Command Filtering and Barrier Lyapunov Function-Based Adaptive Control for PMSMs with Core Losses and All-State Restrictions

With the troubles of core losses and all-state confined to certain limitations which are the innate traits of permanent magnet synchronous motors (PMSMs), this article develops a command filtered adaptive backstepping approach to follow the track of PMSM’s desired rotor position. To begin with, the RBF neural network technique is utilized to get close to the uncharted nonlinear terms which existed in PMSM’s mathematical model. Meanwhile, an advanced adaptive command filter control methodology is constructed to avoid the computing explosion during the process of backstepping design. Furthermore, to make sure that all the state variables are confined into certain ranges, we employed the barrier Lyapunov function (BLF) at every step of the controllers construction. In addition, an error compensatingmechanism is proposed to neutralize filtering errors and only one adaptive law is required. At last, simulation results bear out the superiority of the aforementioned control scheme.


Introduction
Lately, permanent magnet synchronous motors (PMSMs) are employed broadly in real-world utilization.
is proverbial usage is due to the advantageous PMSMs features like straightforward mechanism, petit size, great productiveness, and dependable manipulation. Nevertheless, the PMSM's real-time mathematic model set contains tremendous nonlinearity and multivariables issues which may lead to a challenging mission to acquire optimum control results. So, in order to enhance the PMSM's control effectiveness, many advanced control techniques have been proposed, for instance, PI control [1,2], sliding mode control [3][4][5], adaptive backstepping control [6], and other control schemes [7,8]. Among these methodologies, the backstepping control technique is now becoming a basic foundation to construct controllers for high complexity models since it was designed to obtain asymptotic tracking and global stability. Besides, the load turbulence and time-variant parameters issues can be ripped out of the operation of PMSMs by making use of the adaptive control technique.
However, "certain functions must be linear" and "explosion of complexity" issues which are rooted in the traditional backstepping control methods are very tricky to be dealt with. For one thing, along with the development of radial basis function (RBF) neural networks (NNs) [9][10][11], the nonlinear systems' unknown functions can be approximated by this algorithm based on the adaptive control method and this proposal can rule out the dependency of the accurate mathematical model. For another, in [12][13][14], a command filtered-based backstepping method was developed to tackle the "explosion of complexity" problem and the error compensation technique is employed to make the filtering outcomes more accurate. To be specific, the command filters' input signals are designed as virtual control functions and the filters' outputs can eliminate explosive terms. Moreover, the error compensation mechanism will neutralize the filtering deviations to some extent. But these published methodologies have not considered the state constraints problem of PMSM drive systems.
In fact, according to the PMSM's inherent features, its state variables should be confined to reasonable boundaries. For instance, if the rotor angular velocity, stator current, or other state variables of PMSM are out of the state constraints, the motor's performance will be influenced and may even result in severe security issues. Take stator current as an example, exorbitant current value will lead to serious overheating problems to the rotators which would speed up the aging of insulation materials and decrease the equipment's service lifespan. Accordingly, state limitations are indispensable during the PMSM's controllers construct process. Fortunately, many significant achievements have been obtained in the all-state constrained nonlinear control field, such as [15,16] and from these approaches, the barrier Lyapunov functions (BLFs) are normally utilized to hold off the limitation transgressions.
Apart from state variables restrictions, in order to get perfect control performance of PMSMs, the core loss impact also should be taken into consideration when the driving frequency spikes, which is correlated with the high-speed operation.
e control accuracy will drop sharply with tremendous core losses, and thus the solution of PMSMs with iron losses problem is crucial to actual applications. As far as we know, the core losses and all-state restrictions issues which are rooted in nonlinear high-ordered PMSM drive systems have not been studied by using command filtered adaptive backstepping approach.
So, with these previous observations, we took core losses and all-state restrictions which are the inherent properties of PMSM drive systems into considerations, and then we developed a BLFs-based adaptive command filtered neural control method. Disparate from the conventional control schemes, this method's main novelties are concluded as follows: (1) Unlike [17], the command filters are used to tackle the "explosion of complexity" issue which cannot be neglected in adaptive backstepping control for nonlinear systems. (2) Distinct from [17,18], this paper considers the core losses issue in PMSM's mathematical model and thus makes this method be more applicable in actual usages. (3) Different from [12,19], the errors that arose from command filters are neutralized by compensating signals to diminish their negative influence upon control performance. (4) is article just requires one single adaptive law during the controllers construction process, which can facilitate the calculation compared to [20], so that the control scheme's effectiveness will be improved.
In the posterior section of this paper, simulation figures and a comparison table are displayed to substantiate the effectiveness and robustness of the submitted control approach.

Nonlinear Control Methods.
Many approaches are proposed to enhance the control effectiveness of nonlinear systems. In [21], Zheng et al. utilized a stable adaptive PI control strategy in the discrete-time domain for the PMSM drive system. e PI controller is capable of automatic online tuning of the control gains based on the gradient descent method and the experimental results illustrated the tracking performance is favorable. Li et al. [22] put forward the sliding-mode control method to deal with nonlinear active suspension systems. ey designed an adaptive slidingmode controller to guarantee the reachability of the specified switching surface. Yin et al. designed a backstepping controller for the switch complex nonlinear system in [23]. During the construction process, they developed a state backstepping controller to realize the exponential stability of the observer-backstepping feedback control system. Zhang et al. presented a linear quadratic regulator-based proportion integral differential equivalent controller for PMSM in [24]. e method was implemented through the dSPACE digital signal processor system and the experimental result confirmed its effectiveness.

Approximation Techniques.
To enlarge the practical field of the backstepping method, the nonlinear terms in the nonlinear systems' mathematical model need to be dealt with.
e literature [25][26][27][28] has utilized the fuzzy logic systems (FLS) to approximate unknown nonlinearities in different kinds of scenarios. And the approximation results verified that this technique can well serve its original purpose. In [29][30][31], the authors employed the LSTM and the GRU techniques to predict traffic speed, power load, and traffic flow, respectively. And the experimental results indicate these two kinds of deep recurrent networks are skilled in modeling abilities, which make them more suitable for sequence-based long-term tasks. In [32], Bai et al. utilized a compound autoregressive network to predict multivariate time series. Jin et al. proposed two nonlinear estimation methods to achieve real-time indoor RFID tracking in [33]. In [9], Fu et al. utilized the RBF neural networks to cope with the unknown nonlinear functions. e finite-time adaptive neural controller was proposed via the new command filter backstepping technique, and the tracking error converges to a small neighborhood of the origin in finite time.

Filtering
Algorithms. Some scholars put forward a variety of filtering methods in recent years. In [34], Bai et al. proposed a neuron-based Kalman filter to enhance the control effect of various intelligent terminals and promote the sensing level. ey introduced the neuronunits into the conventional Kalman filter framework and thus the filtering process could be optimized to reduce the effect of the unpractical system model and hypothetical parameters. In [35,36], the authors employed the dynamic surface control (DSC) technique to resolve the "explosion of complexity" problem, and it is a first-ordered filtering method for every step's virtual input during the traditional backstepping 2 Complexity controllers design. But the DSC technique does not take the deviations ascribed to the first-order filters into account, which may induce unwanted influence on the control result.
In [37], the adaptive filtering technique was proposed to filter the complex noise and obtain the true measurements' value and thus the MEMS gyroscope performance can be improved. In [38], the authors proposed a state filteringbased least squares parameter estimation for bilinear systems. Zhang et al. developed a novel state estimation algorithm to enhance the computational efficiency based on delta operator in [39].

PMSM Innate Features and Identification Methods.
In [40,41], the authors take full-state constraints of nonlinear systems into consideration and constructed the barrier Lyapunov functions to ensure the state constraints are not transgressed. In [42], Zhao et al. proposed a health performance evaluation method to detect anomaly occurrences and evaluate the multirotor system's real-time health condition. Ding et al. in [43,44] derived gradient-based and two-stage gradient-based iterative algorithms to generate more accurate parameter estimation to overcome the difficulty of state and input identifications. In [45], Zhang et al. developed joint estimation algorithms for states and parameters of nonlinear systems to make the parameter estimates converge to their true values. With the identification methods in [43][44][45], the parameters of the mathematical models can be obtained accordingly.

Dynamic Model and Preparations
In the d − q frame of axes, the PMSM's dynamic model with core losses from [46] can be described as where θ, ω, n p , J, T L , R s , and R c represent rotor position, rotor angular speed, quantity of pole pairs, rotor momental inertia, load torque, stator resistance, and core loss resistance, sequentially. u d stands for the d-axis voltage and u q represents the q-axis voltage. i d is the d-axis current while i q is the q-axis current. L d and L q present as stator inductors. L ld and L lq represent leakage inductances, while L md and L mq are the notations of magnetic inductances. Finally, λ PM stands for the excitation flux. To simplify the above mathematical equations, we employ the following symbolizations: With the aforementioned symbolizations, the mathematical model set will be converted into e ultimate goal of this paper is to develop the controllers u q along with u d to make the rotor position x 1 track the desired signal x d as perfect as possible while the state variables x i are demanded to meet the premises that e RBF neural network is a feedforward network with three layers of neurons called the input layer, the hidden layer, and the output layer. Figure 1 is the graphic illustration of the structure of the RBF neural network. Additionally, the input layer contains an equal number of nodes to the dimension of the input vector. e hidden layer's nodes number depends on the complexity of the problem. e output layer's nodes number equals the dimension of the output vector. e weight parameter W stands for the link between nodes and it only exists between the hidden and output layers. e self-training law will be given later. e k-means clustering algorithm which is a kind of unsupervised algorithm of RBF neural network will be used in this paper. With the neural network's theory and its parameters' definitions which can be found in [47], we know that any time-consecutive function φ(z) can be estimated by RBF NNs. e estimate functions φ(z) � W * T S(Z) satisfy the premise of R q ⟶ R, in which q stands for input dimensions. Moreover, the NNs' input variable Z needs to be within the domain of Z ∈ Ω Z ⊂ R q and the weight vector W * is formed as where l represents the quantity of NNs nodes. We choose the Gaussian basis [12], we have the definition of command filters listed as

Lemma 1 From
From the above equations, we know that the output variables e 1 and e 2 can be obtained by the input variable α 1 . In order to do so, there are some rules that should be satisfied. Firstly, for all t ≥ 0, the first and second ordered time derivative forms of α 1 should meet the demands of | _ α 1 | < ρ 1 and |€ α 1 | < ρ 2 while ρ 1 , ρ 2 are positive constants. Secondly, the initial conditions of these variables should satisfy that e 1 (0) � α 1 (0), e 2 (0) � 0. Consequently, | _ e 1 |, |€ e 1 |, and |e ... 1 | will be bounded into certain ranges. Additionally, with ξ ∈ (0, 1] and ω n > 0, the deviation between the input and output signals will satisfy that |e 1 − α 1 | ≤ μ. Assumption 1 (see [41]). e desired signal x d and its firstordered time derivative form _ x d should both be smooth, limited, and known. us, they can meet the demands of

Command Filtered Self-Adaptive Neural Network Controllers Construction
During this process, we constructed the self-adaptive command filtered neural network controllers for PMSMs with core losses and all-state restrictions based on the BLFs.
To begin with, we define the error variables as where x d is the desired rotor position trajectory and α i are the input variables of the filters while x i,c represent the filters' output variables, in which i � 1, 2, 3, 4. To neutralize the filtering errors which are the values of x i,c − α i , at every filtering step, we use the error compensation technique and ζ i represent the compensation signals, where i � 1, 2, 3, 4, 5, 6. Additionally, we introduce a tight set Ω v � |v i | < k b i , i � 1, 2, . . . , 6 , in which the constants k b i should be positive. During the next construction process, the error renumeration variables ζ i , the virtual controllers α i , and the real controllers u d and u q will be given.
Step 1. In [15], the barrier Lyapunov function was proposed. So, with the description, we select the first BLF V 1 as Within the compact set Ω v , the first-ordered time derivative form of V 1 should be in which . . , 6. Next, we conceive the virtual controller α 1 and the remunerate variable ζ 1 as where k 1 > 0 is designed to be the control gain and the terms k i > 0 (i � 1, 2, . . . , 6) will be applied in constructing other virtual control laws and compensation signals later on. By using (8), (7) can be converted into the following equation: Step 2. Similarly, we set up the second BLF as where V 2 should be time-consecutive within the compact set Ω v , so we compute its first-ordered time derivative form and apply _ V 1 into it to get the following equation: When it comes to actual applications, T L is presumed to be unknown but should be limited to a certain range. erefore, we assume |T L | ≤ d, in which the constant d should be positive. With Young's inequality theorem, we With the aforementioned description of RBF NNs, we know that it always has a RBF NN W T 2 S 2 (Z) to make f 2 (Z) � W T 2 S 2 (Z) + δ 2 (Z) holds, where δ 2 (Z) is the estimate error. en, for any ε 2 > 0, δ 2 (Z) will satisfy that |δ 2 (Z)| ≤ ε 2 . So, under the premise of l 2 > 0, we have Design the second virtual control signal α 2 along with the remuneration variable ζ 2 as with θ being the approximation of θ, which will be constructed later on. Applying (13) and (14) into (12), we have Step 3. Construct the third BLF as V 3 is continuous within the compact set Ω v , so we compute its first-ordered time derivative form and apply _ V 2 into it, then we get is the estimate error. en, for any ε 3 > 0, δ 3 (Z) will satisfy that |δ 3 (Z)| ≤ ε 3 . With the premise of l 3 > 0, we deduce that Set the third virtual control law α 3 along with the remuneration variable ζ 3 as Complexity Applying (18) and (19) into (17), we have Step 4. Design the next BLF V 4 as Akin to _ V 3 , _ V 4 will be listed as in which f 4 (Z) � b 4 x 4 + b 5 x 3 . Akin to the last step, it always has a neural network W T 4 S 4 (Z) to make that f 4 (Z) � W T 4 S 4 (Z) + δ 4 (Z) holds, where δ 4 (Z) stands for the estimate error. en, for any ε 4 > 0, δ 4 (Z) will satisfy that |δ 4 (Z)| ≤ ε 4 . So, with the premise of l 4 > 0, we can obtain that Now, we set the actual control signal u q and the remunerate variable ζ 4 as en, with the terms of (23) and (24), the inequality (22) will result in Step 5. Set the next BLF V 5 as where V 5 is continuous within the compact set Ω v , so we compute its first-ordered time derivative form and apply _ V 4 into it; then, we have With the premise of l 5 > 0, we can deduce that Design the fourth virtual control signal α 4 along with the remunerate variable ζ 5 as By using (28) and (29), inequality (27) can be transformed as Step 6. Design the sixth BLF V 6 as follows:

Remark 2.
Ensue from the definition of a and b, along with the proper control variables m and k i (i � 1, 2, . . . , 6), small variables ε j (j � 2, 3, . . . , 6), l n (n � 2, 3, . . . , 6), and large parameter r, the rotor position tracking error |z 1 | will be small enough to meet the control requirement. On account of With the definition of α 1 in equation (8), we know that α 1 contains the terms of z 1 as well as _ x d . erefore, the upper limitation of α 1 which is noted as Parallelly, it can be verified that |x 3 | < k c 3 , |x 4 | < k c 4 , |x 5 | < k c 5 , and |x 6 | < k c 6 . At this point, the proof is accomplished.

Simulation Results
To substantiate the control scheme's validity, a simulation has been conducted in this part. e PMSM's parameters with core losses are chosen as Table 1.
As to the RBF neural network, the neurons' quantity is 11 and we choose the activate functions as for i � 2, 3, 4, 5, 6. e activate functions' centers are scattered evenly in scale [− 5, 5], and their widths are all defined as 1.
(a) To control the PMSM driving system with core losses and all-state restrictions, we developed the BLFsbased adaptive command filtered neural network controllers. So, during this simulation, we employed control coefficients as k 1 � 10, k 2 � 7, k 3 � 100,  From these simulation outcomes, we can observe that even under load torque uncertainty, these two approaches can both follow the given trajectory nicely. But it cannot be ignored that the tracking error which is showed in Figure 3(a) is much smaller than the error that was displayed in Figure 3(b). Moreover, the state variables in Figures 6(a) and 7(a) are controlled in the confined ranges, but i oq of Figure 6(b) is varying from − 20 to 60, which overstepped the presupposed current's constraint (Table 2).
Remark 3. Disparate from the DSC control method without taking filtering errors into consideration, in this paper, we  Complexity developed the BLFs-based adaptive command filtered neural control approach. is approach can not only guarantee that the state variables would confine into reasonable bounds but also ensure that the tracing deviation would congest into a smaller range of zero, which will be much more functional and robust in real-world applications.

Conclusion
is paper developed a BLFs-based self-adaptive command filtered neural network control approach to work out the position tracking problem of the PMSM driving system with core losses and all-state restrictions. By merging BLFs into CFC techniques, the issues of "explosion of complexity" and all-state restrictions can be well resolved. Additionally, virtual control laws were constructed to ensure the position tracking error would be limited to a minute range of zero. In the end, simulation performances illustrated the system's adaptability and antidisturbance ability.

Data Availability
e data used to support the findings of this study are partially included within the article. Further information is available from the corresponding author upon request.

Conflicts of Interest
e authors declare that there are no conflicts of interest regarding the publication of this paper.