Adaptive ELM neural computing framework with fuzzy PI controller for speed regulation in permanent magnet synchronous motors

The growth in magnetic materials and semiconductor power devices has made the permanent magnet synchronous motor drive highly significant in various control applications. Basically, a permanent magnet synchronous motor (PMSM) is an asynchronous motor with its field excitation carried out with permanent magnet and possesses a sinusoidal back electromotive force (EMF) waveform. Employing permanent magnets, these motors are likely to generate torque at zero speed. As well, they have smaller frame size for the equal power developed using induction motors. The prominent characteristics of PMSM include: clean, fast, more reliable, less noise, no sparks, possibility for high-performance servo applications, more compact, lighter than induction motors, low electromagnetic interference and more efficient. Speed regulation is an important characteristic to be maintained in respect of PMSM so as to employ it effectively for various servo and control applications. Nevertheless, the performance and efficiency of PMSM are highly affected based on the external load disturbances and also on the parameter deviations of the plant model. This has been handled over the years with certain sliding mode control techniques, nonlinear control approaches and machine learning controller models and to perform an efficient speed control of PMSMs (Hu et al. 2019; Dursun et al. 2018; Tarczewski and Grzesiak 2018).

Related works carried out in speed control of PMSM using various controllers are presented in this section. An adaptive robust finite-time neural control scheme has been proposed for uncertain permanent magnet synchronous motor servo system employing nonlinear dead zone input. Adaptive finite-time controller was designed based on a fast terminal sliding mode control principle (Chen et al. 2017). Novel design of adaptive neuro fuzzy inference system (ANFIS) and modified particle swarm optimization (MPSO) technique-based proportional–integral–derivative (PID) speed controller has been incorporated in PMSM drive to improve its dynamic performance (Yadav and Verma 2017). Direct torque control (DTC) based on artificial neural network (ANN) of a five-phase permanent magnet synchronous motor drive (PMSM) has also been developed (Kamel et al. 2017). In order to solve the problems of stochastic disturbance and input saturation existing in permanent magnet synchronous motors (PMSMs) drive systems, a command filter-based adaptive neural control method has already been proposed (Han et al. 2018). For control of PMSM, adaptive position estimators are required as the parameters of the machines like rotor resistance and inductance changes sometimes and a work was developed with ANFIS- and ANN-based position estimator in a field-oriented control of a permanent magnet synchronous motor drive (Tabrez et al. 2018). For practical applications of PMSM, the switching gain and the value of boundary layer thickness are difficult to select in the complementary sliding mode control. Hence, the Elman neural network estimator was used to estimate the value of the uncertain factors, instead of the switching control in the sliding mode control (Jin and Zhao 2019). An adaptive speed controller based on artificial intelligent technique has been modelled to improve the performance of classical direct torque control (DTC) for permanent magnet synchronous motor (PMSM) drives. Here, back propagation (BP)-based neural network (NN) was employed to tune the parameters of classical proportional–integral (PI) speed controller (Jia and Kim 2018).

In a work, based on the combination of particle swarm optimization (PSO) algorithm and neural network (NN), an adaptive neural network internal model control (NNIMC) has designed for a permanent magnet synchronous motor (PMSM) (Frijet et al. 2018). Permanent magnet synchronous motors become popular in wind turbines and industrial applications. In critical machines, it is necessary to use robust condition monitoring and fault diagnosis algorithms to prevent faults or shutdowns. So, a deep auto encoder-based unsupervised learning method was developed to identify the features of the fault classification algorithm in a self-supervised way, which overcomes the shortage of labelled data (Senanayaka and Robbersmyr 2018). Several control schemes such as fuzzy logic control (FLC), neural network (NN) and sliding mode control have been preferred for speed control of the PMSM. FLC and NN algorithms have been used as effective methods in the control of system affected by the destroyed entrance which is unknown quality and unstraight line (Sakunthala et al. 2017). Comparison among different techniques of speed and position estimation by using proportional and integral controller (PI) tuned by using heuristic, MATLAB PID auto tuner and particle swarm optimization (PSO) method of sensor-less control for PMSM has also been done (Nazelan et al. 2018). Also, an online-trained NN-PI (neural network–proportional–integral) speed controller has been employed in an SVM-DTC (space vector modulation–direct torque control)-based interior permanent magnet synchronous motor (IPMSM) drives (Jia and Kim 2018).

A work explored a new control method for permanent magnet synchronous motor (PMSM) based on a BP neural network and a high gain observer (Gao et al. 2018). Further, design of the rotor position controller of the permanent magnet synchronous motor (PMSM) servo control system has been proposed. A fuzzy neural network (FNN) position controller, which combines the capability of fuzzy reasoning in handing uncertain information and the capability of neural network in learning, was utilized (Liu and Chang 2018). A terminal sliding mode control (TSMC) based on the radial basis functions neural network (RBFNN) for the permanent magnet synchronous motor (PMSM) was done. The designed controller was composed of a RBFNN and a terminal sliding mode controller (Ge et al. 2018). In another work, based on the combination of particle swarm optimization (PSO) algorithm and neural network (NN), a new adaptive speed control method for a permanent magnet synchronous motor (PMSM) has been proposed (Zribi et al. 2018). A high-precision tracking control strategy of permanent magnet synchronous linear motor (PMSLM) has also been developed (Liu et al. 2018).

A work introduced multi-objective particle swarm optimization, based on decomposition and dominance (D² MOPSO) in order to design the permanent magnet synchronous motor fuzzy controller for different objects (Kao et al. 2017). Basically, performance of the permanent magnet synchronous motor drive was reasonably influenced by the speed controller design. For improving the drive’s performance, a proper and continuous tuning of the PID controller is required. Hence, to achieve this objective, and cope with inherent nonlinearity in PMSM drive, adaptive neuro fuzzy inference system (ANFIS) has been proposed as a replacement of conventional PID controller (Yadav and Verma 2018). The speed sensor-less field-oriented control (FOC), of permanent magnet synchronous motor (PMSM) fed by a space vector pulse width modulation (SVPWM), is always been of high importance. The characteristics for the PMSM are considered by two different algorithms: the first one is a model reference adaptive system (MRAS) and the second algorithm utilized adaptive neuro fuzzy inference system (ANFIS) (Salem et al. 2018). Artificial neural network-based control of speed for PMSM has been investigated in both open and closed loops under no-load and loaded condition (Tummala and Dhasharatha 2019). A work proposed a neural network model to improve performance when determining the rotor position in permanent magnet synchronous motors. This method measures the three-phase current at the motor and normalizes the readings for input to the neural network (Chang et al. 2019).

In a paper, an adaptive neural network (NN) control based on command filtered back stepping approach was presented for fractional-order permanent magnet synchronous motor (PMSM) with parameter uncertainties and unknown time delays (Lu et al. 2019). A position estimator for an outer rotor permanent magnet synchronous machine (PMSM) has been presented and evaluated. This proposed estimator employs a machine learning-based neural network algorithm to interpret the signals, which are obtained from linear Hall effect sensors located in the fringe field of the rotor (Wang et al. 2019). In a work, deep recurrent and convolutional neural networks with residual connections are empirically evaluated for their feasibility on the sequence learning task of predicting latent high dynamic temperatures inside PMSMs has been studied (Kirchgässner et al. 2019). A model reference adaptive system (MRAS) observer-based sensor-less control of permanent magnet synchronous motor (PMSM) has been proposed (Lin et al. 2019). Permanent magnet synchronous machines (PMSM) are widely used in the automotive industry for electric vehicle (EV) and hybrid electric vehicle (HEV) propulsion systems, where the trend is to achieve high mechanical speeds. A speed-adaptive control structure that overcomes these stability problems and extends the speed operation range of the PMSM has been presented (Arias et al. 2019).

The parameter identification of permanent magnet synchronous motor (PMSM) is an important and challenging task of power electronic systems, which has an important impact on the control performance of the drive system. Aiming at the parameter identification problem, an adaptive differential evolution algorithm based on hybrid mutation operator (SHDE) has been proposed in this work (Wang et al. 2019). A novel state-space version of the most prominent algorithm used with neural networks, namely the backpropagation algorithm by incorporating the knowledge of the state-space model, has been developed for PMSM control module (Bjaili et al. 2019). Nature-inspired optimization algorithm was developed for adaptive speed control of permanent synchronous motor (PMSM) drive with variable parameters. In the proposed approach, a state feedback controller (SFC) has been utilized for speed control of the PMSM, while online adaptation of its coefficients is made with the help of artificial bee colony (ABC) algorithm (Szczepanski et al. 2019). Operating in the high-speed range is necessary for high-performance permanent magnet synchronous motor (PMSM) drives. This work presented a new flux weakening scheme along with an improved vector control strategy to alleviate the influence of this problem. Control parameters of the anti-windup proportional and integral (AWPI) controller are optimized off-line in relying on an adaptive velocity particle swarm optimization (AVPSO) algorithm (Xu et al. 2019).

Considering the above related works carried out in speed control of PMSM with various heuristic and machine learning controllers, it is observed that each method possesses its own merits and limitations. To overcome the limitations of nonlinear dead zone and saturation occurrences and undue happenings of local and global minima, this paper is intended to develop a new hybrid soft computing framework with ABC, ELM and fuzzy controller to perform effective and efficient speed regulation of the permanent magnet synchronous motor. Advantages of individual soft computing models are hybridized to design a suitable speed controller for PMSM and to achieve an effective control strategy.

The rest of the paper is organized as follows: Sect. 2 presents the mathematical modelling of PMSM, and the developed new adaptive extreme learning machine neural model is detailed in Sect. 3. Section 4 elucidates the design of fuzzy controller with the developed new ELM neural model. Simulation and applicability of the proposed controller model for PMSM are depicted with attained results in Sect. 5. Section 6 provides a discussion on the simulated results and also a comparative analysis to validate the proposed work. Concluding remarks are presented in Sect. 7.

The PMSM drive comprises of a speed controller, current controller and current regulator, pulse width modulation inverter and a position encoder to sense the inputs. Figure 1 shows the basic block diagram of a PMSM speed controller system.

In respect of the block diagram shown in Fig. 1, the parameters used for mathematical modelling of PMSM speed controller system are as follows:

Considering the parameters given, ‘e’ and ‘de’ are the input variables to the fuzzy speed controller and it results in the control current ‘I_cc’. The stator voltage equations of PMSM are given by:

In Eq. (1), V_a, V_b, V_c are the phase voltages, i_a, i_b, i_c are the phase currents, e_a, e_b, e_c specify the electromotive forces (EMFs) of phase windings, and R_sr indicates the motor’s stator resistance. In this equation, the component ‘L_ss-M’ is equal to the synchronous inductance of the motor ‘L_sy’ and is given by:

In Eq. (2), L_ss represents the total phase inductance along with the leakage (L_l) and self- inductance (L_si). The mutual inductance that prevails in the phase windings is given by ‘M’. Equation (1) becomes:

EMFs generated by the permanent magnet are calculated with:where ‘ρ_fr’ represents the flux due to the permanent magnet rotor and θ_r indicates the measured rotor position. The generated electrical torque T_el is given by:

At last, the speed of the rotor and its position is given by:

Extreme learning machine neural network is a third-generation neural network wherein it is designed with only one single hidden layer and the number of hidden neurons present in the hidden layer gets adjusted by itself during the training process. In this paper, a new adaptive version of ELM is constructed wherein the weights and the fixation of hidden neurons in hidden layer are carried out adaptively using artificial bee colony algorithm. This modification in the classic ELM tends to increase the generalization ability and learning ability of the neural network model. This section presents the new adaptive ELM neural network model and the developed algorithm that is to be employed for training process in the design of the controller.

In this paper, Mamdani fuzzy PI controller is designed to replace the conventional PI controller whose input represents the decoded output from the new adaptive extreme learning machine neural architectural model. The fuzzy inference system (FIS) rules are formulated in Mamdani fuzzy PI controller to evaluate the proportional gain and integral gain and related performance characteristics based on the error and derivative of error. The fuzzy control mechanism in this paper is designed with the following steps—formulation of fuzzy membership values and rules to perform a decision making model, measurement of input data and making suitable decision based on the fuzzified value and the final output evaluation of defuzzified value with respect to the inference rules so as to calculate the proportional gain and integral gain and thereby the performance characteristics of the considered system. In Mamdani fuzzy inference model, fuzzy rule is formulated as:

In the above equation, the values ‘P’ and ‘Q’ represent the membership values of the error and derivative of error pertaining to the given inputs of the motor and these are the antecedents and the output ‘R’ is the crisp value of the control current ‘I_cc’ and is the consequents. In this work, the final output from the Mamdani fuzzy module pertains to a fuzzy single value component. The formulated rule in this case becomes:where Z(.) indicates the membership functions for the given inputs. Hence, in Mamdani fuzzy PI controller model, the final output of the system (control current I_cc) is evaluated to be the centroid of all rule outputs and is obtained using:

The developed algorithm using optimized adaptive extreme learning machine neural network-fuzzy model generates the error value and computes the control current for appropriate proportional gain and integral gain in respect of the PMSM model. The proposed ABC adaptive ELM fuzzy PI controller employs Mamdani fuzzy system model, and the developed rule set based on the error (e) and derivative of error (Δe) for the output functions is given by:where ‘n’ = 1, 2, 3,… and ‘i, j’ = 1,2,3,…. and α_n, β_n and γ_n specify the linear parametric constants. The layer-based computational process of new adaptive ELM fuzzy controller model is as given below:

Layer 1: In fuzzification process, Gaussian membership function is employed and membership values are assigned between 0 and 1 for both error and derivative of error. The membership assigned is given by:where \( {\text{MF}}_{li} \) and \( {\text{MF}}_{lj} \) represent the output of the nodes in the first layer and \( \mu_{i} X_{i} \) and \( \mu_{i} X_{j} \) are the membership functions of the Gaussian membership function. The Gaussian membership function is used in this work because of its effective distribution in the membership values.

Layer 2: Here, the product of all the incoming inputs is carried out. In this layer, rule formation is performed and the generalized form is given by:

Layer 3: In this layer, the rule activations are determined and compared with all the activation levels present. The firing strength of the rule is given by:

Layer 4: Using the fuzzy inference rule, the final output is obtained as:where α_i, β_i and γ_i specify the linear parameters that are computed from the learning rules.

Layer 5: The final defuzzified output is computed using the centroid method and is given by:

Based on the above ABC adaptive ELM fuzzy controller design, the final output control current with respect to the proportional and integral gains is calculated, and based on this the performance characteristics like peak time, peak overshoot, settling time and rise time are calculated by performing speed regulation of PMSM. Also, the new optimized ELM neural network model with its tuned weights and hidden neurons following gradient descent learning rule and sigmoidal activation function tunes the fuzzy membership functions and its rules. At the time of ABC-based adaptive ELM training process, the mean square error is evaluated and the learning process continues until the error reaches a possible minimum value. The new modelled ABC adaptive ELM fuzzy PI model enhances the generalization ability and learning capability and thereby empowers the neuro fuzzy model to learn and adapt by itself and attain convergence with better and optimal solutions. Figure 4 shows the model of a designed fuzzy PI controller for PMSM speed drive system.

The proposed optimized adaptive ELM neural network-based fuzzy PID controller is applied for speed regulation of permanent magnet synchronous motor. New adaptive ELM neural model is trained for the weight and bias coefficients and the number of hidden neurons using the ABC algorithm. The complete simulations are implemented MATLAB R2018a environment and are executed in a PC with Intel Core2 Duo Processor with 2.27 GHz speed and 2.00 GB RAM. The input to the new optimized ELM neural model is the error (e) and derivative of error (de), and the output from the developed model is the control current (du) represented by ‘I_cc’. During the progress of training, the developed new model tunes for the optimal values of proportional gain (k_p) and integral gain (k_i) and carries out effective speed regulation of the PMSM system. In respect of basic fuzzy inference system design, the fuzzy membership functions and the consequent rules are modelled by the user with the fundamental priori knowledge. After then, the new adaptive ELM model tunes for optimal membership and consequent rule parameters. Prior to the start of training mechanism, all the required linear and nonlinear parameters of the proposed fuzzy PI controller are required to be updated. Table 2 gives the simulation parametric values assigned for the proposed ABC adaptive ELM fuzzy PI controller model for the training process. The PMSM parameters employed for simulation process are presented in Table 3.

Initially, the training process of adaptive ELM network model is started with the inputs to be the error and its time derivative. For the first phase of training, the weights and bias coefficients are generated randomly and the training proceeds with gradient descent learning rule in evaluating the performance metric mean square error. The mean square error is evaluated using the following equation:

Figure 5 shows the new adaptive ELM fuzzy controller model designed to perform speed regulation of PMSM. On presenting the training input and output data into the optimized ELM model, then fuzzy membership functions and rules are to be formulated. Here tuning is done to determine the membership functions and rules by the optimized ELM neural network model. Adaptive tuning of weights, bias and placement of number of hidden neurons is done in ELM model with the ABC algorithm wherein suitable exploration and exploitation mechanism results in better values for fuzzy PI controller model. The entire algorithmic process is carried out until the considered performance metric mean square error comes to a near possible minimal value. The input and output membership values defined at the initial start of the fuzzy system design are shown in Fig. 6. Three linguistics positive (P), zero (Z) and negative (N) are assigned for the inputs error (e), derivative of error (de) and output control current (du). Table 4 shows the initial fuzzy rules formed using the defined fuzzy linguistics. The error varies between − 100 and + 100, error derivative varies between − 1 and + 1, and the output variable varies between − 200 and + 200.

The newly proposed ABC-based adaptive ELM neural network model is invoked for evaluating the optimal membership function and rule consequent sets of the Fuzzy PI controller. At the time of training, simultaneously the weights and bias of extreme learning machine neural network model get trained for their optimal values using the classic artificial bee colony algorithm. The proposed technique gets initiated and operates so as to update the membership values and rules of the fuzzy PI controller. The training error computed at the time of ABC ELM training in respect of the gain values k_p and k_i is shown in Fig. 7. The training error values for values k_p and k_i at the end of 100^th iteration are 1.96 × 10⁻⁴ and, respectively. Table 5 depicts the PI controller values evaluated during the algorithmic process and is the final optimized value. Figure 8 presents the surface plot of the control current value obtained using the optimized rules from adaptive ELM model.

Figure 9 shows the phase voltages of the PMSM drive when speed regulation is carried out. During the start of the motor, it accelerates from zero speed to 1500 rpm at no load and the current increases steadily to attain its rated value. Once the motor begins to run in normal condition, then the speed of the motor has attained to the required base speed and it attains the required frequency of the current. Here, the motor starting torque is sustained at a constant value. PMSM drive is given a sudden load of 6 N-m at t = 0.25 s to its shaft and it is noted that the rotor speed decreases. This is depicted in Fig. 10. The corresponding current and torque variations are shown in Figs. 11 and 12, respectively. Table 6 presents the performance measures for the optimized ELM fuzzy PI controller of PMSM drive system and that of the other methods considered for comparison.

It is inferred from Table 6 that the proposed optimized ABC-based ELM fuzzy PI controller developed for PMSM speed drive has attained the peak value of 1500 rpm and its rise time is as minimal in comparison with the other methods from the existing literature. This proves the effectiveness of the neural-based fuzzy PI controller for PMSM drive module for the load condition of 6 N-m. This is also inferred from Fig. 13. Figure 13 shows the step response characteristics of the proposed controller and that of the other controllers considered for comparison.

Artificial bee colony optimization algorithm is employed in this paper to locate optimal weight values and bias values for the new adaptive extreme learning machine neural network model. The algorithm operates in a manner so as to minimize the performance metric mean square error. In this case, the number of hidden neurons in ELM neural network model is also tuned for better value so that this also plays a role in minimizing the mean square error. Over the 100 iterations and 24 trial runs made, the optimal number of hidden neurons is identified to be 4 for better performance of ELM model. After then, with these optimal values the fuzzy PI controller part gets initiated and the training progresses in achieving better membership values and fuzzy rules to determine the control current of the PMSM speed drive system. Based on the inputs, error and error derivative the output control current was evaluated and this is shown in Fig. 12 for the proportional and integral gain values of 6.1296 and 1.000, respectively. ABC-based adaptive ELM fuzzy PI controller has designed the speed drive system with load of 6 N-m and has attained better performance in respect of performance measures for step response, gain values, speed variations and mean square error than that of the other methods considered for comparison from the literature (Zhao et al. 2019, Zhao et al. 2020; Deng et al. 2019; 2017a; b).

The validity of the proposed controller is investigated by carrying out statistical analysis of determining the correlation value and coefficient of determination. Table 7 shows the correlation value and coefficient of determination evaluated for the proposed ELM fuzzy PI controller, and both these values are observed to be nearer to 1, proving the efficacy of the proposed technique. The computational time incurred for the run of the proposed control algorithm for PMSM speed drive system is 121 s. Table 8 shows the mean square error evaluated during training and testing process, and at the end of 100th iteration it is 0.0005 and 0.0003, respectively, proving the effectiveness of proposed algorithm.

In this paper, a new optimized adaptive extreme learning machine neural network-based fuzzy PI controller is designed for a permanent magnet synchronous motor speed drive system. Basically, ELM neural model is a multi-layer neural structure with only one single hidden layer, and in this paper the weights and bias coefficients are optimized using the classic artificial bee colony algorithm. This optimization procedure is carried out in order to avoid the stagnation and delayed convergence of adaptive ELM neural network model. Adaptive ELM neural model with its optimal weights and bias integrates with the fuzzy PI controller to design an effective speed regulation for the PMSM drive system. The output control current is attained for the tuned proportional and integral gain values, and the speed response of the drive system proves its efficacy over the other methods compared from the existing literature studies. The detailed statistical analysis made also substantiates the validity of the developed research work. Minimized error values confirm the effectiveness of the developed model. The proposed method is as well validated for its applicability with the statistical analysis performed.