Model Predictive Control of A Speed Sensorless Linear Induction Motor Drive

Transcription

1 Proceedings of the 14th International Middle East Power Systems Conference (MEPCON 10), Cairo University, Egypt, December 19-21, 2010, Paper ID 173. Model Predictive Control of A Speed Sensorless Linear Induction Motor Drive Ahmed Abd Eltawwab Hassan,and Yehia Sayed Mohamed Faculty of Engineering Minia University Minia, Egypt.line address (aahsn@yahoo.com) Abstract - In this paper, the model predictive control (MPC) technique has been used to control the speed of the linear induction motor (LIM) drive. The mathematical model of the LIM has been described in the stationary frame. The longitudinal end effects of the linear induction motor are taken into account as external load.the concept of field orientation is used to decouple the mover speed from the secondary flux. The MPC technique has been designed such that the effect of the uncertainty due to motor parameters variation and load disturbance could be reduced. A simplified LIM model is introduced in the MPC structure so as to minimize the computational load. To decrease the associated maintenance cost, and increase reliability, the most common model reference adaptive system (MRAS) structure is used to estimate the motor speed. An adaptive full-order observer based on LIM equation is used to estimate the primary current and secondary flux. Lyapunov s stability criterion is employed to estimate the motor speed. The same algorithm deduced from Lyapunov s stability criterion is given to estimate the stator resistance, which results in the speed estimation error. Digital simulations are provided to validate the effectiveness of the proposed scheme. The results show that the proposed system possesses the advantages of good transient performance and robustness in face of uncertainties. A performance comparison between the proposed MPC controller and both of sliding mode and PI control schemes is carried out confirming the superiority of the proposed MPC technique. The proposed system has the advantages of increased reliability, and low cost due to the elimination of the mechanical speed sensor. Keywords: Linear induction motor Field orientation Electrical elevator PI controller - model predictive control.. 1. INTRODUCTION Linear induction motor has been widely used in a variety of applications like as transportation, conveyor systems, actuators, material handling, sliding door closers, curtain pullers, robot base movers, and so on. It has several advantages such as high starting thrust, simple mechanical construction, silence, no backlash, and less friction [1-4]. Both the linear and rotary induction motors have similar driving principles. However, the control characteristics of the LIM are more complicated. This is attributed to the time varying motor parameters as a result of change in operating conditions such Takashi Hiyama, and Tarek Hassan Mohamed Faculty of Electrical Engineering & Computer Science, Kumamoto University Kumamoto, Japan. address (tarekhie@yahoo.com) as mover speed, temperature, and rail configuration. Moreover, there are uncertainties existed in practical applications of the LIM [5-7] which usually composed of unpredictable plant parameter variations, external load disturbance, and unmodeled and nonlinear dynamics. Therefore, the LIM drive system must provide high tracking performance, and high dynamic stiffness to overcome the above difficulties. Because of their simplicity and effectiveness, PI controllers are considered as the most widely used controllers which have been employed in the electrical machine control systems [8-9]. However, The use of PI controllers for speed control of induction machine drives is characterized by an overshoot during tracking mode and exhibits poor load disturbance rejection. The fast improvements in power electronic devices and microelectronics has made possible the application of the field oriented control (FOC) technique on the induction motor drives [9-10]. It has been applied successfully to the LIM by aligning the d-axis of the primary current with the secondary flux linkage. However, the main drawbacks of FOC are the sensitivity of the system performance to the parameters variation and inadequate rejection of external disturbances and load changes. The direct torque control (DTC) technique [11] was developed to overcome the drawbacks of the FOC method. The DTC technique has the merits of fast response and less parameter dependency. On contrast, the flux and torque waveforms contain large ripples. Sliding mode control (SMC) is one of the effective means to control the induction motor drives [12-13]. It has many good features such as fast dynamic response, simplicity of design and implementation, and robustness to parameter variations or load disturbance. However, undesirable chattering appears in the control effort which excites unmodeled high frequency plant dynamics and causes unexpected instability. In the past few years, there has been considerable interest in the applications of advanced and intelligent control methods to deal with the nonlinearities and uncertainties of the LIM drive system. Linear quadratic Gaussian method, neural, fuzzy and 318

2 genetic techniques have been employed for this purpose [14-17]. In spite of the success of the previous methods to control the speed or position of the linear motor, new control techniques are needed to face the large uncertainties existed. The electrical dynamic model of the LIM is modified from the traditional model of a three phase, Y-connected induction motor in stationary frame and can be described by the following differential equations [20]: On the other hand, the MPC appears to be an efficient strategy to control many applications in industry [18-19]. It has many advantages such as very fast response, and robustness against load disturbance and parameters uncertainty. Moreover, the MPC controller can provide the optimal solution while respecting the given constraints. From the viewpoints of reliability, robustness, and cost, several approaches have been proposed that addresses the elimination of the mechanical sensors from induction motor control schemes. Using current and voltage measuring devices, the speed of the induction motor can be determined without the need to speed sensors. Several schemes of speed estimators have been proposed [23-25], among them, the MRAS approach has relative simplicity and low computational effort and gives good performance [24]. Several MRAS structures are possible. In this paper, the speed control of the field oriented LIM drive has been developed based on MPC technique, The field orientation principle is used to decouple the mover speed from the secondary flux amplitude. The MRAS technique is employed to estimate the the mover speed. The MPC technique law produces its optimal output derived from a quadratic cost function minimization based on simplified LIM model. The end-effect of LIM is modeled as an external load force dependent on the mover speed[28-29]. The technique calculates the optimal control signal while respecting the given constraints over the mover speed and developed force. The speed sensorless LIM drive with the proposed MPC controller has been tested against parameters uncertainty and load disturbance using computer simulation. The performance of the MPC has been validated versus the SMC and the traditional PI controllers. Simulation results proved that the proposed controller can be applied successfully to control the speed sensorless LIM drive very efficiently. The paper is organized as follows: Section 2 presents the dynamic model of the linear induction motor. Indirect field oriented technique is described in section 3. General consideration about MPC and its cost function are presented in section 4. A description of a common MRAS estimator is found in section 5. The implementation scheme of the sensorless LIM drive together with the MPC controller is described in section 6. Simulation results and general remarks are presented in section 7. Finally, the conclusions are given in section LIM Dynamic Model,, and h : Secondary time constant, : Secondary inductance per phase, : Secondary resistance per phase, : Magnetizing inductance per phase, : Primary winding resistance per phase, : Primary inductance per phase : Mover linear velocity, : secondary flux components, : primary current components, : primary voltage components, : Leakage coefficient, : Pole pitch, : Number of pole pairs. : Differential operator. : electromagnetic force, : external force disturbance, : total mass of the moving element, : viscous friction and iron-loss coefficient The longitudinal end-effect is approximated by Taylor s series and can be taken as an external load force [28-29], (6) are constants. This end-effect increases with the speed of the mover [30-31]. 3. Indirect Field Oriented LIM In the field oriented control method, the dynamics of the highly coupled nonlinear structure of the induction machine becomes linearized and decoupled. The decoupled relationship is obtained by proper selection of state coordinates, under the hypothesis that the rotor flux is kept constant [17]. Therefore, the rotor speed is only asymptotically decoupled from the (1) (2) (3) (4) (5) 319

3 rotor flux, and is linearly related to the torque current only after the rotor flux becomes in the steady state case. The flux model of the LIM can be described in the d-q synchronous frame as: : : secondary flux components, : primary current components, : synchronous linear velocity, :supply frequency. In an ideally decoupled induction motor, the secondary flux linkage axis is forced to be aligned with the d-axis, and the field orientation conditions can be applied. It follows that: (7) (8), and (9) Using equation (9), the desired secondary flux linkage in terms of can be found from equation (7) as (10) Moreover, equation (8) can be combined with equations (9) and (10) to give the feedforward slip velocity signal as follows: (11) The electromagnetic force can be described in the d-q synchronous frame as [17]: is the force constant which is equal to: (12) With the implementation of the field oriented control, equation (12) can be rewritten using equations (9) and (10) as: (13) If the d-axis primary current (flux current component) is kept constant at the rated value, therefore the electromagnetic force is directly proportional to the q-axis current; which can be realized via closed loop control. In this case, if the q-axis current (load current component) is rapidly changed in response to the load variation, this will be followed by a rapid change in the motor developed force and the LIM will exhibit a high dynamic performance. 4. model predictive control Due to its simplicity and effectiveness as a control technique. MPC has proved to efficiently control a wide range of applications in industry such as : chemical process, petrol industry, electromechanical systems and many other applications. The MPC scheme is based on an explicit use of a prediction model of the system response to obtain the control actions by minimizing an objective function. Optimization objectives include minimization of the difference between the predicted and reference response, and the control effort subjected to prescribed constraints. The effectiveness of MPC is demonstrated to be equivalent to the optimal control. It displays its main strength in its computational expediency, real-time applications, intrinsic compensation for time delays, treatment of constraints, and potential for future extensions of the methodology. At each control interval, the first input in the optimal sequence is sent into the plant, and the entire calculation is repeated at subsequent control intervals. The purpose of taking new measurements at each time step is to compensate for unmeasured disturbances and model inaccuracy, both of which cause the system output to be different from the one predicted by the model[18-19]. Figure 1 shows a simple structure of the MPC controller. An internal model is used to predict the future plant outputs based on the past and current values of the inputs and outputs and on the proposed optimal future control actions. the prediction has two main components : The free response which being expected behavior of the output assuming zero future control actions, and the forced response which being the additional component of the output response due to the candidate set of future controls. For a linear systems, the total prediction can be calculated by summing both of free and forced responses, reference trajectory signal is the target values the output should attain. The optimizer is used to calculate the best set of future control action by minimizing the cost function J, the optimization is subject to constraints on both manipulated and controlled variables [21,22]. The general object is to tighten the future output error to zero, with minimum input effort. The cost function to be minimized is generally a weighted sum of square predicted errors and square future control values, e.g. in the Generalized Predictive Control (GPC) : (14) 320

4 are the lower and upper prediction horizons over the output, is the control horizon, are weighting factors. The control horizon permits to decrease the number of calculated future control according to the relation: for. represents the reference trajectory over the future horizon. Constraints over the control signal, the outputs and the control signal changing can be added to the cost function: (15) Solution of equation (14) gives the optimal sequence of control signal over the horizon while respecting the given constraints of equation (15). Model Predictive Control have many advantages, in particularly it can pilot a big variety of process, being simple to apply in the case of multivariable system, can compensate the effect of pure delay by the prediction, inducing the anticipate effect in closed loop, being a simple technique of control to be applied and also offer optimal solution while respecting the given constraints. On the other hand, this type of restructure required the knowledge of model for the system, and in the present of constraints it becomes a relatively more complex regulator than the PID for example, and it takes more time for on-line calculations,,, and I, J are unit matrix and skew symmetric matrix respectively where: The full order adaptive observer for the primary current and secondary flux can be deduced using equation (16) as follows : signifies the estimated value. (17) Since the primary current can be measured easily, then it is selected as the error feedback value. Subtracting equation 17 from 16 and assuming = would result : and G is the observer gain matrix. (18) The error between the states, and can be used to derive a speed adaptive control mechanism which adjusts the estimated speed. Past outputs Past controls Model Model Free responce Reference trajectory Forced Total response response _ A Lyapunov s function can be selected as [27]: (19), Q, F are both positive symmetric matrices, and The derivative of with respect to time is as follows: Future controls Cost function J Optimizer Future errors Constraints Fig. 1 A simple structure of the MPC controller 5 Adaptive speed and primary resistance estimations Equations (1-4) are used as the reference model and can be used rewritten in matrix form as: :,,,, (16),, and (20 ) Using the Lyapunov s stability theory [26], we can construct a mechanism to adapt the mechanical speed from the asymptotic convergence s condition of the current estimation errors : (21) Also, according to the same Lyapunov s theory, the primary resistance R s can be estimated as: (22) k Pv, k Iv, k PR, k IR are PI parameters of speed and stator resistance adaptive estimators respectively, and is the integral Laplace operator. The block diagram which illustrates the use of MRAS to estimate the mover speed and the primary resistance of the LIM is shown in Fig. (2). 321

5 u s B T P[i s,λ r ] LIM [i s,λ r ] T λr C i s i s - e 3-phase v * _ v Rectifier MPC controller Optimizer Linearized model L i q * C PWM inverter S a S b S c Hysteresis Current controller * * * i a i b i c LIM 3-phase currents AG v f(i s -i s ) Fig. 2 Estimation of the mover speed and primary resistance of the LIM using MRAS technique. 6. System configuration The block diagram of the indirect field oriented LIM drive system including the proposed MPC controller is shown in Fig. 3. The indirect field oriented LIM drive system consists of LIM, current controlled voltage source inverter, hysteresis current controller, field orientation mechanism, and coordinate translators. On the other hand, the primary currents and stator votages are obtained using coordinate translation of measured primary currents and voltages respectively, and used as input signals of MRAS observer to give the estimated speed v which used for closed loop control and compared with the reference speed. The estimated and reference speeds are fed to the model predictive controller in order to obtain the force current command. The flux current command is set at rated value. The force and flux current commands are used to obtain the slip command using equations (11). This latter is added to the actual speed, and the sum is integrated to obtain the field angle. Therefore the commanded phase currents are obtained using coordinate translation of, and. The 3-phase primary currents are measured and fed to hysteresis current controller. The current controlled pulse width modulation with hysteresis controller regulates the actual primary phase currents to closely follow the sinusoidal commanded currents. Using indirect field oriented technique, the transfer function of the motor can be deduced using equation (5) as: Transfer function (23) For easy implantation, the simplified linearized model of the LIM described by equation (23) is employed in the structure of the MPC controller. i d * Slip Callculator n p v e Π / h Coordinate translator θ e MRAS Fig. 3 Block diagram of the indirect field oriented Linear induction motor drive 7. Results and Discussions V Vαs, Vβs iαs, iβs Computer simulations have been carried out in order to validate the effectiveness of the proposed scheme. The Matlab / Simulink software package has been used for this purpose. The data of the LIM used for simulation procedure are [17]: 3-phase, Y-connected, 8-pole, 3-kW, 60-Hz, 180-V, 14.2 A. The motor detailed parameters are listed below in table.1. The parameters of the MPC controller are set as follows: Prediction horizon = 60, control horizon = 40, Weights on manipulated variables = 0, Weights on manipulated variable rates = 0.1, Weights on the output signals = 100, Sampling interval = sec. Constraints are imposed over the developed force, and motor speed as : Max. developed force = 1000 N. Min. developed force = 0 N. Max. mover speed = 1.5 m/sec. Min. mover speed = -1 m/sec. The parameters of MRAS observer are: K Iv = and K Pv = 200. K IR = and K PR = 5. Firstly, the dynamic response of the system is investigated under the condition of load disturbance effect. Figure (4) 322

6 shows the simulation results of the proposed scheme in this case assuming nominal motor parameters. The LIM is assumed to start at t=0 and accelerated up to 1 m/sec in the first 0.1 second, then the motor speed is kept constant at this value during the next 0.8 second, and decelerated till zero speed is reached during the next 0.1 sec (short acceleration and deceleration times are suitable for the used small LIM ). The results from the top to the bottom are: the reference, estimated and actual speeds, d-q secondary flux components, 3-phase primary currents, developed force, and the external load force (The load force is assumed to be stepped from 350 N to 700 N at t = 0.5 second. In additional to that force produced by the end effect (putting.). It has been noticed that the reference and both of estimated and actual speeds are aligned and good tracking performance has been achieved in spite of the load disturbance. Also the figure indicates that the actual d-axis secondary flux is equal to the set value ( wb) while the actual q-axis flux is kept zero during the simulation period. This means that the field orientation condition has been realized which leads to high dynamic performance drive. The figure reports also that the developed force follows the increase of the load disturbance. Similarly, the primary phase currents respond quickly to the speed and load variations. secondary resistance was increased by 15% in the LIM model ( ), while it is kept at its nominal value in both of the controller and the slip calculator, also, the mover mass amount was increased by 50% ( ) only in the motor model, where and represent the nominal values of and. And the stator resistance of the motor had a step change and the nominal value of. This case was studied at law speed (0.2 /s). Figure. 5 depicts the speed response of the MPC controller in this case of uncertainty at half load ( ), it has been indicated that very fast response has been achieved using the MPC controller. In addition, we can see that, the estimated stator resistance can trace the real motor resistance using the proposed MRAS, this effects significantly on the speed response especially at the change moments of the stator resistance of the motor as shown in Fig.6. Fig. (5) Dynamic responses of the proposed system under parameters mismatch condition: a) estimated and actual speeds, b) estimated value of the stator resistance, c) developed force, and e)the force represents the end effect.. Fig. (4) Dynamic response of the proposed system with load disturbance and the end effect: a) estimated and actual speeds, b) d-q secondary flux components, c) 3-phase primary currents, d)developed force, and e)load force plus force represents the end effect.. Secondly, the robustness of the MPC controller against parameter uncertainty was validated, in this case, the Fig. (6) Focusing on dynamic responses of the proposed system at stator resistance changes. 323

7 The tracking performance of the sensorless LIM drive together with the MPC controller is investigated at low speed (0.1 m/s). Also, the load force is assumed to be stepped from 350 N to 700N. at t = 0.5 second plus that force represents the end effect. Figure 7-a shows the actual speed of the MPC response compared to the SMC ( with detailed parameters listed in [13]) and PI (Ki =7, Kp =0.6 ) responses in that case of study. It has been noticed that with the MPC controller, good tracking performance has been achieved even at the time of the load disturbance. This is because the MPC provides feedback compensation for the load disturbance. In contrast, both SMC and PI controllers need a period of time in order to attain the steady state value either from start or after the load disturbance took place ( SMC controller needs about sec. and PI controller needs about 0.18 sec.). Also, Fig.7-b illustrates the estimated speeds of the three systems. It has been noticed that with the MRAS observer, the reference, estimated and actual speeds are aligned at low reference speed (0.1 m/s). Fig. 7 MPC response versus SMC and PI responses at low speed. 8. Conclusion This paper investigates sensorless robust speed control of a linear induction motor drive based on the model predictive control technique. The field orientation principle is used to asymptotically decouple the mover speed from the secondary flux. The complete nonlinear dynamic model of the system has been described in the stationary frame. The end effect is considered in the dynamic model of the LIM. The MRAS technique has been used to estimate the mover speed and primary resistance. Digital simulations have been carried out in order to validate the effectiveness of the proposed scheme. The proposed scheme has been tested through mismatched parameters and load force disturbance at both high and low speeds. Simulation results show that the proposed MPC controller response has many advantages such as: very fast response, robustness against parameter uncertainties and load changes, well tracking of speed trajectory at all speeds and has almost no current and force ripples. In additional, the proposed MRAS observer of the motor speed and primary resistance produces good speed estimation at high and low reference speeds, and under the effect of motor parameters variation. A performance comparison between the proposed controller and both of sliding mode control and a conventional PI control schemes is carried out. It is clear from the results that the MPC controller response is much faster than that of the SMC or the PI responses and able to deal with load changes more efficiently. REFERENCES [1] I. Takahashi, and Y. Ide," Decoupling control of thrust and attractive force of a LIM using a space vector control inverter", IEEE Trans. Indust. Appl, Vol. 29, No.1, 1993, pp [2] I. Boldea, and S. 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