Robust adaptive neural network control for switched reluctance motor drives

ABSTRACT This article presents a robust adaptive neural network controller for switched reluctance motor (SRM) speed control with both parameter variations and external load disturbances. The radial basis function neural network with the technology of minimal learning parameters is employed to approximate an ideal control law which includes the parameter variations and external disturbances. Furthermore, a proportional control term is introduced to improve the transient performance and chattering phenomena of the SRM drive system. The asymptotic stability of the proposed controller is guaranteed through rigorous Lyapunov analysis. A main advantage of the proposed control scheme is that it contains only one adaptive parameter that needs to be updated on-line. This advantage result in a much simpler adaptive control algorithm, which is convenient to implement in switched reluctance drives. Finally, the simulations and experiments are carried out to demonstrate the effectiveness of the proposed control scheme.


Introduction
Switched reluctance motor (SRM) drive is a promising competitor of variable speed drive system due to its simple construction, high efficiency and fault tolerance capability. It has been used in many industry applications, including electric vehicle, wind power generation, aviation industry, household appliances and so on [1][2][3]. However, the severe nonlinearity of electromagnetic torque caused by the double salient structure and discrete commutation from one phase to another makes the speed of SRM difficult to control. Besides, the speed performance of an SRM is quite sensitive to motor parameter variations and external load disturbance. Therefore, many investigations have been undertaken over the past few decades in order to obtain the high quality speed performance of SRM by applying all kinds of control techniques, such as the PID control [4], feedback linearization control [5], internal model control [6], adaptive control [7], variable structure control [8], intelligent control [9][10][11][12][13][14][15] and so on.
The intelligent control has the advantage of strong capability of handing uncertain and nonlinear information. Thus, using intelligent algorithm in SRM modelling and control have been extensively studied in the past few years. In [9], a nonlinear model of SRM is established by using the least square and vector machine technique. In [10], the authors develop a backpropagation neural network (NN) to describe the current i(ψ, θ) and torque T(θ , i) characteristics. In [11], a two-layer recurrent NN is trained and applied to identify the damper parameters from operating data. But the NN-base control systems require numerous sample data from experiments or finite-element analysis to train the NN off-line, which inevitably takes much time. Moreover, during the motor operation, the NN needs to be stored in processor to guarantee the real-time of control system, which requires a large amount of computational burden for the algorithm execution. This problem can be partially solved by combining the intelligent algorithm and proportional-integral-derivative (PID) control. In [12], a self-tuning fuzzy PID controller which can adaptively adjust controller parameters on-line is developed to regulate the speed of SRM drive system. In [13], based on different fuzzy inference rules, the fuzzy PID controller is designed to solve the position servo control of SRM direct drive volume control hydraulic press. To eliminate the effect of the dead zone of SRM, an improved dead-zone compensation fuzzy logic controller is presented in [14]. However, the expert's control knowledge and operator's experience are always needed to summarize the fuzzy logic rules in these methods. In order to solve this problem, the adaptive technique is incorporated with intelligent algorithms. Based on Lyapunov's stability theory, an adaptive TSK-fuzzy controller with a compensated control is proposed to control the speed of SRM in [15]. In [16], an adaptive fuzzy cerebellar model articulation controller is designed to improve the speed performance of SRM. One main advantage of these schemes is that the parameters of controller were adjusted on-line according to adaptive rules and, therefore, guarantee the robustness of SRM drive system for external load disturbance and motor's parameter variations. Although these control methods have significant advantages, there still exists a weakness that is the number of on-line updated parameters depending on the number of fuzzy rule bases or the number of NN nodes. In order to improve the approximation accuracy, the number of fuzzy rule bases or the number of NN nodes will increased enormously in general. As a result, the on-line learning time will become prohibitively large and then the huge computational burden prohibit the use of this control scheme for practical implementations. Fortunately, this problem was first solved by minimal learning parameters (MLP) technique in [17]. A main feature of this technique is that only one parameter is needed to be estimated on-line. Inspired by the work [17], a new direct adaptive fuzzy control is proposed for nonlinear strict-feedback systems in [18]. However, there is very little work on MLP technique for SRM drive systems.
Therefore, a robust adaptive neural network (RANN) controller is developed to regulate the speed of SRM drive system in this article. A radial basis function (RBF) NN with the MLP technology is employed to approximate the desirable control law of SRM drive system. The parameter variations and external load disturbances of the system are considered to guarantee the robustness of the proposed controller. And the stability of the proposed RANN controller is guaranteed by using rigorous Lyapunov analysis. The key features of the proposed control scheme are that, (1) the number of adaptive parameters that needs to be updated on-line is reduced to one, and (2) the transient performance and chattering phenomena of the system is improved by introducing a proportional control term. These features guarantee that the computational burden of the proposed control scheme is drastically reduced and the proposed control scheme facilitates the implementation in actual SRM drive system. Finally, simulation and experimental results are provided to demonstrate the effectiveness of the proposed control scheme.
The remainder of this article is assigned as follows: the SRM model and problem formulation are presented in Section 2; the RANN controller is designed and then analysis its stability in Section 3; Section 4 presents the numerical simulation results and the experimental results are showed in Section 5; Section 6 contains conclusions.

SRM model and problem formulation
In the SRM, only the stator has windings and the rotor consists simply of steel laminations stacked without permanent magnets and conductors. The cross section of a typical three-phases 12/8 poles SRM and its power converter are shown in Figure 1. With the Faraday's law, the pth phase voltage balance equation of the threephases SRM can be expressed as where ψ p , v p , R p , i p and θ are stator phase flux linkage, stator phase voltage, stator phase resistance, stator phase current and rotor position, respectively. If we ignore the effect of mutual inductance, the stator flux linkage can be described as where L p (θ , i j ) is the phase self-inductance. The most general expression for the electromagnetic torque at any position can be represented as where T e is the electromagnetic torque, W c = i 0 ψ(θ, i) di is the co-energy, W s = ψ 0 i(θ, ψ) dψ is the stored field energy. By neglecting the saturation field effects in SRM, the self-inductance L p becomes independent of the current i p . In this case, according to (3), the electromagnetic torque T e can be expressed as where K t (θ , i p ) > 0 denotes the torque coefficient. The electromagnetic dynamic equation of the machine and loads can be expressed as where J, k w , and w are the moment of inertia, the viscous frictional and the rotor speed, respectively. T L is the external load torque.

Problem formulation
In practical applications, the SRM operates under different working conditions, which will lead to changes in the parameters of the motor. Hence, the real parameter consists of two parts, the nominal value and their variations. Substituting (4) into (5), we havė where In this article, the control objective is to design a robust adaptive controller using NN for the speed regulation of SRM drive system with the system uncertainty d L , so that the motor speed ω can be tracking the speed reference ω * , while the speed tracking error z 1 tends to zero when the time t tends to infinity. The block diagram of SRM drive system is depicted in Figure 2. As shown in Figure 2, the current command u RANN and duty cycle signals d p are generated by the speed controller and current controller, respectively. The threephase currents are detected by current sensors and the commutation signals s p are provided by rotor position sensors. The proposed controller and a PI controller are employed in outer loop and inner loop, respectively.

RANN control design
In this section, the RANN controller is designed for SRM drive system with the technique of MLP.

RBF neural network
In control engineering, RBF NNs are usually used as a tool for approximating the nonlinear function because of their excellent capabilities in function approximation. RBF NNs consist of two layers, which are hidden layer and output layer. The network structure is shown in Figure 3.
The universal approximation properties show that, if node number q is chosen sufficiently large, then RBF NN can approximate any unknown nonlinear function f (X) to any desired accuracy over the compact set [19]. In this article, we use the following RBF NN to approximate a nonlinear function f (X) where X = [x 1 , x 2 , . . . , x M ] T is the input vector of the RBF NN, y q is the output of the NN, W = [w 1 , w 2 , . . . , w q ] T is the weight vector of the NN and S = [s 1 (X), s 2 (X), . . . , s q (X)] T is the basis function that is usually considered Gauss function . . , c iM ] T is the centre vector fo the receptive field, and σ i is the width of the Gauss function. It has been prove that the RBF NN in (7) can approximate any unknown continuous function f (X) over the compact set x ⊂ R M to arbitrary accuracy as (9) where W * is the optimal weights, ε(X) is the smallest approximation error. The optimal weight vector W * is an "artificial" quantity required only for analytical purposes. Typically, W * is defined as the value of W that minimizes |ε(X)| for all X ∈ X ⊂ R M , i.e.
The norm of optimal weight vector W * of RBF NN and the smallest approximation error ε(x) are bounded and these can be described as where W m and ε m are positive constants.

Design of RANN controller
Define the speed error and its differentiation as follows where ω * is the desired value of speed. Let the ideal control law be where k 1 > 0 is a design parameter. Substituting (13) into (6), we have where Z = [z 1 z 2 ] T . Equation (14) indicates that the motor speed ω can track the speed reference ω * asymptotically. The ideal control law (13) cannot be carried out because the lamped uncertainty d L of the system is unknown. According to the equation (9), the RBF NN can be used to approximate the ideal control law (13). Define Z and f (Z) =Ŵ T S(Z) as the input vector and output variable of the RBF NN, respectively. Using the RBF NN to approximate the ideal control law (13), we have where ε is the minimum approximation error and W * is the optimum weight vector and can be described as In order to reduce the number of on-line updating weights of RBF NN, the norm of the optimal weights is considered as an on-line updating parameter [18]. This allows for a n-nodes NN, only one parameter needs to be updated on-line. As a result, the computational burden of the RBF NN-based controller is greatly reduced.
Define the ξ as follows To update the weighting vector of RBF NN on-line, the adaptive law of control system can be developed aṡ whereξ is the estimate of ξ , γ and a are design parameters, k 0 > 0 is a small design constant.
Combing (11), (15) and (18), the RANN control is proposed as where sgn(·) is sign function that is expressed as In the real applications, a good transient response is desired. However, including the RBF NN, all of the adaptive NN algorithms need some time to approximate the objective function. This means that a large ε m is needed at the beginning to improve the transient response. But a large ε m will cause a severe chattering of the control law. In order to improve the transient response and chattering phenomena of the proposed control scheme, a proportional control term is introduced into the proposed control law. Then, the proposed RANN controller is rewritten as where λ > 0 is the deign parameter.

Simulations
In this section, the simulations are carried in Matlab/Simulink software to evaluate the performance of the proposed RANN controller. Further, the comparative studies are conducted between the proposed RANN controller and conventional PI controller. A three-phase 1.5 kW 12/8 poles SRM is used in the simulation and experimental investigation in this paper. Detailed parameters of the SRM are presented in Table 1. The simulation model of the 12/8 poles SRM is obtained by taking the off-line measurement of electromagnetic characteristics of this motor. The measured flux-linkage and static torque characteristics are described in Figure 4. The tuned parameters of inner-loop current controller are K pc = 0.3 and K ic = 0.1. The satisfactory performance parameters K p = 0.1 and K i = 0.4 of the conventional PI controller were obtained through fine tuning at a motor rated speed. Note that all of the reference speeds in simulations are set for with 1500 r/min/s slope.
Firstly, three different combinations for λ and ε m are set up to verify the beneficial effects of introducing proportional control term. Secondly, the comparisons between the proposed RANN controller and conventional PI controller are conducted at different speed references (100 r/min and 1500 r/min) with external load T L = 1 Nm to test their wide range speed response capability. Thirdly, two parameter variations that the friction coefficient k ω : 0.01 → 0.02 and the moment of inertia J: 0.005 → 0.01 are exerted to evaluate the robustness of proposed controller. Finally, an external step load T L which is changed from 1 Nm to 9.55 Nm is applied at steady state to verify the anti-disturbance performance of the proposed controller. The speed response ω, speed error z 1 , phase flux linkage ψ p , phase currents i p and control effort u RANN are shown in each subplot.
With λ = 0 and ε m = 0.1, Figure 5(a) shows that the maximum transient speed error is approximately 79 r/min. Small ε m lead to a large speed error. In Figure 5(b), the maximum speed error is obviously lower because large ε m caused a compensation. However, the larger ε m also caused undesired chattering of the control input u. Therefore, it is difficult to achieve satisfactory results without introducing proportional term. Figure 5(c) presents the simulation  results of the proposed RANN control with λ = 0.02 and ε m = 0.1. It can be seen that the RANN controller after introducing proportional term not only has small transient speed error but also effectively reduces chattering. Figure 6(a,b) depict the simulation results of the proposed controller and conventional PI controller at speed command 1500 r/min. In Figure 6, the maximum speed error of the proposed controller and PI controller are 41.01 and 45.55,r/min, respectively. And the steady-state speed error of the proposed controller and conventional PI controller are 0.4 and 0.7 r/min, respectively. The simulation results of the proposed controller and conventional PI controller at speed command 100 r/min is shown in Figure 7(a,b). In Figure 7, the maximum speed error of the proposed controller and conventional PI controller are 40.09 and 45.57 r/min, respectively. The steady-state speed error of the proposed controller and conventional PI controller are 1.13 and 1.22 r/min, respectively. From Figures 6 and 7, in addition, we can see that the settling time of the proposed controller is much less than that of the conventional PI controller. The results indicate that the proposed RANN controller has better speed tracking capability compared with the conventional PI control over a wide speed range. Figure 8(a,b) plot the simulation results of the proposed RANN controller and conventional PI controller under parameter variations in J and k ω , respectively. As shown in Figure 8(a), the maximum speed error and the steady speed error of the proposed controller are 44.52 and 0.27 r/min, respectively. Figure 8(b) presents the simulation results of the conventional PI controller, where the maximum speed error and the steady speed error are 51.68 and 0.65 r/min, respectively. Figure 8 indicates that the proposed controller can track the speed reference vary well in the presence parameter variations. That is because that the proposed RANN controller has a good adaptive capability for parameter variations. Figure 9(a,b) present the simulation results obtained using the proposed controller and conventional PI controller at 1500 r/min command speed with a step external load T L = 1 Nm → 9.55 Nm applied in 1.5 s. Figure 9(a) shows the simulation results of proposed controller, where the maximum speed dip is 50.92 r/min, and the speed error backs to the range  of 0.5 r/min only through 0.3 s. Figure 9(b) presents the simulation results of the conventional PI controller under the same conditions, where the maximum speed dip is 51.34 r/min, and the speed error recovers to the range of 0.5 r/min through 1.0 s. The results indicate that the proposed controller has the better robustness against the external load disturbance than the conventional PI controller.
In general, the proposed control scheme has a excellent speed regulation performance, especially when the parameters and external load change, still can maintain high performance speed control. Hence, the  effectiveness of the proposed RANN controller is verified by simulation. The proposed control scheme will be implemented on a practical SRM to demonstrate its practicality in the following section.

Experiments
In this section, the experimental results are presented with a three-phase 12/8 poles 1.5 kW SRM. Detail motor parameters are listed in Table 1. The SRM test bench is shown in Figure 10. A magnetic particle brake acts as a load on the SRM test system. The desired load torque can be obtained by regulating the excited current of magnetic particle brake. Control algorithm and data logging are programmed in DS1103 which produced by dSPACE company. The SRM is driven by an asymmetrical half-bridge dc-dc converter, which consists of six IGBT gates. Hall-effect sensors are used for current and voltage measurements, respectively. The sampling frequencies in the experimental study are set to be 1 and 10 kHz for outer loop (speed control) and inner loop (current control), respectively. The experimental data were gathered and processed by computer. All of the reference speeds in experiments are set for with 1000 r/min/s slope.
The investigations were performed as follows to demonstrate the wide range speed tracking capability and the robustness against the external load disturbance of the proposed RANN controller. Firstly, the proposed RANN controller is operated at two speed   references (100 and 1500 r/min) to confirm the full range speed tracking capability. Secondly, for testing the robustness, the proposed controller is started with load T L = 1 Nm, and an external load T L = 5 Nm is applied at 2.5 s when the system operated in steady state. In the experimental studies, the conventional PI control is also carried out in the same condition to compare their speed performance with proposed RANN controller. Figures 11 and 12 present that the experimental results obtained at 100 and 1500 r/min reference speeds  with 1 Nm external load. The speed response ω and speed error z 1 are shown in each subplot. Figure 13 shows that the experimental results for the control system with external step load T L = 1 Nm → 9.55 Nm applied at 2.5 s in the steady state. The speed response ω and the speed error z 1 of the proposed controller and conventional PI controller are shown in Figure 13(a,b), respectively. In order to quantitatively evaluate the control performance of the proposed RANN controller and PI controller, the experimental RMSEs are summarized in Table 2. The performance index, RMSE, is defined as follows: where N is the number of the sampled points. Table 2 clearly indicates that the RANN controller is superior than the conventional PI control under different operational speeds. Moreover, it is observed from Figure 13, that the speed dips of the proposed controller and conventional PI controller are all approximately 38 r/min. But the measured speed of proposed controller recovers the speed reference (1000 r/min) more quickly than conventional PI controller. This experimental results indicated that the proposed controller has a good robustness to against the external disturbance.
In summary, the performances of conventional PI control are unsatisfactorily in a wide range of speeds, parameter variations, especially in the case of external load disturbances. In contrast, the proposed RANN controller exhibits the similar satisfactory control performance in each case. This is because that the conventional PI control with fixed parameters is weak in resisting external disturbance and has not the adaptability to parameter variations, whereas the proposed RANN controller has the adaptability and the better robustness due to exploiting the universal approximation ability and the on-line learning ability of adaptive NN. In the proposed control scheme, the parameter variations and external load disturbances of the SRM drive system can be compensated by the adaptive RBF NN.

Conclusions
In this article, a robust adaptive RBF NN control scheme has been developed for SRM drive system with parameter variations and external load disturbances. The desirable control law including parameter variations and external load disturbances is approximated by using an RBF NN. The proposed controller is derived in the sense of Lyapunov function, and thus, the control system can be guaranteed to be semi-global asymptotic stability. The adaptive law of the proposed controller is derived by combining MLP technique, which makes only one parameter that needs to be updated on-line for n-nodes NN. As a result, the proposed controller is very easy to be implemented in digital processor. The comparative studies are carried out by simulations and experiments to confirm the control performance of the proposed controller. The simulation and experimental results demonstrate that the proposed controller has superior adaptive ability for parameter variations and robustness to against external disturbance. In the future research, we will consider the four quadrant speed control and improve the efficiency of SRM drive system.

Disclosure statement
No potential conflict of interest was reported by the authors.

Funding
This work was supported by National Natural Science Foundation of China [grant number 51407021].