Predictive-adaptive sliding mode control method for reluctance actuator maglev system

The control performance of the reluctance actuator maglev system is seriously affected by inherent nonlinearities (e.g. hysteresis, eddy current, flux leakage, etc.) and external disturbances. To this end, this paper proposes an enhanced unknown system dynamics estimator (USDE)-based sliding mode control (USDE-SMC) method by a novel predictive-adaptive switching (PAS) controller (USDE-SMC-PAS). First, a USDE is incorporated into SMC to compensate for the uncertainties, including the external disturbance, the parametric uncertainty, and the model mismatch of inherent nonlinearities; second, an adaptive switching (AS) controller is used to reduce chattering; finally, the switching controller is modified by PAS to enhance the dynamic response ability and uncertainty rejection ability with reduced chattering. In the PAS controller, the switching gain comprises a filtered estimation error part and a residual error part, which are obtained by a first-order filter and an adaptive law, respectively. Consequently, the strong nonlinearity and uncertainty compensation ability, high levitation accuracy, high dynamic response, and reduced chattering property can be achieved simultaneously. The stability properties of USDE-SMC, USDE-SMC with an AS controller (USDE-SMC-AS), and USDE-SMC-PAS in the closed-loop system are investigated by the Lyapunov theorem. Simulations and experiments are provided to validate the effectiveness and superior performance of the proposed method.


Introduction
Reluctance actuators (i.e. electromagnetic actuators) have the advantages of large thrust and no contact, so they are widely used in magnetic levitation systems (MLSs) [1]. When the MLSs are used in semiconductor equipment, high levitation accuracy and high dynamic response are necessary for the control to satisfy the requirements of both the production quality and the production efficiency [2]. However, high-performance control of reluctance actuators is challenging due to inherent nonlinearities [3] (e.g. hysteresis, eddy current, flux leakage, etc.) and external disturbances.
There are many advanced methods in the control of MLSs, including output feedback control [4], linear active disturbance rejection control method [5], adaptive backstepping control method [6], model predictive control method [7], disturbance observer based control [8], adaptive fuzzy control method [9,10], and sliding mode control (SMC) [11] and so on. Among them, SMC is a method with simple structure but good control performance [12][13][14]. Recently, many SMC methods have been used for the control of MLSs, such as fractional-order SMC [15], PID-based SMC [16],highorder SMC [17], super-twisting SMC [18], and neu-ral network adaptive SMC [19]. To enhance control performance, many researchers added additional compensation methods to SMC to deal with uncertainties. Due to their strong generalization abilities, fuzzy logic systems (FLSs) [20] and neural networks (NNs) [21] have been embedded into SMC for uncertainty compensations, and they were successfully applied in MLSs. In NNs and FLSs, the weights need to be updated online, while this will impose a heavy computational burden and cannot guarantee fast convergence of parameters [22]. On the other hand, the disturbance observer (DOB) methods, which can estimate and compensate for the uncertainties in the control synthesis, can also deal with the problem [23]. In [24], DOB was embedded into SMC to compensate for the nonlinear effects and model uncertainties in nanopositioning stages. Besides, nonlinear disturbance observer [25], adaptive disturbance observer [26], extended disturbance observer [27], generalized proportional integral observer [28], and unknown system dynamics estimator (USDE) [29] have also been developed for SMC to deal with unknown dynamics and disturbances. The switching controller in the SMC methods with uncertainty compensations must suppress the residual compensation error. However, due to the difficulty of obtaining accurate uncertainty bounds, excessive switching gain is unavoidable to ensure robustness, which may introduce obvious chattering, lead to lower position control accuracy, and even cause mechanical and electronic damage to MLS.
Recently, adaptive SMC methods, which can update the switching gain in a dynamic adaption, are famous for balancing the chattering and the robustness. In [30], a threshold-based adaptive law, which is updated by the distance of the system state to a discontinuity surface, was proposed. In [31], an adaptive SMC strategy without under-estimation and over-estimation was proposed, which stops increasing (resp. decreasing) the control gain when the tracking error decreases (resp. increases). In [32], the switching gain of SMC was updated by an integral/exponential adaptation law for chattering reduction. In [33], the upper bound of uncertainty was designed by a NN, and the adaptive NN updated the switching gain of SMC. In [34], a new adaption law with an arbitrarily small vicinity of the sliding manifold was presented for the chattering reduction, which is inversely proportional to the sliding variables. In [35], the uncertainty upper bound was designed as a first-order polynomial, and thus, the switching gain is adaptively updated by the polynomial with adaptive coefficients. However, although the existing adaptive SMC methods can ensure that the switching gain converges to the vicinity of the actual boundary value, there will be a significant difference between the switching gain and the true boundary value during the dynamic process. The dynamic response performance thus will be sacrificed.
Due to the fact the switching controller is mainly used to deal with the compensation error, in cases that the estimation error of USDE can be embedded in the switching gain, it may be possible to improve the dynamic response performance. To this end, this paper proposes an enhanced USDE-based SMC (USDE-SMC) method by a predictive-adaptive switching (PAS) controller (USDE-SMC-PAS). The switching gain of the PAS controller is composed of a filtered estimation error part and an adaptive residual error part to obtain a higher dynamic response and stronger robustness. The main contributions lie in: The rest of this paper is organized as follows. In Section II, the mathematical model of an MLS is introduced; the new control scheme is proposed in Section III; the simulations are performed in Section IV; the experiments are conducted in Section V; finally, Section VI concludes the paper.

Modelling of maglev system
The structure of the maglev plant is shown in Fig.1. It comprises a levitation reluctance actuator and a disturbance injection reluctance actuator. The levitation reluctance actuator contains a bottom E-shaped mover  and an I-shaped target, and the disturbance injection reluctance actuator includes a top E-shaped mover and an I-shaped target. Silicon-iron materials laminate the E-shaped movers and the I-shaped target. The excitation currents are i and i d for the levitation and disturbance injection actuators, respectively. The parameters are set as Table 1.
Assume that the magnetic field in the reluctance actuator is uniformly distributed. It can be obtained according to the Maxwell equation that where l Fe is the length of the flux linkage, H is the magnetic field intensity, μ 0 denotes the air permeability, and x denotes the levitation position. Using H = B μ 0 μ r [36], one obtains where B n (x) represents the flux caused by the nonlinearities including hysteresis, eddy current, and flux leakage. Therein, it can be obtained that l Fe μ r 2x, and thus l Fe μ r can be omitted. According to Maxwell's stress tensor, the relationship between the output force F and the flux density B can be calculated as Denoting K (x) = μ 0 AN 2 8x 2 , the actual dynamic model of MLS can be given as where δ m is the deviation of the parameter m, and F d represents the external disturbance. A lumped uncertainty, including the external disturbance F d , the parametric uncertainty δ m , and the model mismatch caused by the inherent nonlinearities, can be defined as Linearizing the MLS by using where [x 1 x 2 ] T = [xẋ] T .

Controller design
In an MLS control system, the tracking error and its derivative with respect to time are given by where r is a given trajectory. Based on the tracking error e and the derivativeė, a sliding mode variable S and its derivative with respect to timeṠ are designed as where λ is a positive control gain.

The design of USDE-SMC
To compensate for the lumped uncertainty L d , a USDE is used. Herein, the filter variables x 2 f and u f are defined as where κ is a positive constant that determines the bandwidth of the low-pass filter 1 1+κL (L represents the Laplace variable). The following lemma can be obtained.
Lemma 1 [37] Considering system (6) and the filter variables in (9), an invariant manifold for any positive constant κ can be defined as where γ satisfies lim Proof Differentiating γ with respect to time and considering (6) and (9), one obtainṡ Choosing a Lyapunov function as V u = 1 2 γ 2 , the derivative of V u with respect to time is expressed asV u = γγ . According to Assumption 1, it can be deduced thatV It is easy to prove that lim κ→0 lim γ t→0 = 0, which indicates that γ = 0 is an invariant manifold [38].
Therefore, L d can be estimated aŝ Adding a filter 1 1+κL into (6), one obtains Defining Comparing both (12) and (14), it can be deduced that L d f =L d . Therefore, the estimation error e L can be deduced as Furthermore, one obtains [38] Comparing (15) and (10), one obtains e L = γ . Due to γ is bounded, we can know that |e L | is also bounded by One obtains that lim κ→0 lim e L t→0 = 0. This proves that the higher the bandwidth of the low-pass filter, the better the compensation performance. The control law of the USDE-SMC is designed as where is a positive coefficient, ρ is the switching gain, −L d represents the USDE compensator, and −ρsgn (S) denotes the switching controller. (6) with the control law (18) are uniformly ultimately bounded (UUB) [35], if ρ ≥ |e L |.

The design of USDE-SMC-AS
Defining K as the upper bound of the compensation error e L , where K ≥ |e L |. The control law of the USDE-SMC-AS is designed as where u as represents the adaptive switching controller.
In addition,K is the estimation of K , updated bẏ where and ϒ are positive constants, andK (0) > 0. It can be concluded thatK ≥ 0, due tô

The design of USDE-SMC-PAS
Considering that the switching controller after USDE compensation is mainly used to deal with the compensation error, it is possible to introduce the USDE estimation error into the switching controller. Therefore, we propose a filtered estimation error-based uncertainty upper bound and use it to design a new adaptive switching controller (i.e. PAS controller) to improve the disturbance rejection ability and the dynamic response performance.
Defining e L f as the filtered variable of e L , it can be deduced that  (15) and (35), one obtains Therefore, A filtered estimation error-based dynamic upper bound of the compensation error e L can be defined as where e L f and ( > 0) represent the filtered estimation error part and the residual part, respectively. Based on the dynamic bound, the control law of the USDE-SMC-PAS is designed as where u e , u c , and u pas represent the equivalent controller, the USDE compensator, and the PAS controller. In addition, the estimation of (i.e.ˆ ) can be updated bẏ where and ϒ are positive constants, andˆ (0) > 0. It can be concluded thatˆ ≥ 0, due tô Algorithm 1: The steps of the USDE-SMC-PAS algorithm 1 % * * * * * * Initialization of the parameters * * * * * * %; 2 Setting the sampling interval T s and the terminal time T end ; 3 Setting the controller parameters ; 4 Setting the reference trajectory r ; 5 Setting the initial value of the computing time T as 0 ; 6 % * * * * * * Algorithm executed at sampling time of T s seconds * * * * * * %; 7 while T < T end do 8 Acquire the gap sensor signal and transfer it to the position value x.; 9 Calculate the tracking error by e = r − x ; The steps of the USDE-SMC-PAS algorithm in the actual control process can be represented by pseudocode as shown in Algorithm 1. The PAS controller, utilizing a filtered estimation error and an adaptive law, has the computational complexity of O(1), which means that it possesses a low computational burden. (6) with the control law (39), the adaptive law (40), the estimator (12), and the filter (37) are UUB [35].

Simulation
The MLS dynamic model introduced in Section 2 was simulated. The simulations were conducted on a personal computer with the setup of Intel Core i7, 3.3 GHz CPU, and 16 GB of RAM. The simulation code was written in Matlab 2016a with a fixed step size [40][41][42]. The parameters of the simulation model are shown in Table 1. In the simulation, we took into account the effects of eddy current and flux leakage. The quantified resolution of the sensor is 20 nm, the same as the gap sensor in the experiments. In addition, the energy loss and electrical delay are not considered.
Three simulations were performed to evaluate the performance of the control algorithms, including Case I: Sinusoidal trajectory tracking; Case II: Step trajectory tracking; Case III: Sinusoidal disturbance rejection. The uncertainties and their derivations in all cases are bounded. The sinusoidal trajectory of Case I is shown as the black curve in Fig. 3, where the trajectory amplitude is 2×10 −4 mm, the trajectory bias is 4×10 −4 mm, and the trajectory frequency is 0.5 Hz. The step trajectory of Case II is shown as the black curve in Fig. 4, where the height of an individual step is 1×10 −4 mm, the lowest position is 2×10 −4 mm, the highest position is 6×10 −4 mm, and the step time is 1 s. The trajectory of Case III is shown as the black The maximal error (MAE) and the root mean square error (RMSE) are used to evaluate the control accuracy of the system, defined as Therein, r (t k ) and x(t k ) are the desired position and the actual position, respectively, and n k is the number of the sampled data. In addition, t k = kT s , where T s denotes the sampling interval (T s = 3 × 10 −4 s). All algorithms have the same parameters as follows.  In Case I, the MLS is levitated to track a sinusoidal trajectory. The tracking results of Case I are shown in Fig. 3, including tracking position (position), control current (current), and switching gain (SW gain). The numerical results are recorded in Table 2. USDE-SMC-PAS achieves the best tracking accuracy with MAE and RMSE of 7.58 μm and 2.97 μm, respectively, while USDE-SMC-AS has the worst accuracy with MAE and RMSE of 48.29 μm and 25.54 μm, respectively. In Case II, the MLS is levitated to track a step trajectory. The results of Case II are shown in Fig. 4 and Table 2. USDE-SMC-PAS obtains the best dynamic response with the maximum and minimum settling time of 0.24 s and 0.21 s, respectively. In comparison, USDE-SMC-AS has the worst dynamic response, with the maximum and minimum settling time of 0.87 s and 0.31 s, respectively. In addition, in Case I and Case II, both USDE-SMC-AS and USDE-SMC-PAS have attenuated the chattering phenomenon, while chattering is very obvious in USDE-SMC. The tracking results of Case I and Case II show that PAS can help USDE-SMC-PAS reduce chattering and obtain better tracking accuracy and dynamic response performance than USDE-SMC and USDE-SMC-AS.
In Case III, a sinusoidal disturbance is injected. The results of Case III are shown in Figs. 5 and Table 2. USDE-SMC-PAS has the best disturbance rejection ability. Its MAE and RMSE in Case III are 0.46 μm and 0.21 μm, respectively. In comparison, USDE-SMC and USDE-SMC-AS are relatively worse. In Case III, both USDE-SMC-AS and USDE-SMC-PAS show reduced chattering phenomenon, while USDE-SMC shows very obvious chattering. The disturbance rejection results of Case III show that PAS can help The convergence trends of the switching gains ρ in (18),K in (24), and e L f +ˆ in (39) are also plotted in the bottom parts of Figs. 3-5. Simulation results of filtered estimation error e L f in USDE-SMC-PAS are plotted in Fig. 6. It can be found that the switching gain of USDE-SMC-PAS is mainly affected by the filtering estimation error in all cases. As a result, the switching gain of USDE-SMC-PAS can better capture the feature of the residual uncertainty than USDE-SMC-AS and USDE-SMC. Finally, USDE-SMC-PAS has a greater ability to handle residual uncertainty relative to USDE-SMC-AS and has less chattering than USDE-SMC.

Experiments and validation
Experiments were conducted on a one-dimensional MLS, as shown in Fig. 7. The MLS contains a maglev plant, a Speedgoat (Performance), and two drivers (TA115, Trust Automation. Inc.). In the maglev plant, the reluctance actuator used to produce levitation force comprises the bottom E-shaped mover and the I-shaped target, while the reluctance actuator used to generate disturbance force contains the top E-shaped mover and the I-shaped target. The gap x between the bottom Eshaped mover and the I-shaped target is measured by a gap sensor ( HEIDENHAIN encoder with 20 nm resolution) and fed back to Speedgoat. In the MLS, the gap Sinusoidal trajectory tracking experiments and step trajectory tracking experiments were carried out in Case I and Case II, respectively, to evaluate the ability of USDE-SMC-PAS in dealing with the inherent nonlinearities in an MLS. The motion process and the numerical results of Case I are shown in Fig. 8 and Table 3, respectively. It can be found that the MAE and RMSE of USDE-SMC-PAS are 9.68 μm and 4.57 μm, respectively, and the chattering is slight. When compared to USDE-SMC and USDE-SMC-AS, USDE-SMC-PAS can smoothly track the sinusoidal trajectory with higher accuracy. The tracking results indicate that USDE-SMC-PAS has well-tracking accuracy when the MLS is guided to a sinusoidal trajectory. The motion process and the numerical results of Case II are shown in Fig. 9 and Table 3, respectively. It can be found that the maximum settling time and minimum settling time of USDE-SMC-PAS are 0.30 s and 0.15    Fig. 10 and Table 3. It can be found that the system with USDE-SMC-PAS can smoothly preserve high levitation accuracy with the smallest MAE and RMSE values when dealing with the disturbance. However, USDE-SMC obviously has chattering phenomena, and its levitation accuracy is bad when the disturbance is injected. USDE-SMC-AS can also reduce chattering, but its improvement of disturbance suppression ability is worse than USDE-SMC-PAS. The results reveal that PAS can help USDE-SMC-PAS reduce the chattering and improve the disturbance rejection ability.
The convergence trends of the switching gains ρ in (18),K in (24), and e L f +ˆ in (39) are also plotted in the bottom parts of Figs. 8-10. In addition, experimental results of filtered estimation error e L f are plotted in Fig. 11. USDE-SMC has a fixed switching gain set larger than the upper bound of the uncertainty to obtain sufficient robustness, and thus significant chattering is inevitable. USDE-SMC-AS can adaptively update the switching gain according to the sliding variable, and thus its chattering can be reduced. However, since it is difficult for traditional AS methods to accurately estimate the uncertainty, the disturbances rejection ability of USDE-SMC-AS is not significantly improved. USDE-SMC-PAS embeds the filtered estimation error of USDE into the adaptive switching gain, where the residual part is adaptively updated according to the sliding surface. The switching gain of USDE-SMC-PAS can fit the uncertainty more accurately. Therefore, USDE-SMC-PAS has a better dynamic response, higher trajectory tracking accuracy, and stronger distur- In addition, simulation results of velocityẋ in USDE-SMC-PAS are plotted in Fig. 12, and experimental results of velocityẋ in USDE-SMC-PAS are plotted in Fig. 13. The velocities in the simulation and the velocities in the experiment have similar charac- teristics, including peak size and noise level. However, there is a difference between the simulated current and the experimental current. This is mainly because the reluctance actuator inductance and heat dissipation are not considered in the simulation.

Conclusion and future works
This paper proposes a USDE-SMC-PAS to control the MLS with inherent nonlinearities and external disturbances. The filtered estimation error of USDE and the adaptive part comprise a PAS controller for improving the robustness and dynamic response of MLS. The stability properties are investigated by constructing Lyapunov functions. Simulations were implemented in Matlab 2016a, and experiments were conducted on a reluctance MLS setup. Results of simulations and experiments show that USDE-SMC-PAS can smoothly preserve high levitation accuracy with high dynamic responses when tracking sinusoidal trajectory and step trajectory. In addition, USDE-SMC-PAS shows better robustness with reduced chattering than USDE-SMC and USDE-SMC-AS when dealing with disturbance. With the high control performance and sufficient robustness, the USDE-SMC-PAS method is worth to be recommended to other control applications where inherent nonlinearities and external disturbances are apparent.
In addition, we only use a simple sliding mode surface in this method, so there is much room for improvement in the control performance in USDE-SMC-PAS. In the future, we will investigate the combination of USE and PAS with other advanced sliding mode methods, such as terminal SMC, super-twisted SMC, and higher-order SMC, in the hope of further improving the control performance of the algorithm. Data Availability All data generated or analysed during this study are included in this published article.