Suspension Strategy of Maglev Vertical Axis Wind Turbine Based on Sliding Mode Adaptive Neural Network Predictive Control

The magnetic levitation system of the maglev vertical axis wind turbine is presented in this paper. The design and implementation of the magnetic levitation controller are discussed, and the nonlinear mathematical model of the magnetic levitation system is established. However, the magnetic levitation system is extremely susceptible to disturbance. To suppress the external disturbance and parameter perturbations, a sliding mode adaptive neural network predictive control method is presented, which is composed of the sliding mode control, an adaptive neural network and a model predictive control. The sufficient simulation and experimental results show that the proposed suspension method reduces the impact of the external disturbance and improves the dynamic response speed.


I. INTRODUCTION
The wind power generation in low-wind-speed areas is increasingly on the agenda with the development of energy strategies, and low-wind-speed wind power is gradually attracting attention [1], [2], [3]. Unfortunately, low-windspeed areas often have the disadvantage of frequent changes in wind speed and wind direction; thus the application of vertical axis wind turbines (VAWTs) has received extensive attention. Meanwhile, the unavoidable friction during the rotation of VAWTs essentially reduces the wind energy utilization coefficient. For this reason, it is a good method to introduce a magnetic levitation system (MS) into VAWT [4], [5]. However, the MS is a fragile and weakly damped system, while the VAWT needs some damping to maintain stable operation during the rotation process. Therefore, how to effectively reduce the power consumption, adjust the damping reasonably, and ensure the stability of the system require additional research. For this reason, we propose a novel maglev vertical axis wind turbine (MVAWT) that The associate editor coordinating the review of this manuscript and approving it for publication was Gerardo Flores .
adopts the maglev disc motor as the MS and reduces the suspension air gap to 1-2mm, which can effectively control suspension and damping. Therefore, the stable operation of MS is fundamental in MVAWT research.
Nevertheless, due to the characteristics of high nonlinearity, strong coupling, and inherent instability of the MS [6], [7], [8], [9], [10], it is a challenge to achieve stable suspension, especially as the strong disturbance in low-windspeed areas will bring great challenges to MS. At present, the traditional suspension strategy employs the linear control theory to construct the control law [11], [12], [13], [14], [15], [16]. However, as the operating point often changes and deviates from the operating point, effective control cannot be obtained for this MS. Hence, some intelligent nonlinear control strategies, such as sliding mode control [17], [18], neural network control [19], [20], and fuzzy logic control [21], [22] were introduced into the suspension strategy and achieved corresponding performance improvements. Sun et al. [18] proposed an adaptive sliding mode control for MS based on the RBF neural network, but the chattering problem was not well solved. Wai and Lee [19] presented an adaptive neuro-fuzzy control scheme based on the state observer, but the system convergence speed was not discussed. Su et al. [22] presented a fuzzy control scheme with the Takagi-Sugeno model for MS, but the nonlinear model of the MS was simplified and the experiments were not provided to certify it. Although the above-mentioned control methods have better control performance, they are not conducive to engineering implementation because they require large calculation time. In addition, they have no constraints on the input and output states, which makes the robustness of the MS poor and threatens the overall safe operation when the suspension air gap is small. For the MVAWT studied in this paper, reliability, robustness and rapidity are the most important performance indicators. Therefore, model predictive control (MPC) with constraint optimization, simple structure, and multi-objective optimization becomes an ideal suspension strategy, especially as it has been widely employed in the field of power electronics in recent years [23], [24], [25], [26], [27], [28], [29]. In [27], the MPC based on two-level state feedback was designed to ensure the safe and reliable operation of MS. The simulation results showed that this method has better air-gap control ability. In [28], the generalized MPC method based on input and output data was proposed to adjust the parameters of the controller and ensure the system stability while suppressing the vibration caused by the elastic track. In [29], the application of MPC based on a linear parameter-varying model was presented to control the active magnetic bearing subject to input and state constraints. The simulation analysis verified the effectiveness of the proposed control method, but the application of this method required linearization of the magnetic levitation system. However, the suspension strategies based on the MPC in the above studies were all linearized, and the nonlinear characteristic of the MS was ignored, while most of them adopt one-step MPC, which greatly restricts the application of MPC in MS.
To address the above problems and reduce the influence of external disturbance and chattering on the MS, a sliding mode adaptive neural network predictive control method (SMANNPC) composed of cascade control strategy and an additional auxiliary control is proposed. In the cascade control method, the adaptive neural network predictive control (ANNPC) is applied to the outer loop of suspension pressure and the PID controller is applied in the inner loop of suspension current. The adaptive neural network (ANN) is utilized to approximate the nonlinear MS, which can establish the suspension neural network (SNN) model. Next, the MPC is designed to obtain the optimal control output in the finite time domain, which significantly improved the control performance and realize multi-step prediction. Its global stability with the tracking error converges to zero is proven using the Lyapunov method. Moreover, the auxiliary controller is constructed to compensate for the performance degradation due to model mismatch. Sufficient simulation and experimental results are exhibited to demonstrate better practical performance and anti-disturbance performance. The contribution of this paper lies in three aspects. First, the MVAWT with suspension pressure as suspension objective utilizes the hybrid excitation method and smaller suspension air gap to drastically reduce power consumption. Second, a new suspension control method is proposed, which can ensure that the control system can still obtain satisfactory control performance under harsh conditions. Third, the experimental results show that the proposed method can achieve better control performance than existing methods and show superior robustness.
The remainder of this paper is organized as follows: the dynamic model is established in Section II. In Section III, the suspension control method is presented. Section IV and Section V provide the simulation and experimental results to demonstrate the control performance of the proposed magnetic levitation system. Lastly, the conclusion is presented in Section VI.

II. THE DYNAMIC MODEL OF MS
As shown in Fig. 1(a), the MVAWT mainly contains a permanent magnet synchronous generator, MS, magnetic levitation converter, pressure sensor, etc. The MS consists of a maglev disc motor made up of disc stator and rotor. The disc stator is a hybrid excitation magnet that contains a permanent magnet and electromagnet. As shown in Fig. 1(b), the suspension force can be regulated by adjusting the excitation current of the electromagnet via the magnetic levitation converter based on a four-quadrant H-bridge chopper circuit. The equivalent magnetic circuit model of the hybrid excitation magnet is shown in Fig. 1(c).
The equivalent magnetic circuit model of the hybrid excitation magnet is shown in Fig. 1(c), where R 0 and R 2 denote the air gap reluctance, R 1 and R 3 denote the internal reluctance of the permanent magnet, H c and L p are the coercive force and thickness of the permanent magnet, respectively, N is the number of turns of the electromagnet coil, mg is the weight of the rotating body of MVAWT, F is the suspension force generated by the hybrid excitation magnet, f d (t) is the external disturbance exerted on the MS.
The total magnetic flux can be expressed as follows where i(t) is the excitation current of the electromagnet (or called suspension current). As shown in Fig. 1(c), the structure of the hybrid excitation magnet is symmetric at both ends; thus, R 0 = R 2 = δ/(µ 0 S), R 1 = R 3 = L p /(µ r µ 0 S), in which, µ 0 is the vacuum permeability, µ r is the relative permeability, S is the magnetic pole area, and δ is the air gap length, i.e. the distance between the disk stator and rotor of the maglev disk motor.
According to the energy balance method, and by substituting (1) into, the suspension force can be obtained as follows where, W is the magnetic field energy in volume V , B is the air gap flux density, and H is the air gap magnetic field intensity.

The mechanical equation of MS in the vertical direction is
where, P is the resultant force exerted on the MS in the vertical direction, i.e. the pressure exerted on the pressure sensor by the rotating body of the MVAWT. The excitation voltage u(t) of the electromagnet is where, R and (t) are the resistance and flux linkage of the electromagnet coil, respectively. According to (2) -(4), the dynamic model of MS can be obtained as follows

III. DESIGN OF SUSPENSION CONTROL STRATEGY
The traditional linear control method cannot suppress external disturbance satisfactorily; hence the MS cannot obtain satisfactory control performance. Therefore, to solve this problem, a suspension control strategy shown in Fig. 2 is designed in this section. The ANNPC is applied to the outer loop of suspension pressure, and the PID controller is applied to the inner loop of suspension current. The output signal of the outer loop of suspension pressure is the input current reference of the inner loop of suspension current. Moreover, the auxiliary controller is constructed to compensate for performance degradation due to model mismatch.

A. THE AUXILIARY CONTROLLER
The suspension performance can be deteriorated due to model mismatch and uncertain model parameters. Therefore, an auxiliary controller is designed to compensate for the performance degradation problem of ANNPC caused by the model mismatch. According to the dynamic mathematical model of the MS, the derivation of the pressure P can be obtained as follows: Then, the sliding mode surface is designed as S c 1 e m , where c 1 ∈ R + is the positive gain, and e m = P * − P is the tracking error, P * is the expected suspension pressure value of P, and e m is ensured to converges 0 when MS is on the sliding mode surface (i. e., S = 0).
The derivation of the sliding mode surface S can be calculated as follows: The exponential reaching law is selected asṠ = −νS − tanh(s), then, the sliding mode variable structure control law is where ν and ∈ R + represent the constant reaching coefficient and exponential reaching coefficient, respectively. tanh(s) = e ιs −e −ιs e ιs +e −ιs is hyperbolic tangent function that is employed instead of sgn(s) to reduce chattering, and ι ∈ R + is the positive gain.
In order for the auxiliary controller monitor the system state and make up the system deviation in time, the evaluation function is selected as = e m +ė m . Then, the auxiliary control law is designed as follows: The auxiliary controller can adaptively detect the system state and make up for system errors caused by model mismatch or large external disturbances in time.

B. THE SNN MODEL
In order to improve the control performance of the MS and avoid linearization when MPC is used as the optimized controller, the ANN is employed to establish the SNN model.

1) CONSTRUCTING THE SNN MODEL
The inputs of the input layer is The hidden layer comprises eight neurons; its input s j (k) and the output y j (k) of the j-th neuron are where ω ij (k) is the connection weight between the ith input vector and the jth neuron, θ j is the bias vector of the jth neuron, and f 1 (·) = tanh(·) is the activation function of the output layer.
The output layer involves one neuron; its input s(k) and output y(k) are where ω j (k) and ξ denote the connection weight and bias vector of the output layer, respectively, f 2 (·) is the activation function of the hidden layer, and P m (k + 1) is the response output value of SNN model.

2) TRAINING THE SNN
The specific steps of using the ANN algorithm to establish the SNN model are as follows: i) Initialize the connection weights ω ij (k) and ω j (k), and pass the input vector x i (k) to the input layer successively.
ii) Calculate the output y j (k) and y(k) by going forward, and compute the error as e(k) = y(k) − y r (k) = P m (k + 1) − P(k), where y r (k) is the reference value to y(k).

3) CONSTRUCTING THE UPDATE LAW OF CONNECTION WEIGHTS
For the given input x i (k), the weight vectors ω j (k) and ω ij (k) of the ANN are updated as follows: where q(k) = y r (k)/(8ω j (k)), and g 1 (·) = f −1 1 (·). The discrete Lyapunov function is defined as V (k) = e 2 (k) to validate that this update law can build the SNN model. The change in the Lyapunov function V is Substituting (12) into (14), the error e can be obtained as (15) and (13), there is V (k) = −e 2 (k − 1) < 0. Remark 1: By constructing the above updated law, the error between reference output y r (k) and actual output y(k) can quickly converge to 0.
To prevent the singularities caused by the zero value of the q j (k), x i (k) and ω j (k), the weight update laws ω j (k) and ω ij (k) can be modified as follows The values of ϑ 1 , ϑ 2 , and ϑ 3 are the small positive numbers.

C. THE ADAPTIVE NEURAL NETWORK PREDICTIVE CONTROLLER
The role of the optimizer is to obtain the optimal control output, i.e. the reference suspension current i opt , through the rolling optimization of suspension cost function J (k) with the following constrained finite-horizon optimization, which is selected as follows: + 1), . . . , i(k + N u − 1)] T is the future control increment. α and λ are the pressure weighting factor and the current weighting factor, respectively; N p is the prediction horizon, N u is the control horizon (N p = N u = d), and d means the prediction steps.
To obtain and maintain the actual optimal control and reduce the number of iterations required to achieve the optimum, the Newton-Raphson method is utilized as the optimized controller, and the predictive value of the suspension current at the (k + 1)-th moment is where (k) and (k) are the Jacobian matrix and Hessian matrix, respectively, and i(k +1) is just the suspension current reference i opt (k). By calculating the current deviation e i = i opt − i, the control law of suspension current inner loop can be obtained as follows: (18) where, K p , K i and K d are the proportional coefficient, integral coefficient and differential coefficient, respectively. By taking the first-order and second-order derivatives of (11), respectively, (k) and (k) can be obtained as follows: The proposed SMANNPC consisted of an auxiliary controller and an ANNPC. Hence, according to Equations (9) and (18), the control law of SMANNPC can be obtained as follows:

D. STABILITY ANALYSIS
The stability performance is essential to the successful application of SMANNPC [30][31]. In this section, the stability of SMANNPC is intensively investigated by employing terminal constraints of MPC. Theorem: Consider the constrained finite-horizon optimal control problem represented by Equation (16), and the control law designed as Equation (17); then, the asymptotic stability of the proposed SMANNPC can be guaranteed.
2) The i(k) = [i(k), i(k + 1), . . . , i(k + N u − 1))] T is postulated as the optimal control found by the optimization procedure at the k-th moment. Now, let us present the suboptimal control i s (k + 1) assumed at the (k + 1)-th moment. The control sequence i s (k+1) is obtained on the basis of the control at time k. Hence, for the suboptimal control i s (k + 1), the cost function can be defined as follows: Then, the difference in cost function J s (k + 1) and J (k) can be calculated as follows (23) Therefore, by taking (23), Theorem can be proven. Remark 2: As the proposed SMANNPC requires solving a constrained nonlinear optimization problem, the problems of feasibility and stability are extremely important. The feasibility and stability of the SMANNPC are proven by Theorem. The tracking performance and control accuracy of MS will be improved as the number of prediction steps of SMANNPC increases. However, it will increase the time consumption of MS and cause the dynamic response performance to deteriorate. Therefore, in order to ensure the overall performance of MS, two prediction steps are selected and compared with the traditional one-step prediction.

IV. SIMULATION ANALYSIS
A simulation platform was built to verify the control performance of the proposed control method. The ANNPC and PID controller were applied to the outer loop of suspension pressure and the inner loop of suspension current of the SMANNPC-PID, respectively, while the auxiliary controller output was superimposed on the cascade control output. The SMANNPC-PID was compared with ANNPC2-PID, ANNPC1-PID, and PID-PID in this simulation experiment. The parameters of the MS are listed in Table 1; among them the mass m of the rotating body of MVAWT is 130kg, the suspension force generated by the permanent magnet of the hybrid excitation magnet is 1014N, and the initial pressure is about 260N. The expected suspension pressure of the MS is set to 50N. The switching frequency of the MS is 10kHz.   The control parameters of the proposed method can be divided into two, one is the parameters for the ANNPC and PID control, and the other is the parameters for auxiliary controller. After sufficient tuning, the parameters of ANNPC and PID are obtained: The other are auxiliary control parameters, which are selected as: c 1 =10, ν=5, =10, ι=1, a=0.5.

A. SIMULATION FOR DYNAMIC RESPONSE PERFORMANCE
The control effects of SMANNPC-PID, ANNPC2-PID, ANNPC1-PID, and PID-PID were compared to examine the VOLUME 10, 2022  dynamic response performance of the proposed suspension strategy.
It is shown in Fig. 3 that the system with the PID-PID had a settling time of more than 3s. The system with the ANNPC1-PID needed more than 2s and showed vibrations in the results. The system with the ANNPC2-PID had a settling time of more than 1.3s, and the system with the proposed SMANNPC-PID had a settling time of less than 0.6s. The static error in the SMANNPC-PID was smaller than that in the ANNPC2-PID. Furthermore, compared with ANNPC1-PID, ANNPC2-PID, and PID-PID, the dynamic response speed of SMANNPC-PID was increased and the suspension current was much smoother. This shows that the proposed control method has stronger control capability and faster tracking performance.

B. SIMULATION FOR ANTI-DISTURBANCE PERFORMANCE
In order to verify the robustness and anti-disturbance capability of the proposed suspension strategy, the nonlinear periodic disturbance f d (t) = 100sin(2t − 100) + 50sin(4t − 50) was added to MS at 4s and removed at 17s. The simulation results under this disturbance f d (t) are shown in Figs. 5-6.
As shown in Fig. 5, the suspension pressure of PID-PID changed constantly with the variation of external disturbance, which had a maximum static error of more than 8N. The suspension pressure fluctuation of ANNPC1-PID exceeded 5N and that of ANNPC2-PID was more than 3N. Meanwhile, the suspension pressure fluctuation of SMANNPC-PID was less than 1N. Moreover, when the external disturbance was removed, the static error of PID-PID still existed, and ANNPC1-PID had a suspension pressure  fluctuation of more than 5N. The suspension pressure fluctuation of ANNPC2-PID was more than 3N, while that of ANNPC2-PID was less than 1N and could be restored to its steady state within 0.2s. As shown in Fig. 6, the SMANNPC-PID had the maximum suspension current fluctuation of 10A, while the PID-PID had the minimum suspension current fluctuation of 9.5A. Therefore, when MS is subjected to external disturbance, the SMANNPC-PID can better restrain the disturbance and improve the robustness of the MS.

V. EXPERIMENTAL RESULTS
In order to further examine the practical performance of the proposed control method, a physical prototype of the 1kW MVAWT was developed, and the experimental platform shown in Fig. 7 was built. On this basis, this paper adopted dSPACE 1202 and the maglev converter as the suspension controller. The expected suspension pressure was set to 180N with an initial value of 240N, the control frequency was 10kHz, and the supply voltage was 36V. Two sets of experiments were carried out to validate the tracking performance and the anti-disturbance capability.
For the experiments, the control gains and parameters are carefully determined as follows:

A. EXPERIMENT FOR DYNAMIC RESPONSE PERFORMANCE
An experiment was employed to examine the dynamic response performance of the SMANNPC-PID, ANNPC2-PID, and PID-PID. The initial position of the suspension pressure was 240N, and, from the time t = 2s, the corresponding control laws were utilized to drive the MS to the reference position of 180N. In the PID-PID, the PID control parameters of outer loop of the suspension pressure were 100, 50, and 100, respectively. And the PID control parameters of inner loop of the suspension current were 14, 5, and 10, respectively. The experiment results are shown in Figs. 8-9. It can be seen from Fig. 8 that the system with the SMANNPC-PID method had a settling time of less than 0.5s without obvious suspension pressure fluctuation. The system with the ANNPC2-PID method had a settling time of more than 7s and the suspension pressure fluctuation is less than 7N, while, for that with PID-PID, the settling time was more than 4s and the suspension pressure fluctuation was more than 10N. Compared with PID-PID and ANNPC2-PID, the SMANNPC-PID could reach a stable state in the shortest time and suppress overshoot well due to the synergy between the auxiliary controller and ANNPC. Moreover, it is shown in Fig. 9 that the system with the PID-PID method had a suspension current fluctuation of more than 1.5A and a current ripple of more than 1A. Although the suspension current fluctuation of ANNPC2-PID was less than 1A, it had a current ripple of more than 2A. In contrast, the system with the SMANNPC-PID had no obvious suspension current fluctuation and had a current ripple of less than 0.7A. Compared with ANNPC2-PID, the SMANNPC-PID can effectively reduce the suspension current fluctuation of the MS while quickly reaching the desired position, which can improve the working efficiency of the system. These experimental results demonstrate that the proposed control method had faster dynamic response capability and better tracking performance.

B. EXPERIMENT FOR ANTI-DISTURBANCE PERFORMANCE
An experiment was utilized to examine the anti-disturbance performance of the three control methods. A nonlinear  external disturbance with amplitude of 30N was added at 14s and removed at 21s. The experimental results are shown in Figs. 10-11. As shown in Fig. 10, when external disturbance was added, the system with the SMANNPC-PID method had a suspension pressure fluctuation of less than 6N, and the settling time was less than 2s. The suspension pressure fluctuation of the ANNPC2-PID method exceeded 8N with a settling time of more than 3s. In contrast, the PID-PID method had a suspension pressure fluctuation of more than 10N and required a settling time of more than 5s. Although PID-PID can suppress the external disturbance, it had large suspension pressure fluctuation and takes more time to restore the original equilibrium position. In addition, both ANNPC2-PID and SMANNPC-PID could restore the equilibrium state in a shorter time, but SMANNPC-PID had stronger antidisturbance ability and faster settling time. It is clear in Fig. 11 that, compared with ANNPC2-PID and PID-PID, the suspension current of SMANNPC-PID could react quickly in a short time when external disturbance was applied to the MS.
The ANNPC2-PID could use the ANN to establish an accurate SNN model, when the system deviated from the equilibrium position, it could reach the new equilibrium position quickly. However, the fast response of the system leaded to a large suspension current ripple, which may cause chattering in the system and will degrade system performance. While in SMANNPC-PID, due to the additional compensation effect of the auxiliary controller, the system could reach the steady state in a shorter time without obvious suspension pressure fluctuation and suspension current ripple. Since the integral term was added and the continuous hyperbolic tangent function was used instead of the sign function as the switching control law in the auxiliary controller of SMANNPC-PID, when the external disturbance was added to this system, the proposed SMANNPC-PID could quickly suppress the external disturbance and reduce the suspension current ripple.
The numerical values are summarized in a statistical table in Table 2. By summarizing the numerical values in Table 2, it can be seen that the presented suspension control method can achieve better control performance and better disturbance rejection capability under the condition of smooth control current and external disturbance.

VI. CONCLUSION
This paper studied the MS in MVAWT. To improve the suspension control performance of MS, the SMANNPC was presented using an auxiliary controller, an adaptive neural network, and a model predictive control to handle external disturbance and unmatched uncertainties in the system. The stability of the proposed control method was proven by the Lyapunov theorem. Compared with the PID controller and ANNPC, the simulation and experimental results show that the proposed nonlinear robust controller had a non-overshooting, robust, fast dynamic response and could effectively reduce the impact of external disturbance on the system. Furthermore, it could solve the problems of external disturbance and parameter perturbation with a faster adjustment speed while meeting the control quality requirements of MVAWT.
In the nonlinear systems with external disturbance and parameter perturbation, the SMANNPC can adaptively adjust the system parameters, showing better control performance. However, when this control method is applied to other nonlinear systems, the control parameters need to be readjusted according to the actual working conditions, which will take more time and reduce work efficiency. Next, we will study SMANNPC further, so that it can be easily generalized to other nonlinear systems.
YIXI CHEN was born in 1997. He received the B.S. degree in electrical engineering and its automation from Qufu Normal University, Rizhao, China, in June 2019, where he is currently pursuing the M.S. degree in control science and engineering with the College of Engineering.
His current research interests include renewable power generation, power electronics, and magnetic levitation technology.
YANG ZHANG received the B.S. degree in automation from Qufu Normal University, Rizhao, China, in June 2019, where she is currently pursuing the M.S. degree in control science and engineering with the College of Engineering.
Her research interests include containment control, event-triggered control, and nonlinear multiagent systems control.
BIN CAI received the Ph.D. degree in electrical engineering from Xi'an Jiaotong University, Xi'an, China. He is currently a Professor with the College of Engineering, Qufu Normal University, Rizhao, China. His research interests include wind power generation and the applications of artificial intelligent in the renewable energy control fields, power electronics and HVDC-flexible transmission technology. VOLUME 10, 2022