rs x1 + pLq x2 x3 Ld Ld pLd x1 x3 − p f x3 − Rs x2 − Lq Lq Lq p Ld − Lq p f − f x3 + x1 x2 + x2 − Td J J J J (5) The variables to be controlled are current Id and mechanical speed Ω. Y (x ) = y1 (x ) y 2 (x ) = h1 (x ) h2 ( x ) = x1 x2 = Id (6) The parameters used in these equations are defined as Follows: Ud Uq : Stator voltages in the dq axes; I d I q : Stator current in the dq axes; Rs : Stator resistance ; Ld Lq : Stator inductances in the dq axes; Φ : flux created by the rotor magnets; f Ω : Mechanical speed of motor; P : Number of pole pair; J : Inertia momentum; f : The damping coefficient; F : Frequency. III. MODELING OF THE VOLTAGE INVERTER The voltage inverter can convert the DC power to the AC (DC / AC). This application is widespread in the world of power conversion today. The connection matrix is given by (7). VaN 2 - 1 - 1 Sa E VbN = ⋅ − 1 2 - 1 Sb 6 VcN − 1 - 1 2 Sc (7) The inverter is controlled by the technique Pulse Width Modulation (PWM) generated by a carrier which is triangular. It is used for generating a signal which controls the switches, the PWM control signal delivers a square-wave, it is generated by the intersection of two signals, the reference signal, which is generally sinusoidal low frequency, and the carrier signal IV. FIELD ORIENTED CONTROL (FOC) By analyzing the system of equations (1), we can observe that the model is nonlinear and it is coupled. The objective of the field oriented control of PMSM is transformed machine three-phase axis variables into two-phase axis in order to obtain the same decoupling between the field and torque that exists naturally in dc machines; that is to say a linear and decoupled. This strategy is to maintain the flow of armature reaction in quadratic with the rotor flux produced by the excitation system [5]. Since the main flow of the PMSM is generated by the rotor magnets, the simplest solution to a permanent magnet synchronous machine is to keep the stator current in quadratic with the rotor flow ( Id is zero and reduce the stator current to the only component Iq ), that gives a maximum torque controlled by a single current component Iq and to regulate the speed by the current Iq through the voltage Vq . This verifies the principle of the DC machine [4]. V. SLIDING MODE CONTROL (SMC) The switching of the variable structure control is done according to state variables, used to create a "variety" or "surface" so-called slip. The sliding mode control is to reduce the state trajectory toward the sliding surface and make it evolve on it with a certain dynamic to the point of balance. When the state is maintained on this surface, the system is said in sliding mode. Thus, as long as the sliding conditions are provided as indicated below, the dynamics of the system remains insensitive to variations of process parameters, to modeling errors [1],[4],[5]. The objective of the sliding mode control is: • Synthesize a surface, such that all trajectories of the system follow a desired behavior tracking, regulation and stability. • Determine a control law which is capable of attra-cting all trajectories of state to the sliding surface and keep them on this surface. The behavior of systems with discontinuities can be formally described by the equation: x ( t ) = f ( x , t, U ) (8) Where: x : Vector of dimension n, x ∈ ℜ n . t : Time. U : Control input of a dynamical system, u ∈ ℜ . m f : The function describing the system evolution over time.
  3. 3. So we seek that the two functions f + and f − converge towards the surface of commutation S and which have the characteristic to slip on it. We say that the surface is attractive [5],[13] . + x( t )= f ( x,t ,U ) = f ( x ,t ) if S( x ,t )>0 f −( x,t ) if S( x,t )<0 A. The choice of desired surface We take the form of general equation given by J.J.Slotine to determine the sliding surface given by: S ( x) = ( ∂ + λx ) r −1e( x ) ∂t (14) (9) Where: e( x ) : Error vector; e( x) = x ref − x . λ x : Vector of slopes of the S. r : Relative degree, equal to the number of times he derives the output for the command to appear. B. Convergence condition. The Lyapunov function is a scalar function positive for the state variables of the system, the control law is to decrease this function, provided it makes the surface attractive and invariant. En defining the Lyapunov function by: Fig.1 Trajectories of f + and f - in the case of sliding mode. S=0 mathematically, this represented as: [2] lim S > 0 lim S < 0 et s → 0− s → 0+ (10) (11) The function is used, generally, to ensure stability of nonlinear systems. It is defined, like its derivative as follows [1],[3],[4],[13] : x = f ( x,t )+ g( x,t )U x ,u S ( x)S ( x) < 0 (16) This can be expressed by the following equation : lim S > 0 s →0 et − lim S < 0 s → 0+ (17) C. DETERMINATION OF THE CONTROL In sliding mode, the goal is to force the dynamics of the system to correspond with the sliding surface S(X) by means of a command defined by the following equation: u(t ) = ueq (t ) + u N (18) f and g are n- dimensional continuous functions Where in (12) (15) For the Lyapunov function decreases, it is sufficient to ensure that its derivative is negative. This is verified by the following equation: V ( x) < 0 Hence, the condition of attractiveness to obtain the sliding mode: S( x ).S( x ) < 0 1 2 S ( x) 2 The vector f is in a direction towards V ( x) = and t. t ,x Is an n- dimensional column vector, ( x ∈ ℜn ) and ( u ∈ ℜm ). When we are in the sliding mode, the trajectory remains on the switching surface. This can be expressed by [1],[3],[4], [7],[11],[13] : S( x,t ) = 0 et S( x,t ) = 0 (13) The design of sliding mode control requires mainly the three following stages: In which: U: is called control magnitude, Ueq :is called the equivalent components which is used when the system states are in the sliding mode; Un: is called the switching control which drives the system states toward the sliding mode, the simplest equation is in the form of relay: un = ksgnS(x) ;k 0 (19) k : High can cause the ‘chattering‘ phenomenon. When the switching surface is reached, (13) we can write:
  4. 4. U =Ueq with u N =0. (20) VI. THE ‘CHATTERING’ PHENOMENON ELIMINATION The high frequency oscillation phenomenon can be reduced by replacing the function ‘sgn’ by a saturation function [15]. k S(x) si u n = ksgn(S(x)) si S(x) S(x) (21) Fig. 4 Results of simulation of the adjustment by sliding-mode during variation parametric 0 IX. INTERPRETATION Different simulations allow us to see that: disturbance rejections are very good and a very low response time. This control strategy provided a stable system with a practically null static error and a decoupling for the technique suggested by maintaining Id to zero. X. CONCLUSION Fig. 2 Schematic of the overall simulation VII. SIMULATION RESULTS For the validation of the structure of the sliding mode control we made simulations using MATLAB / Simulink. Figure (3) shows the results obtained with the strategy of three surfaces: In this paper, we presented the performances of the sliding mode control for the PMSM. The decoupling technique is based on the Field oriented control to PWM tension inverter. The proposed controller provides high-performance dynamic characteristics and is robust with regard to plant parameter variations. So the controller works well with robustness in a large extent. MACHINE PARAMETERS Rs = 0.6 ; Ld = 1.4mH ; Lq = 2.8mH ; f = 0.12wb ;P = 4; J = 1.1⋅10 − 3 kgm 2 ; f = 1.4 ⋅10 − 3 Nm/rds − 1;F = 50 HZ . REFERENCES [1] MASSOUM Ahmed, Contribution to the Order Singularly Disturbed a Permanent Magnet Synchronous Machine: Variable Structure Control (VSC), Neuro-Fuzzy Control, Ph.D. thesis, University of Djilali Liabes, SBA, Algeria, 2007. [2] SOSSE ALAOUI Mohammed Chakib, Control and sliding mode observer of a pumping system and a manipulator, Ph.D. Thesis, University Sidi Mohamed Ben Abdellah, Fez, Morocco, 2009. Fig. 3 Results of simulation by sliding-mode control At t = 0.3 (s), the reference speed varies from Wr = 100 (rad / s) to Wr = -100 (rad /s), followed by an external load torque disturbance Td=8 (Nm) for periods [0.1s] between t = 0.1 (s) to t = 0.2 (s) and t = 0.4 (s) to t = 0.5 (s). VIII. ROBUSTNESS TESTING To highlight the importance of the technique of sliding mode control, we will test the robustness of our machine. [3] VIDAL Paul-Etienne, Non-linear control of an asynchronous machine has dual power supplies, Ph.D. Thesis, Thesis, National High School Electrical, electronics, computer, On hydraulic and Telecommunications, Toulouse, France, 2004. [4] BELABBES Baghdad, Linearizing control of a permanent magnet synchronous motor, memory magister, djilali liabes University, SBA, Algeria, 2001.
  5. 5. [5] ATTOU Amine, control by Sliding mode of a synchronous machine with a permanent magnet, memory master, University of Djilali Liabes, SBA, Algeria, 2011. [6] Marwa Mohamed Moustafa EZZAT, nonlinear control of A PERMANENT MAGNET SYNCHRONOUS MOTOR WITHOUT MECHANICAL SENSOR, Ph.D. Thesis, CENTRAL SCHOOL NANTES,France, 2011. [7] Sid Ahmed El Mahdi ARDJOUN, Mohamed ABID, Abdel Ghani AISSAOUI, Abedelatif NACERI, A robust fuzzy sliding mode control applied to the double fed induction machine, INTERNATIONAL JOURNAL OF CIRCUITS, SYSTEMS AND SIGNAL PROCESSING, Issue 4, Volume 5, 2011. [8] Harib, K.H.; Khousa, E.A., Ismail, A, Field Oriented Motion Control of a 3- Phase Permanent Magnet Synchronous Motor,IEEE; Electric Power and Energy Conversion Systems (EPECS) 2011. [9] Rui Guo; Xiuping Wang, Junyou Zhao; Wenbo Yu, Fuzzy Sliding Mode Direct Torque Control for PMSM,Fuzzy Systems and Knowledge Discovery (FSKD), 2011 Eighth International Conference,IEEE 2011. [10] A. Asri, D. Ishak, S. Iqbal, M. Kamarol, A Speed Sensorless Field Oriented Control of Parallel- Connected Dual PMSM, IEEE International Conference on Control System, Computing and Engineering, 2011. [11] Xiuli Yu Shimin Wei and Lei Guo, Cascade Sliding Mode Control for Bicycle Robot, IEEE CONFERENCE PUBLICATIONS , Volume: 1,Page(s): 62 – 66, Publication Year 2010. [12] Vittek, J., Bris, P., Stulrajter, M., Makys, P., Comnac, V., Cernat, M. Chattering Free Sliding Mode Control Law for the Drive employing PMSM Position Control,Optimization of Electrical and Electronic Equipment, 2008. OPTIM 2008. 11th International Conference ,IEEE 2008. [13] A. Kechich, B. Mazari, I.K. Bousserhane, Application of nonlinear sliding-mode control to permanent magnet synchronous machine, International journal of applied Engineering research / ISSN 0973-4562 Vol.2, pp. 125–138, No.1 (2007). [14] Ming MENG, “Voltage Vector Controller for Rotor Field-Oriented Control of Induction Motor Based on Motional Electromotive Force , Industrial Electronics and Applications, 2007. ICIEA 2007. 2nd IEEE Conference, 2007. [15] Ahmed MASSOUM ,Mohamed Karim FELLAH, Abdelkader MEROUFEL, Patrice WIRA,Baghdad BELLABES, Sliding Mode Control For A Permanent Magnet Synchronous Machine Fed By Three Levels Inverter Using A Singular Perturbation Decoupling, Journal of Electrical Electronics Engineering, vol. 5, no. 2, 2005. ᜚