Adaptive backstepping controller design based on neural network for PMSM speed control

ABSTRACT


INTRODUCTION
The permanent magnet synchronous motor (PMSM) is employed for a longue period in many industrial applications.It is characterised by numerous significant advantages such as: compact design, great efficiency, high ratios values between power/weight and torque/inertia.But, behind these advantages the PMSMs has most known advantages which are the high cost and time-varying magnetic characteristics [1]- [3].Furthermore, their dynamic model presents a high nonlinearity in raison of the coupling between the motor torque and d-q axis currents.The parameters of PMSM, i.e. electrical/mechanics parameters, like stator resistance and friction coefficient are often not measured with accuracy.Of course, the external load torque value in the majority of cases is unknown and cannot be known precisely [1]- [2].So, with these constraints the controllawscomputingbecame very difficult and may not be robust to PMSM speed control under severe operated conditions in some real industrial applications like high speed and precision requirement.
The fieldoriented control, known by vector control, is considered among the most important techniques used in closed loop control of alternative currentmachinesfor variable speed applications.It can suppress dynamic model nonlinearity especially the decoupling existing between torque and flux, so they can be regulated each one seperatly [1]- [5].With this control strategy the PMSM can be shown like a separately excited direct currentmotorwhile it can maintain the AC motor advantages over DC motors [1], [2], [4], [5].However, the vector control performances may be deteriorated due to mator parameters variation, in particular the stator resistance and winding inductances which increase with temperature increasing and the magnetizing saturation effect [2], [4]- [11].Recently, great attraction has been shown to the identification of parameters variation of the PMSM during normal operationg drive.This refreshing an important scientific activity to devlop new nonlinear adaptive control strategies in order to enhance the drive performances achieving speed and flux tracking with accuracy [6], [7], [9]- [17].
Due the great evolution in nonlinear and adaptive control, several techniques have been successfully implemented for industrial applications in the past decades.Among these strategies that have been applied in electric drives systems, especially AC motors, is the backstepping method [9]- [16], [18].It is an orderly and recursive design algorithm destined for nonlinear feedback control, convenable with a wide type of feedback nonlinear systems which can be linearised and feedback linearisable nonlinear systems presenting invariant uncertainty, and it ensures total regulation and tracking performances.This strategy may reduce some limitations existing in other nonlinear methods [9]- [13].It provides the choice of procedure design to taking into account uncertainties and high nonlinearities and can avert inutilsconcellations.The basic procedure of backstepping control developpement depends on recursive selection of some adequate functions in terms of state variables like pseudo-control inputs for subsystems of the global system.In each step end of the backstepping, a new pseudo-control variable is obtained which is expressed as a function of the pseudocontrol variables from the previous steps.At the end of the procedure, a feedback computing for the real control input is obtained and realizes the objective design using a global Lyapunov function wich is expressed as sum of Lyapunov functions of each step [9]- [16].
In the literature, several researches have showing the introduction of artificial intelligence (AI) techniques, especially neural networks, in various industry applications to improve the robustness of the nonlinear adaptive control techniques.Neural networks (NNs) have been employed like a function approximation to represent functions with weighted sums of nonlinear terms [19]- [21].They have proven a good potentiality for complex systems modelling and as adaptive controller for nonlinear systems [19]- [28].Exploiting their universal approximation potentienality, the intelligent controller associated with NNs may be constructed without knowledge requirement of the systems.A detailed state-of-art about the different applications of NNs and their employement in the power electronics devices and variable speed applications has been presented in [24].The authors of [19] have been proposed an intelligent adaptive-backstepping controller design using a hidden-layer RNN for the mover position control of linear induction motor, in which the method of gradient-descent is utiliszed to determine the NN parameter-training algorithms.F. J. Linet al., [22], a PI-NNs controller with single NN is introduced for Maglev position control, where a single neuron network unit is integrated in the vector control closed-loop for LIM servo-drive.
In this paper, the developpement of nonlinear adaptive backstepping controller exploiting the vector control using neural network for estimation of electrical parameters for PMSM speed control is proposed.The presented controller is developed to construct the control scheme, which is unsensitive to changing of the parameters and load variations.The rest of this paper is organized as follows.Section 2 describes the motor dynamic model and the application of the vector control for permanent magnet synchronous motor.Section 3 reviews different step designs of the adaptive backstepping controller for PMSM speed control.The principle of the neural network estimator of electrical parameters associated to the nonlinear adaptive backstepping is detailled in section 4. Section 5 gives different simulation tests and results.Finally, some conclusions are given in section 6.

MATHEMATICAL MODEL OF THE PMSM
The model of a non sailentPMSM can be expressed in the synchronous (d-q) frame through the Park transformation as: [1], [2], [5], [29]- [32]: Where Thus, the dynamic model of a non sailent PMSM can be described as [29]- [32]: Where d i and q i denote the d-q axis currents, d v and q v represent the d-q axis voltages, s R is the stator resistance, d L , q L denoted-and q-axes stator inductances, P represents the pole pairs, J denotesthemoment of inertia, B is the coefficient of friction, l T is the load torque,  is the mechanical angular speed, f  is the permanent magnetic flux linking the stator.According to the model given in (4), it can be observer that the control of the rotor speed can be done by regulating the q-axisvoltage q v [1], [2].When the d-axis current d i is keeped equal to zero, then the PMSM dynamic model can be shown as [1], [2], [29]- [35] q f em q q d q q q f q q s q i J

NONLINEAR ADAPTIVE BACKSTEPPING CONTROLLER FOR PMSM SPEED CONTROL
The backstepping technique is a methodical approach to build Lyapunov equation and unsensitive controller, so that the system is evenly asymptotic stable [9]- [16].In the design, constructing suitable function enables system design to achieve expected purpose.The main objective of designing this controller is to compute the input control voltages of the PMSM to attend precisely speed tracking performance.Utiliszing stability theory according to Lyapunov and adaptive backstepping strategy, with the appropriate Lyapunov function, we can obtain a specific controller.As there are various disturbed parameters in different combinations according to various situations, and then the adaptive update law can be designed correspondingly [11]- [16].In order to track the speed of PMSM, the development of the adaptive nonlinear controller involves the following steps.
Step 1: In this first step, a speed tracking error can be defined as Where *  is the desired reference trajectory of the rotor speed, and the speed error dynamic is given by (7).
Adaptive backstepping controller design based on neural network for PMSM speed control (E.Sabouni) As the speed error needs to be reduced to zero, the q-axe current component q i is identified as the first virtual control element to stabilize the motor speed.To determine the stabilizing function, the following Lyapunov function is shoosenas We differentiate (8) to get Where, 1 k is positive gain.The tracking speed objectives are achieved if a stabilizing function is defined as * q i are the references currents.By substituting (10) and ( 11) into (9) yields Thus, the virtual control is asymptotically stable.Since the load torque l T is unknown we must use its estimate value l T ˆ in (10).Thus, let us define Step 2: Now we go to next step and try to make the signals * d i and * q i behave as desired.So, we define the following error signals involving the desired variables Then the error (7) can be written as Where To stabilize the current components d i and q i , we define now the current error dynamics as Now, since the actual control inputs d v and q v have appeared in the above equations, we can go to the final step where both control and parameter updating laws are determined.In the most practical cases, it's very known that the PMSM inductances d L and q L frequently vary with the saturation effect in the stator and rotor cores, so it is very necessary to identify their values using an adaptive method to update the applied control laws.
 and 3  denote adaptive positive constants, and the errors on the estimated inductances.We differentiate the Lyapunov function V 2 in (19) and substitute all error dynamics to get.
If the d-q axes control voltages are selected as: Adaptive backstepping controller design based on neural network for PMSM speed control (E.Sabouni) Then, the (21) can be simplified to: So, we can get the following expression: From ( 26), the following update laws can be given as: Therefore, the derivative of Lyapunov function (25)

ADAPTIVE BACKSTEPPING WITH NEURAL NETWORKS ESTIMATOR DESIGN FOR PMSM SPEED CONTROL
The great progress shown in adaptive nonlinear control is followed with a remarkable increasing of the introduction of the neural networks in complex systems identification [20].With the assistance of neural networks, the linearity-in-the-parameter assumption of nonlinear function and the determination of regression matrices can be avoided.As a consequent of this association that in the last decade, a several scheme based on backstepping design schemes which combine between the backstepping technique and adaptive NNs are proposed [19], [20], [21], [24]- [28].In this section, a neural network for parameter estimation of the adaptive nonlinear backstepping controller is investigated.This approach consists to employ the recurrent neural networks to improve the parameters adaptation of the nonlinear adaptive backstepping.The principle of this approach is based on the controller proposed in the above section where the updating laws of the motor parameters and load disturbance ( d L , q L and l T ) is replaced by a neural networks estimator.The new estimator has the same input variables as shown in ( 27), ( 28) and ( 29) which have been undergone to a learning phase in order to get an intelligent estimator.The Figure 1 illustrates the principle scheme of the adaptive backstepping controller based on the neural networks estimator for PMSM speed control.A threelayer recurrentneuralnetwork as shown in Figure 2 which is comprised of an input layer, a hidden layer, and an output layer, is used to estimate the PMSM inductances To design the neural networks estimator, the NN function is adopted in order to approximate the continuous Where

SIMULATION RESULTS
To demonstrate the effectiveness and the robustness of the PMSM speed control using the described intelligent adaptive control scheme, some simulations tests have been carried out using MATLAB/Simulink software.Numerous simulations were performed, and sample results are shown here.The machine parameters are given in Table I in the Appendix.The configuration of the adopted control scheme is drawn in Figure 2. The parameter used in simulation are chosen as .Twodifferent operating conditions for the permanent magnet synchronous motor are simulated to illustrate the operation of the proposed adaptive backstepping with neural network estimator, and the influence of the parameters uncertainties into PMSM drive system.shows that, the proposed adaptive backstepping based on NNs estimator scheme can achieve good tracking performance even in relation to load torque and electric parameters uncertainties.In Figure 3 (a), the speed response of the intelligent adaptive controller is observed that an accurate tracking performance and more robustness are obtained against speed reference reversals (minimal rising time, small undershoot against load torque application, with negligible overshoot and zero steady-state error).It could be noted that, with the presentedadaptivecontrolalgorithm, the current level ind-axis is kept to zero which proves a proper crosscoupling and an unperturbed field orientation ( 0 ).The q-axis stator current swiftly reaches to its new value regarding to load torque values.This shows the capability of the adaptive controller in terms of disturbance rejection.Figure 3 (c) and Figure 3 (d) show the electromagnetic torque of the motor.It is observer that the electromagnetic torque em T increases when the load disturbance is applied and it takes the form of the q-axis stator current q i .Furthermore, Figures 4 (a Test 2: Now, the proposed adaptive backstepping with neural network parameter adaptation has been tested under motoring mode at constant speed operating point.In this test, the PMSM control drive is subjected to a fixed desired speed equal to 100rd/s and load torque which is increased from 1.2N.m to 3.6N.m at 1sec and its return to its initial value at 2sec.An increasing in the d-axis and q-axis inductance has

CONCLUSION
In this presented study, we have presented an adaptive nonlinear controller using backstepping technique with a neural networks parameter estimator in order to offer a choice of design to compensate parameters uncertainties and load disturbance estimation.This study has successfully demonstrated the application of the nonlinear adaptive backstepping control for the speed regulation of a permanent magnet synchronous motor based on field orientation control.The control laws were derived using the motor model incorporating the parameters variation and the external disturbances.By recursive procedure, pseudo-control states of the PMSM drive have been determined and stabilizing laws are calculated using the theory of Lyapunov stability.Variations of the electrical parameters and load torque have been estimated using a neural network employing the equation quantities developed by adaptive backstepping.The performance of the studied nonlinear adaptive control associated to neural network has been investigated in simulation using Matlab/Simulink software.The simulation results show its effectiveness and robustness at tracking a reference speed under electric parameters uncertainties and load torque disturbance.

APPENDIX
Int J Pow Elec & Dri Syst ISSN: 2088-8694  Adaptive backstepping controller design based on neural network for PMSM speed control (E.Sabouni) 1941

Step 3 :
To develop the control and parameter updating laws, we define a new Lyapunov candidate function which include the current error variables 2 e , 3 e and the parameter estimation errors NN weights are adjusted adaptively by using the stable on-line learning algorithm derived from Lyapunov theory and the sigmoid activation function     is defined as:

Figure 1 .Figure 2 .
Figure 1.Structure of the neural network estimator

Test 1 : 2 has.
In this test, the effectiveness of the adaptive nonlinear controller with NN estimator has been tested under reversing speed operation mode.This first test consists of step change on speed reference from 100rd/s to -100rd/s where the change is made at 5sec.The PMSM drive system is started with a constant load torque set at .2N.m 1 then an additive load torque of .4N.m The performance of PMS motor drive using adaptive backstepping-NN estimator scheme under this operating condition test is depicted in Figures3 (a)-(d).The rotor speed and the d-qaxes motor current ( d i and q i ) are depicted in Figure 3 (a) and Figure 3(b), respectively.The Figures 3 (a)-(b) clearly )-(c) show the estimated values of the electrical parameters and load disturbance ( d L ˆ, q L ˆ and l T ˆ) and their real values.It is clearly shown that the estimated values of these parameters follow properly the measured values (real d L , q L and l T ) with a minimal error (error converges to zero), during different drive conditions.