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

Authors: Xiaoling Wang; Jinpeng Yu
Journal: Complexity
 (2021)

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## Abstract

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 compensating mechanism 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.


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## Body

## 1. Introduction

Lately, permanent magnet synchronous motors (PMSMs) are employed broadly in real-world utilization. This 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 [3–5], 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) [9–11], 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 [12–14], 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. The 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)
This 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.


## 2. Related Works

### 2.1. 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. The 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. They designed an adaptive sliding-mode 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]. The method was implemented through the dSPACE digital signal processor system and the experimental result confirmed its effectiveness.


### 2.2. 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. The literature [25–28] 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 [29–31], 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. The 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.


### 2.3. 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. They 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 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 filtering-based 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].


### 2.4. 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 [43–45], the parameters of the mathematical models can be obtained accordingly.


## 2.1. 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. The 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. They designed an adaptive sliding-mode 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]. The method was implemented through the dSPACE digital signal processor system and the experimental result confirmed its effectiveness.


## 2.2. 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. The literature [25–28] 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 [29–31], 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. The 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.


## 2.3. 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. They 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 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 filtering-based 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].


## 2.4. 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 [43–45], the parameters of the mathematical models can be obtained accordingly.


## 3. Dynamic Model and Preparations

In thed−q frame of axes, the PMSM’s dynamic model with core losses from [46] can be described as(1)dθdt=ω,dωdt=npλPMJioq+npLmd−LmqioqiodJ−TLJ,dioqdt=RcLmqiq−RcLmqioq−npLdLmqωiod−npλPMLmqω,diqdt=−RsLlqiq+RcLlqioq+1Llquq,dioddt=RcLmdid−RcLmdiod+npLqLmdωioq,diddt=−RsLldid+RcLldiod+1Lldud,where θ, ω, np, J, TL, Rs, and Rc represent rotor position, rotor angular speed, quantity of pole pairs, rotor momental inertia, load torque, stator resistance, and core loss resistance, sequentially. ud stands for the d-axis voltage and uq represents the q-axis voltage. id is the d-axis current while iq is the q-axis current. Ld and Lq present as stator inductors. Lld and Llq represent leakage inductances, while Lmd and Lmq are the notations of magnetic inductances. Finally, λPM stands for the excitation flux. To simplify the above mathematical equations, we employ the following symbolizations:(2)x1=θ,x2=ω,x3=ioq,x4=iq,x5=iod,x6=id,a1=npλPM,a2=npLmd−Lmq,b1=RcLmq,b2=−npLdLmq,b3=−npλPMLmq,b4=−RsLlq,b5=RcLlq,c1=RcLmd,c2=−npLqLmd,c3=−RsLld,c4=RcLld,d1=1Llq,d2=1Lld.With the aforementioned symbolizations, the mathematical model set will be converted into(3)x˙1=x2,x˙2=a1Jx3+a2Jx3x5−TLJ,x˙3=b1x4−b1x3+b2x2x5+b3x2,x˙4=b4x4+b5x3+d1uq,x˙5=c1x6−c1x5−c2x2x3,x˙6=c3x6+c4x5+d2ud.The ultimate goal of this paper is to develop the controllersuq along with ud to make the rotor position x1 track the desired signal xd as perfect as possible while the state variables xi are demanded to meet the premises that xi<kcii=1,2,3,4,5,6, in which kci>0.The RBF neural network is a feedforward network with three layers of neurons called the input layer, the hidden layer, and the output layer. Figure1 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. The hidden layer’s nodes number depends on the complexity of the problem. The output layer’s nodes number equals the dimension of the output vector. The weight parameter W stands for the link between nodes and it only exists between the hidden and output layers. The self-training law will be given later. The 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 φz can be estimated by RBF NNs. The estimate functions φ^z=W∗TSZ satisfy the premise of Rq⟶R, in which q stands for input dimensions. Moreover, the NNs’ input variable Z needs to be within the domain of Z∈ΩZ⊂Rq and the weight vector W∗ is formed as W∗=Φ1∗,…,Φl∗T, where l represents the quantity of NNs nodes. We choose the Gaussian basis function SZ=p1Z,p2Z,…,pnZT/∑i=1npiZ and piZ=exp−Z−ηiTZ−ηi/γi2 for i=1,2,…,n, where ηi=ηi1,ηi2,…,ηinT is the center vector and γi is the Gaussian function’s width. So, with the above definitions, the inequality WiSik2≤lii=1,…,n holds.Figure 1
The structure of RBF neural network.Lemma 1.
From [12], we have the definition of command filters listed as(4)e˙1=ωne2,e˙2=−2ξωne2−ωne1−α1.
From the above equations, we know that the output variablese1 and e2 can be obtained by the input variable α1. 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 α1 should meet the demands of α˙1<ρ1 and α¨1<ρ2 while ρ1, ρ2 are positive constants. Secondly, the initial conditions of these variables should satisfy that e10=α10, e20=0. Consequently, e˙1, e¨1, and e⃛1 will be bounded into certain ranges. Additionally, with ξ∈0,1 and ωn>0, the deviation between the input and output signals will satisfy that e1−α1≤μ.Assumption 1.
(see [41]). The desired signal xd and its first-ordered time derivative form x˙d should both be smooth, limited, and known. Thus, they can meet the demands of xd≤Y0<kc1 and x˙d≤Y1, in which Y0 and Y1 are positive constants.


## 4. 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(5)z1=x1−xd,z2=x2−x1,c,z3=x3−x2,c,z4=x4−x3,c,z5=x5,z6=x6−x4,c,v1=z1−ζ1,v2=z2−ζ2,v3=z3−ζ3,v4=z4−ζ4,v5=z5−ζ5,v6=z6−ζ6,where xd is the desired rotor position trajectory and αi are the input variables of the filters while xi,c represent the filters’ output variables, in which i=1,2,3,4. To neutralize the filtering errors which are the values of xi,c−αi, at every filtering step, we use the error compensation technique and ζi represent the compensation signals, where i=1,2,3,4,5,6. Additionally, we introduce a tight set Ωv=vi<kbi,i=1,2,…,6, in which the constants kbi should be positive. During the next construction process, the error renumeration variables ζi, the virtual controllers αi, and the real controllers ud and uq will be given.Step 1.
In [15], the barrier Lyapunov function was proposed. So, with the description, we select the first BLF V1 as(6)V1=12logkb12kb12−v12.
Within the compact setΩv, the first-ordered time derivative form of V1 should be(7)V˙1=Kv1v˙1=Kv1z2+x1,c−x˙d−ζ˙1,in which Kvi=vi/kbi2−vi2, i=1,2,…,6. Next, we conceive the virtual controller α1 and the remunerate variable ζ1 as(8)α1=−k1z1+x˙d,ζ˙1=−k1ζ1+ζ2+x1,c−α1,where k1>0 is designed to be the control gain and the terms ki>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:(9)V˙1=−k1Kv1v1+Kv1v2.Step 2.
Similarly, we set up the second BLF as(10)V2=V1+J2logkb22kb22−v22,where V2 should be time-consecutive within the compact set Ωv, so we compute its first-ordered time derivative form and apply V˙1 into it to get the following equation:(11)V˙2=V˙1+JKv2v˙2=−k1Kv1v1+Kv1v2+Kv2a1x3+a2x3x5−TL−Jx˙1,c−Jζ˙2.
When it comes to actual applications,TL is presumed to be unknown but should be limited to a certain range. Therefore, we assume TL≤d, in which the constant d should be positive. With Young’s inequality theorem, we have −Kv2TL≤1/2ε12Kv22+1/2ε12d2 with ε1>0.
So, rewrittenV˙2 as(12)V˙2≤−k1Kv1v1+Kv1v2+Kv2a1x3−Jζ˙2+f2Z+12ε12d2,in which f2Z=a2x3x5−Jx˙1,c+1/2ε12Kv2, Z=x1,x2,x3,x4,x5,x6,xd,x˙dT. With the aforementioned description of RBF NNs, we know that it always has a RBF NN W2TS2Z to make f2Z=W2TS2Z+δ2Z holds, where δ2Z is the estimate error. Then, for any ε2>0, δ2Z will satisfy that δ2Z≤ε2. So, under the premise of l2>0, we have(13)Kv2f2Z=Kv2W2TS2Z+δ2Z≤W22Kv22S2TS22l22+l222+Kv222+ε222.
Design the second virtual control signalα2 along with the remuneration variable ζ2 as(14)α2=−1a1k2z2+Kv22+Kv2θ^S2TS22l22+Kv1kb22−v22,ζ˙2=−1Jk2ζ2−a1ζ3−a1x2,c−α2,with θ^ being the approximation of θ, which will be constructed later on. Applying (13) and (14) into (12), we have(15)V˙2≤−k1Kv1v1−k2Kv2v2+Kv2a1v3+12ε12d2+W22−θ^Kv22S2TS22l22+l222+ε222.Step 3.
Construct the third BLF as(16)V3=V2+12logkb32kb32−v32,V3 is continuous within the compact set Ωv, so we compute its first-ordered time derivative form and apply V˙2 into it, then we get(17)V˙3≤−∑i=12kiKvivi+Kv2a1v3+Kv3b1x4−x˙2,c−ζ˙3+f3Z+W22−θ^Kv22S2TS22l22+l222+ε222+12ε12d2,in which f3Z=−b1x3+b2x2x5+b3x2. Akin to Step 2, it always has a RBF NN W3TS3Z to make f3Z=W3TS3Z+δ3Z holds, where δ3Z is the estimate error. Then, for any ε3>0, δ3Z will satisfy that δ3Z≤ε3. With the premise of l3>0, we deduce that(18)Kv3f3Z=Kv3W3TS3Z+δ3Z≤W32Kv32S3TS32l32+l322+Kv322+ε322.
Set the third virtual control lawα3 along with the remuneration variable ζ3 as(19)α3=−1b1k3z3+Kv32+Kv3θ^S3TS32l32+a1Kv2kb32−v32−x˙2,c,ζ˙3=−k3ζ3+b1ζ4+b1x3,c−α3.
Applying (18) and (19) into (17), we have(20)V˙3≤−∑i=13kiKvivi+b1Kv3v4+∑i=23Wi2−θ^Kvi2SiTSi2li2+∑i=23li22+∑i=23εi22+12ε12d2.Step 4.
Design the next BLFV4 as(21)V4=V3+12logkb42kb42−v42.
Akin toV˙3, V˙4 will be listed as(22)V˙4≤−∑i=13kiKvivi+b1Kv3v4+Kv4d1uq−x˙3,c−ζ˙4+f4Z+∑i=23Wi2−θ^Kvi2SiTSi2li2+∑i=23li22+∑i=23εi22+12ε12d2,in which f4Z=b4x4+b5x3. Akin to the last step, it always has a neural network W4TS4Z to make that f4Z=W4TS4Z+δ4Z holds, where δ4Z stands for the estimate error. Then, for any ε4>0, δ4Z will satisfy that δ4Z≤ε4. So, with the premise of l4>0, we can obtain that(23)Kv4f4Z=Kv4W4TS4Z+δ4Z≤W42Kv42S4TS42l42+l422+Kv422+ε422.
Now, we set the actual control signaluq and the remunerate variable ζ4 as(24)uq=−1d1k4z4+12Kv4+Kv4θ^S4TS42l42+b1Kv3kb42−v42−x˙3,c,ζ˙4=−k4ζ4.
Then, with the terms of (23) and (24), the inequality (22) will result in(25)V˙4≤−∑i=14kiKvivi+∑i=24Wi2−θ^Kvi2SiTSi2li2+∑i=24li22+∑i=24εi22+12ε12d2.Step 5.
Set the next BLFV5 as(26)V5=V4+12logkb52kb52−v52,where V5 is continuous within the compact set Ωv, so we compute its first-ordered time derivative form and apply V˙4 into it; then, we have(27)V˙5≤−∑i=14kiKvivi+Kv5c1x6−ζ˙5+f5Z+∑i=24Wi2−θ^Kvi2SiTSi2li2+∑i=24li22+∑i=24εi22+12ε12d2,in which f5Z=−c1x5−c2x2x3=W5TS5Z+δ5Z, δ5Z≤ε5. With the premise of l5>0, we can deduce that(28)Kv5f5Z=Kv5W5TS5Z+δ5Z≤W52Kv52S5TS52l52+l522+Kv522+ε522.
Design the fourth virtual control signalα4 along with the remunerate variable ζ5 as(29)α4=−1c1k5z5+Kv52+Kv5θ^S5TS52l52,ζ˙5=−k5ζ5+c1ζ6+c1x4,c−α4.
By using (28) and (29), inequality (27) can be transformed as(30)V˙5≤−∑i=15kiKvivi+c1Kv5v6+∑i=25Wi2−θ^Kvi2SiTSi2li2+∑i=25li22+∑i=25εi22+12ε12d2.Step 6.
Design the sixth BLFV6 as follows:(31)V6=V5+12logkb62kb62−v62.
Similarly,(32)V˙6≤−∑i=15kiKvivi+c1Kv5v6+Kv6d2ud−x˙4,c−ζ˙6+f6Z+∑i=25Wi2−θ^Kvi2SiTSi2li2+∑i=25li22+∑i=25εi22+12ε12d2.in which f6Z=c3x6+c4x5=W6TS6Z+δ6Z, δ6Z≤ε6. With the premise of l6>0, we have(33)Kv6f6Z=Kv6W6TS6Z+δ6Z≤W62Kv62S6TS62l62+l622+Kv622+ε622.
Another actual control lawud along with the remunerate variable ζ6 will be constructed as(34)ud=−1d2k6z6+12Kv6+Kv6θ^S6TS62l62+c1Kv5kb62−v62−x˙4,c,ζ˙6=−k6ζ6.
Substituting (33) and (34) into (32), we get(35)V˙6≤−∑i=16kiKvivi+∑i=26θ−θ^Kvi2SiTSi2li2+∑i=26li22+εi22+12ε12d2,where θ=maxW22,W32,W42,W52,W62 and θ˜=θ^−θ. Define the final BLF as(36)V=V6+12rθ˜2.
Then, we compute the V’s first-ordered time derivative form as(37)V˙≤−∑i=16kiKvivi+∑i=26li22+εi22+1rθ˜−∑i=26rKvi2SiTSi2li2+θ^.+12ε12d2.
From (37), we choose the self-adaptive signal (also serve as self-training law) θ^. as(38)θ^.=∑i=26rKvi2SiTSi2li2−mθ^,in which the constants m and r should be positive.Theorem 1.
From this aforementioned design process, we have proposed the real controllers (uq and ud), the self-adaptive signal θ^., and the remunerate variables ζi,i=1,2,3,4,5,6, and thus, with the PMSM driving systems satisfying Assumption 1 in set Ωv, by selecting reasonable system parameters and by guaranteeing all-state variables within limitations, we can verify that the tracking errors would be bounded into a small range of origin.Proof.
The detailed proof will be elaborated in the next section.


## 5. Stability Analysis

To illustrate the system’s constancy, replacing (38) into (37),(39)V˙≤−∑i=16kiKvivi+∑i=26li22+εi22+12ε12d2−mθ˜θ^r.It has been verified thatlogkbi2/kbi2−vi2<vi2/kbi2−vi2 in the set vi<kbi in [48]. Plus, by using −θ˜θ^≤−θ˜2/2+θ2/2, the inequality (39) will be converted into(40)V˙≤−∑i=16kilogkbi2kbi2−vi2−mθ˜22r+∑i=26li22+εi22+12ε12d2+mθ22r≤−aV+b,in which a=min2k1,2k2/J,2k3,2k4,2k5,2k6,m and b=∑i=26li2/2+εi2/2+1/2ε12d2+mθ2/2r. With (40), we can infer that logkbi2/kbi2−vi2 and θ˜ will be all inside of the confined bounds. For inequality (40), multiplying eat at both sides, we have dVteat/dt≤beat. Next, at the time range of 0,t, we integrate this inequality and then it can be transformed into(41)Vt≤V0−bae−at+ba≤V0+ba.From inequality (41), we have vi<kbi and θ˜ will be all limited to certain bounds. Moreover, we can infer that limt⟶∞logkbi2/kbi2−vi2≤2b/a and limt⟶∞v1≤kb11−e−2b/a.Remark 1.
For compensating signals, we set the Lyapunov function as(42)V0=12ζ12+J2ζ22+12ζ32+12ζ42+12ζ52+12ζ62.
By employing Lemma1, we can obtain(43)V0.=ζ1ζ˙1+Jζ2ζ˙2+ζ3ζ˙3+ζ4ζ˙4+ζ5ζ˙5+ζ6ζ˙6≤−k1ζ12+ζ1ζ2+ζ1μ−k2ζ22+a1ζ2ζ3+a1ζ2μ−k3ζ32+b1ζ3ζ4+b1ζ3μ−k4ζ42−k5ζ52+c1ζ5ζ6+c1ζ5μ−k6ζ62≤−k1−1ζ12−k2−12−a1ζ22−k3−a12−b1ζ32−k4−b12ζ42−k5−c1ζ52−k6−c12ζ62+1+a1+b1+c12μ2≤−a0V0+b0.in which a0=min2k1−1,2/Jk2−1/2−a1,2k3−a1/2−b1,2k4−b1/2,2k5−c1,2k6−c1/2, and b0=1+a1+b1+c1/2μ2. Therefore, with these two equations, we have(44)limt⟶∞ζi≤1+a1+b1+c1a0μ,where i=1,2,…,6. Since v1=z1−ζ1, we can infer that z1≤v1+ζ1<kb11−e−2b/a+1+a1+b1+c1/a0μ.Remark 2.
Ensue from the definition ofa and b, along with the proper control variables m and kii=1,2,…,6, small variables εjj=2,3,…,6, lnn=2,3,…,6, and large parameter r, the rotor position tracking error z1 will be small enough to meet the control requirement. On account of z1=x1−xd as well as xd≤Y0, we have x1<kb1+1+a1+b1+c1/a0μ+Y0≤kc1. With the definition of α1 in equation (8), we know that α1 contains the terms of z1 as well as x˙d. Therefore, the upper limitation of α1 which is noted as ι1 holds. From x1,c−α1≤μ, we have x1,c≤μ+ι1≤λ1. Additionally, with v2=z2−ζ2, we get z2≤v2+ζ2<kb2+1+a1+b1+c1/a0μ. So, with z2=x2−x1,c, it is obvious that x2≤z2+x1,c<kc2. Parallelly, it can be verified that x3<kc3, x4<kc4, x5<kc5, and x6<kc6. At this point, the proof is accomplished.


## 6. Simulation Results

To substantiate the control scheme’s validity, a simulation has been conducted in this part. The PMSM’s parameters with core losses are chosen as Table1.Table 1
Parameters of PMSM with core losses.
Ld=0.00977HLmd=0.007HλPM=0.0844WbLq=0.00977HLmq=0.008HJ=0.002Kg⋅m2Lld=0.00177HRs=2.21Ωnp=3Llq=0.00177HRc=200ΩWe set all-state variables’ original conditions to zero and choosexd=0.5sint+0.5sin0.5t as the desired signal. So, θ=maxW22,W32,W42,W52,W62\\=0 . Moreover, the limitations of PMSM’s state variables are x1≤2, x2≤15, x3≤30,x4≤30,x5≤15,x6≤20. Additionally, we selected the load toque as TL=1,0≤t<151.5,t≥15. All these parameters of PMSM are obtained based on former experiences in [17, 20, 24]. Certainly, they can be obtained by some identification methods, such as gradient estimation algorithms [43], two-stage gradient-based iterative estimation method [44], or recursive parameter estimation methods [45].As to the RBF neural network, the neurons’ quantity is 11 and we choose the activate functions asSx=p2x,p3x,…,p6xT/∑i=26pix and pix=exp−x−ηiTZ−ηi/γi2 for i=2,3,4,5,6. The 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 BLFs-based adaptive command filtered neural network controllers. So, during this simulation, we employed control coefficients ask1=10, k2=7, k3=100, k4=50, k5=20, k6=30, r=0.05, m=0.02, l2=l3=l4=l5=l6=0.25, kb1=1, kb2=10, kb3=20, kb4=20, kb5=10, kb6=15, ξ=0.9, and ωn=2000.(b)
To show the superiority of the introduced methodology in this article, we also established the dynamic surface adaptive neural network controllers to control this system. To be comparable, we utilized identically the same parameters which are displayed in (a).Next, Figures2–7 illustrate these two simulations outcomes, in which Figures 2(a)–7(a) display the BLFs-based command filtered control (CFC) approach which was introduced in the frontal part of this article, while Figures 2(b)–7(b) demonstrate the dynamic surface control (DSC) scheme under the same circumstance. Figure 2 simulate the curves of x1 and xd. Figure 3 display the tracking error z1. Figures 4 and 5 are the curves of uq and ud, respectively. Figures 6 and 7 are the performances of state variables x2, x3, x4, x5, and x6.Figure 2
(a)x1 and xd for BLFs-based CFC. (b) x1 and xd for DSC.
(a)(b)Figure 3
(a) Tracking error for BLFs-based CFC. (b) Tracking error for DSC.
(a)(b)Figure 4
(a)uq for BLFs-based CFC. (b) uq for DSC.
(a)(b)Figure 5
(a)ud for BLFs-based CFC. (b) ud for DSC.
(a)(b)Figure 6
(a)ω, ioq, iq for BLFs-based CFC. (b) ω, ioq, iq for DSC.
(a)(b)Figure 7
(a)ω, iod, id for BLFs-based CFC. (b) ω, iod, id for DSC.
(a)(b)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 Figure3(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 ioq of Figure 6(b) is varying from −20 to 60, which overstepped the presupposed current’s constraint (Table 2).Table 2
The comparison performances between CFC and DSC.
ItemCFCDSCAbsolute percentage of tracking error (%)0.050.2Variation range ofω[−0.5, 2][−3, 3]Variation range ofiq[−10, 25][−10, 30]Variation range ofioq[−20, 20][−20, 60]Variation range ofid[−1, 3][0, 1]Variation range ofiod[−1, 2][−1, 2]Remark 3.
Disparate from the DSC control method without taking filtering errors into consideration, in this paper, we developed the BLFs-based adaptive command filtered neural control approach. This 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.


## 7. Conclusion

This 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.



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*Source: 6673568-2021-02-10.xml*