Neural Network Backstepping Controller Design for Uncertain Permanent Magnet Synchronous Motor Drive Chaotic Systems via Command Filter

In this study, an adaptive neural network (NN) command filtered control (CFC) method is proposed for a permanent magnet synchronous motor (PMSM) system with system uncertainties and external disturbance by means of a backstepping technique. At every backstepping step, a novel command filter is proposed, and the complicated virtual input and its derivative together can be approximated by this filter. The “explosion of complexity” problem in conventional backstepping design can be avoided because we do not need to calculate the derivative of the virtual input repeatedly. NNs are used to model system uncertainties and disturbances. Finally, an adaptive NN CFC is designed, and the convergence of the tracking error and the boundedness of all signals involved can be guaranteed. Finally, a simulation study is presented to verify the theoretical results.

1. Introduction

In the past several decades, adaptive backstepping control (ABC) has been used by more and more scholars due to its powerful ability in controlling non-linear systems. The ABC approach has some interesting properties. For example, it can achieve global ability and does not need a large amount of control energy. To increase the robustness of ABC, some other control methods, such as adaptive fuzzy control (AFC), adaptive neural network (NN) control, sliding mode control (SMC), etc., have been developed; the research results can be seen in references [1–10], and the references therein. However, the ABC approach has a drawback: the “explosion of complexity” problem, which is generated by differentiating the immediate virtual input repeatedly. Some efforts have been made to solve this problem, for example, in Liu et al. [6], virtual inputs were approximated by fuzzy systems, and in Ahn et al. [2], a sliding surface was used to avert the repeated calculation of the derivatives. Another approach, more powerful than these methods, is command filtered control (CFC), which was introduced by Farrell et al. [11] and Dong et al. [12], where several interesting results were presented to show that errors are of $◦\left(\frac{1}{W}\right)$, with W being the frequency. To drive the tracking error toward a sufficiently small value, one can use large W. However, too large a W value usually means that too much control energy is used. Thus, some other control methods based on CFC have been developed, for example, in references [13–18], noting that the dimensions of the virtual signal should be enlarged to involve the desired signal and its derivative. Yet, the above-mentioned literature only studied the estimation of some command derivatives, i.e., the results do not correspond to the ABC design. Thus, it is meaningful to develop more approaches to solve the above problem.

For more than 30 years, the control of chaotic systems has been paid increasing attention as an important field in non-linear scientific research and has gradually become widely used in engineering and other fields. The permanent magnet synchronous motor (PMSM) has attracted widespread attention due to its rapid dynamics, wide speed range, and simple structure. However, because the PMSM is a multivariate non-linear system and the system exhibits phenomena, such as Hope bifurcation, limit cycles, and chaotic attractors when the system parameters are in some ranges, the control of PMSM systems is still a challenging problem. Chaos in the PMSM system can destroy the stability of the system and even crash it, so it is very important to control this chaos. At present, there are many methods to control chaos in PMSM, such as the OGY method, delayed feedback control method, sliding mode control method, ABC, AFC, and so on [19–23]. In the actual application process, the OGY method requires certain system parameters, some of which cannot be achieved in actual control, whereas the delayed feedback control method has achieved good results in the PMSM chaos control, but the delay is very difficult. In Yu et al. [23], the AFC method was used, and in each step, fuzzy systems were used to model system uncertainties to avoid the repeated calculation of the virtual signal and its derivative. In Sun et al. [24], an internal motion model was used to control PMSM systems with uncertainties. In Yang et al. [25], an AFC CFC method was used where the system uncertainties are not considered. In Niu et al. [26], an output feedback CFC method was proposed for PMSM. In Zou et al. [27], command filtering-based AFC was introduced for PMSM where input saturation is considered. Some related work can be seen in references [28–32]. However, in these studies, fully unknown system models are not considered.

Based on the above discussion, we will introduce an NN CFC method for PMSM systems with fully unknown system models. We combine ABC with CFC and propose a one-order filter to approximate the virtual signal and its derivative at each backstepping step. As the last step, a robust controller is designed, and adaptation laws are also presented. Compared with related works, our contributions are as follows. (1) A one-order filter is introduced. In each step, it can be used to approximate the virtual signal together with its derivative. In addition, the proposed filter has very good approximation ability, and the error can be made as small as possible. By doing this, the “explosion of complexity” problem is avoided. Compared with some related methods, for example, in references [23, 28, 31], our methods are simpler and can be implemented earlier. (2) The proposed control signals with adaptation laws have a very concise form relative to some related methods, for example, dynamic surface control.

This paper is arranged as follows. The description of the PMSM, the controller design, and the stability analysis are presented in section 2. Section 3 gives the simulation results of the proposed method. Finally, section 4 gives the conclusions of this paper.

2. Main Results

2.1. Problem Description

The mathematical model of a PMSM with a smooth air gap can be expressed as [23]

$\begin{array}{ll}\{\begin{array}{l}\frac{d\omega }{dt}=\sigma (i_q-\omega )-{\tilde{T}}_L,\\ \frac{d{i}_q}{dt}=-i_q-i_d\omega +\delta \omega +{\tilde{u}}_q,\\ \frac{d{i}_d}{dt}=-i_d+i_q\omega +{\tilde{u}}_d,\end{array}& (1)\end{array}$

where ω, i_d, i_q are system variables representing the angular velocity and shaft current of the motor, respectively, and ${\tilde{T}}_L$, ũ_q, and ũ_d represent load torque and shaft voltage. When there are no external inputs, denoting x = ω, y = i_q, z = i_d, and putting an input u(t) to the third equation, the PMSM system (1) can be written as

$\begin{array}{ll}\{\begin{array}{l}\stackrel{·}{x}=\sigma (y-x),\\ \stackrel{·}{y}=-y-xz+\delta x,\\ \stackrel{·}{z}=-z+xy+u.\end{array}& (2)\end{array}$

Let the tracking error be ${ϵ}_1=x-x^c$, with $x^c\in {R}$ being a referenced signal. Our purpose is to implement a suitable control signal u such that ϵ₁(t) becomes as small as possible.

2.2. Backstepping Control Signal Design

The backstepping control procedures can be divided into the following three steps.

Step 1. It follows from (2) that:

$\begin{array}{ll}\stackrel{·}{x}=\sigma y+\Delta g_1(x),& (3)\end{array}$

where Δg₁(x) = −σx is assumed to be unknown. Then, Δg₁(x) is estimated by using NN as

$\begin{array}{ll}\Delta g_1=W_1^T{\vartheta }_1=W_1^{*T}{\vartheta }_1+{\epsilon }_1& (4)\end{array}$

with $W_1^T$ being the adjustable parameter of the NN, $W_1^{*T}$ being the optimal parameter, and ε₁ being the optimal estimation error [33, 34]. Then, we can use the following virtual input

$\begin{array}{ll}{\rho }_1=-\frac{1}{\sigma }\left[k_1{ϵ}_1+W_1^T{\vartheta }_1(x)+{\widehat{\epsilon }}_{1}arctan\left(\frac{{\widetilde{ϵ}}_1}{{\alpha }_1}\right)-{\stackrel{·}{x}}^c\right]& (5)\end{array}$

with ${\widehat{\epsilon }}_1$ being the estimation of ε₁, k₁, α₁ > 0, and ${\widetilde{ϵ}}_1$ being a compensated tracking error. Thus, (3), (4), and (5) imply

$\begin{array}{ll}\begin{array}{l}{\stackrel{\bullet }{ϵ}}_1=\sigma y+\Delta g_1-{\stackrel{·}{x}}^c\\ =-k_1{ϵ}_1+\sigma (y-{\rho }_1)+W_1^{*T}{\vartheta }_1(x)+{\epsilon }_1-W_1^T{\vartheta }_1(x)\\ -{\widehat{\epsilon }}_{1}arctan\left(\frac{{\widetilde{ϵ}}_1}{{\alpha }_1}\right)\\ =-k_1{ϵ}_1+\sigma (y^c-{\rho }_1)-{\tilde{W}}_1^T{\vartheta }_1(x)+{\epsilon }_1\\ -{\widehat{\epsilon }}_{1}arctan\left(\frac{{\widetilde{ϵ}}_1}{{\alpha }_1}\right)+\sigma {ϵ}_2,\end{array}& (6)\end{array}$

with ${\tilde{W}}_1^T=W_1^T-W_1^{*T}$ being the NN approximation error and ${ϵ}_2=y-y^c$ being filtered error. We can define the following signal:

$\begin{array}{ll}{\widetilde{ϵ}}_1={ϵ}_1-{\zeta }_1& (7)\end{array}$

where ζ₁ can be obtained by solving:

$\begin{array}{ll}{\stackrel{\bullet }{\zeta }}_1=-k_1{\zeta }_1+\sigma (y^c-{\rho }_1)+\sigma {\zeta }_2& (8)\end{array}$

with ζ₂ being defined in the next step and y^c being the solution of the following equation:

$\begin{array}{ll}{\stackrel{·}{y}}^c=-{\varpi }_2(y^c-{\rho }_1)& (9)\end{array}$

with ϖ₂ > 0. To solve (9), we can set y^c(0) = 0. We can use the following adaptation law:

$\begin{array}{ll}{\stackrel{\bullet }{W}}_1=a_{11}{\widetilde{ϵ}}_1{\vartheta }_1(x)-a_{11}{a}_{12}{W}_1& (10)\end{array}$

and

$\begin{array}{ll}{\stackrel{\bullet }{\widehat{\epsilon }}}_1=a_{41}{\widetilde{ϵ}}_{1}tanh\left(\frac{{\widetilde{ϵ}}_1}{{\alpha }_1}\right)-a_{41}{a}_{42}{\widehat{\epsilon }}_1& (11)\end{array}$

respectively, with a₁₁, a₁₂, a₄₁, a₄₂ > 0.

Step 2. According to the second equation of (2), we have

$\begin{array}{ll}\stackrel{·}{y}=-xz+\Delta g_2& (12)\end{array}$

where Δg₂ = −y + δx is unknown. We can use the NN to approximate it as

$\begin{array}{ll}\Delta g_2=W_2^T{\vartheta }_2=W_2^{*T}{\vartheta }_2+{\epsilon }_2.& (13)\end{array}$

Define

$\begin{array}{ll}\{\begin{array}{l}{ϵ}_2=y-y^c,\\ {\widetilde{ϵ}}_2={ϵ}_2-{\zeta }_2,\end{array}& (14)\end{array}$

with

$\begin{array}{ll}{\stackrel{\bullet }{\zeta }}_2=-k_2{\zeta }_2+z^c-{\rho }_2+{\zeta }_3& (15)\end{array}$

with ζ₂(0) = 0,

$\begin{array}{ll}{\stackrel{·}{z}}^c=-{\varpi }_3(z^c-{\rho }_2),& (16)\end{array}$

$\begin{array}{ll}{\rho }_2=-k_2{ϵ}_2-W_2^T{\vartheta }_2-{\widehat{\epsilon }}_{2}arctan\left(\frac{{\widetilde{ϵ}}_2}{{\alpha }_2}\right)+{\stackrel{·}{y}}^c-\sigma {\widetilde{ϵ}}_1,& (17)\end{array}$

where k₂, α₂ > 0. W₂ and ${\widehat{\epsilon }}_2$ are updated by

$\begin{array}{ll}{\stackrel{\bullet }{W}}_2=a_{21}{\widetilde{ϵ}}_2{\vartheta }_2({\bar{x}}_2)-a_{21}{a}_{22}{W}_2& (18)\end{array}$

and

$\begin{array}{ll}{\stackrel{\bullet }{\widehat{\epsilon }}}_2=a_{51}{\widetilde{ϵ}}_{2}tanh\left(\frac{{\widetilde{ϵ}}_2}{{\alpha }_1}\right)-a_{51}{a}_{52}{\widehat{\epsilon }}_2& (19)\end{array}$

with a₂₁, a₂₂, a₅₁, a₅₂ > 0. Then we know

$\begin{array}{ll}\begin{array}{l}{\stackrel{\bullet }{ϵ}}_2=z+\Delta g_2-{\stackrel{·}{y}}^c\\ =-k_2{ϵ}_2+z-{\rho }_2+W_2^{*T}{\vartheta }_2+{\epsilon }_2-W_2^T{\vartheta }_2\\ -{\widehat{\epsilon }}_{2}arctan\left(\frac{{\widetilde{ϵ}}_2}{{\alpha }_2}\right)-\sigma {\widetilde{ϵ}}_1\\ =-k_2{ϵ}_2+z^c-{\rho }_2-{\tilde{W}}_2^T{\vartheta }_2+{\epsilon }_2-{\widehat{\epsilon }}_{2}arctan\left(\frac{{\widetilde{ϵ}}_2}{{\alpha }_2}\right)\\ -\sigma {\widetilde{ϵ}}_1+{ϵ}_3.\end{array}& (20)\end{array}$

Step 3. According to the last equation of (2), we have

$\begin{array}{ll}\stackrel{·}{z}=\Delta g_3+u& (21)\end{array}$

with Δg₃(x) = −z + xy being unknown. It can be approximated by

$\begin{array}{ll}\Delta g_3=W_3^T{\vartheta }_3=W_3^{*T}{\vartheta }_3+{\epsilon }_3.& (22)\end{array}$

We can implement the controller as

$\begin{array}{ll}u=-k_3{ϵ}_3-W_3^T{\vartheta }_3-{\widehat{\epsilon }}_{2}arctan\left(\frac{{\widetilde{ϵ}}_3}{{\alpha }_3}\right)+{\stackrel{·}{z}}^c-{ϵ}_2& (23)\end{array}$

with k₃, α₃ > 0. Define

$\begin{array}{ll}\{\begin{array}{l}{ϵ}_3=z-z^c,\\ {\widetilde{ϵ}}_3={ϵ}_3-{\zeta }_3,\end{array}& (24)\end{array}$

with

$\begin{array}{ll}{\stackrel{\bullet }{\zeta }}_3=-k_3{\zeta }_3-{\zeta }_2.& (25)\end{array}$

The parameters are updated by

$\begin{array}{ll}{\stackrel{\bullet }{W}}_3=a_{31}{\widetilde{ϵ}}_3{\vartheta }_2-a_{31}{a}_{32}{W}_3,& (26)\end{array}$

$\begin{array}{ll}{\stackrel{\bullet }{\widehat{\epsilon }}}_3=a_{61}{\widetilde{ϵ}}_{3}tanh\left(\frac{{\widetilde{ϵ}}_3}{{\alpha }_3}\right)-a_{61}{a}_{62}{\widehat{\epsilon }}_3& (27)\end{array}$

with c₃₁, c₃₂, c₆₁, c₆₂ > 0. As a result,

$\begin{array}{ll}\begin{array}{l}{\stackrel{\bullet }{ϵ}}_3=\Delta g_3-{\stackrel{·}{z}}^c+u\\ =-k_3{ϵ}_3+W_3^{*T}{\vartheta }_3(\bar{x})+{\epsilon }_3-W_3^T{\vartheta }_3(\bar{x})-{\widehat{\epsilon }}_{3}arctan\\ \left(\frac{{\widetilde{ϵ}}_3}{{\alpha }_3}\right)-{ϵ}_2\\ =-k_3{ϵ}_3-{\tilde{W}}_3^T{\vartheta }_3(\bar{x})+{\epsilon }_3-{\widehat{\epsilon }}_{3}arctan\left(\frac{{\widetilde{ϵ}}_3}{{\alpha }_3}\right)-{ϵ}_2.\end{array}& (28)\end{array}$

Let us give the following reasonable assumption and lemma.

Assumption 1. The NN approximate error is bounded, i.e., there exists ${\epsilon }_i^{*}$ such that ${\epsilon }_i\le {\epsilon }_i^{*}$.

Lemma 1. [15] If α > 0 and κ = 0.27846, then we have

$\left|y\right|-ytanh\left(\frac{y}{\alpha }\right)\le \kappa \alpha .$

Then, we have

$\begin{array}{ll}\begin{array}{l}{\stackrel{\bullet }{\widetilde{ϵ}}}_1=-k_1{ϵ}_1+\sigma (y^c-{\rho }_1)-{\tilde{W}}_1^T{\vartheta }_1+{\epsilon }_1-{\widehat{\epsilon }}_{1}arctan\left(\frac{{\widetilde{ϵ}}_1}{{\alpha }_1}\right)\\ +\sigma {ϵ}_2-{\stackrel{\bullet }{\zeta }}_1\\ =-k_1{ϵ}_1-{\tilde{W}}_1^T{\vartheta }_1+{\epsilon }_1-k_1{\zeta }_1-{\widehat{\epsilon }}_{1}arctan\left(\frac{{\widetilde{ϵ}}_1}{{\alpha }_1}\right)\\ +\sigma {ϵ}_2-\sigma {\zeta }_2\\ =-k_1{\widetilde{ϵ}}_1+\sigma {\widetilde{ϵ}}_2-{\tilde{W}}_1^T{\vartheta }_1+{\epsilon }_1-{\widehat{\epsilon }}_{1}arctan\left(\frac{{\widetilde{ϵ}}_1}{{\alpha }_1}\right),\end{array}& (29)\end{array}$

$\begin{array}{ll}\begin{array}{l}{\stackrel{\bullet }{\widetilde{ϵ}}}_2=-k_2{ϵ}_2+z^c-{\rho }_2-{\tilde{W}}_2^T{\vartheta }_2+{\epsilon }_2-{\widehat{\epsilon }}_{2}arctan\left(\frac{{\widetilde{ϵ}}_2}{{\alpha }_2}\right)\\ -\sigma {\widetilde{ϵ}}_1+{ϵ}_3-{\stackrel{\bullet }{\zeta }}_2\\ =-k_2{ϵ}_2-{\tilde{W}}_2^T{\vartheta }_2+{\epsilon }_2-{\zeta }_3-{\widehat{\epsilon }}_{2}arctan\left(\frac{{\widetilde{ϵ}}_2}{{\alpha }_2}\right)\\ -\sigma {\widetilde{ϵ}}_1+{ϵ}_3-k_2{\zeta }_2\\ =-k_2{\widetilde{ϵ}}_2-{\tilde{W}}_2^T{\vartheta }_2+{\epsilon }_2+{\widetilde{ϵ}}_3-{\widehat{\epsilon }}_{2}arctan\left(\frac{{\widetilde{ϵ}}_2}{{\alpha }_2}\right)-\sigma {\widetilde{ϵ}}_1,\end{array}& (30)\end{array}$

$\begin{array}{ll}\begin{array}{l}{\stackrel{\bullet }{\tilde{e}}}_3=-k_3{ϵ}_3-{\tilde{W}}_3^T{\vartheta }_3+{\epsilon }_3-{\stackrel{\bullet }{\zeta }}_3-{ϵ}_2-{\widehat{\epsilon }}_{3}arctan\left(\frac{{\widetilde{ϵ}}_3}{{\alpha }_3}\right)\\ =-k_3{\widetilde{ϵ}}_3-{\tilde{W}}_3^T{\vartheta }_3(\bar{x})+{\epsilon }_3-{\widetilde{ϵ}}_2-{\widehat{\epsilon }}_{3}arctan\left(\frac{{\widetilde{ϵ}}_3}{{\alpha }_3}\right).\end{array}& (31)\end{array}$

Theorem 1. When $|x_i^c-z_i|\le b$ with b > 0, then (8), (15), and (25) imply

$\begin{array}{ll}\parallel \zeta \parallel \le \frac{c}{2\widehat{k}}(1-e^{-2\widehat{k}t})& (32)\end{array}$

with $\zeta ={[{\zeta }_1,{\zeta }_2,{\zeta }_3]}^T\in {{R}}^3$, $\widehat{k}=\frac{1}{2}min\{k_1,k_2,k_3\}$, and c = b + σ.

Proof. Let $V_1=\frac{1}{2}\parallel \zeta {\parallel }^2$. It follows from (8), (15), and (25) that

$\begin{array}{ll}\begin{array}{l}{\stackrel{\bullet }{V}}_1=-{\displaystyle \sum _{i=1}^3}{k}_i{\zeta }_i^2+\sigma {\zeta }_1(y^c-{\rho }_1)+{\zeta }_2(z^c-{\rho }_2)\\ \le -2\widehat{k}\parallel \zeta {\parallel }^2+A\parallel \zeta \parallel \\ \le -4\widehat{k}{V}_1+\sqrt{2}a\sqrt{V_1}.\end{array}& (33)\end{array}$

As a result, it follows from (33) that (32) satisfies.     ■

Theorem 2. Consider (2) satisfying Assumption 1. Virtual inputs are given by (5) and (17) under the filters (8), (9), (15), (16), and (25). The adaptation laws are (10), (18), (26), (11), (19), and (27). Then, the controller (23) ensures the convergences of ${\widetilde{ϵ}}_1$, ${\widetilde{ϵ}}_2$ and ${\widetilde{ϵ}}_3$ to a small region.

Proof. Define

$\begin{array}{ll}V=\frac{1}{2}{\displaystyle \sum _{i=1}^3}{\widetilde{ϵ}}_i^2+{\displaystyle \sum _{i=1}^3}\frac{1}{2c_{i1}}{\tilde{W}}_i^T{\tilde{W}}_i+{\displaystyle \sum _{i=1}^3}\frac{1}{2c_{i+3,1}}{\tilde{\epsilon }}_i^2& (34)\end{array}$

with ${\tilde{\epsilon }}_i={\widehat{\epsilon }}_i-{\epsilon }_i^{*}$. Then, (29), (30), (31), Assumption 1, and Lemma 1 imply

$\begin{array}{ll}\begin{array}{l}{\displaystyle \sum _{i=1}^3}{\widetilde{ϵ}}_i{\stackrel{\bullet }{\widetilde{ϵ}}}_i={\displaystyle \sum _{i=1}^3}{\tilde{e}}_i\left[{\epsilon }_i-{\widehat{\epsilon }}_{i}arctan\left(\frac{{\tilde{e}}_i}{{\alpha }_i}\right)\right]-{\widetilde{ϵ}}_1{\tilde{W}}_1^T{\vartheta }_1-{\widetilde{ϵ}}_2{\tilde{W}}_2^T{\vartheta }_2\\ -{\widetilde{ϵ}}_3{\tilde{W}}_3^T{\vartheta }_3-{\displaystyle \sum _{i=1}^3}{k}_i{\tilde{e}}_i^2\\ \le {\displaystyle \sum _{i=1}^3}\left[|{\tilde{e}}_i|{\epsilon }_i^{*}-{\tilde{e}}_i{\widehat{\epsilon }}_{i}arctan\left(\frac{{\tilde{e}}_i}{{\alpha }_i}\right)\right]-{\widetilde{ϵ}}_1{\tilde{W}}_1^T{\vartheta }_1\\ -{\widetilde{ϵ}}_2{\tilde{W}}_2^T{\vartheta }_2-{\widetilde{ϵ}}_3{\tilde{W}}_3^T{\vartheta }_3-{\displaystyle \sum _{i=1}^3}{k}_i{\tilde{e}}_i^2\\ ={\displaystyle \sum _{i=1}^3}\left[|{\tilde{e}}_i|{\epsilon }_i^{*}-{\tilde{e}}_i{\widehat{\epsilon }}_{i}arctan\left(\frac{{\tilde{e}}_i}{{\alpha }_i}\right)-{\tilde{e}}_i{\epsilon }_i^{*}arctan\left(\frac{{\tilde{e}}_i}{{\alpha }_i}\right)\right]\\ -{\widetilde{ϵ}}_2{\tilde{W}}_2^T{\vartheta }_2-{\widetilde{ϵ}}_1{\tilde{W}}_1^T{\vartheta }_1\\ +{\displaystyle \sum _{i=1}^3}{\tilde{e}}_i{\epsilon }_i^{*}arctan\left(\frac{{\tilde{e}}_i}{{\alpha }_i}\right)-{\displaystyle \sum _{i=1}^3}{k}_i{\tilde{e}}_i^2-{\widetilde{ϵ}}_3{d}^{*}arctan\left(\frac{{\widetilde{ϵ}}_3}{{\alpha }_3}\right)\\ -{\widetilde{ϵ}}_3{\tilde{W}}_3^T{\vartheta }_3\\ \le {\displaystyle \sum _{i=1}^3}\left[-{\tilde{e}}_i{\widehat{\epsilon }}_{i}arctan\left(\frac{{\tilde{e}}_i}{{\alpha }_i}\right)+{\tilde{e}}_i{\epsilon }_i^{*}arctan\left(\frac{{\tilde{e}}_i}{{\alpha }_i}\right)\right]\\ -{\displaystyle \sum _{i=1}^3}{k}_i{\tilde{e}}_i^2-{\widetilde{ϵ}}_2{\tilde{W}}_2^T{\vartheta }_2\\ -{\widetilde{ϵ}}_3{\tilde{W}}_3^T{\vartheta }_3-{\widetilde{ϵ}}_1{\tilde{W}}_1^T{\vartheta }_1+\kappa {\displaystyle \sum _{i=1}^3}{\alpha }_i{\epsilon }_i^{*}\\ =-{\displaystyle \sum _{i=1}^3}{\tilde{e}}_i{\tilde{\epsilon }}_{i}arctan\left(\frac{{\tilde{e}}_i}{{\alpha }_i}\right)-{\displaystyle \sum _{i=1}^3}{k}_i{\tilde{e}}_i^2-{\widetilde{ϵ}}_2{\tilde{W}}_2^T{\vartheta }_2\\ -{\widetilde{ϵ}}_3{\tilde{W}}_3^T{\vartheta }_3-{\widetilde{ϵ}}_1{\tilde{W}}_1^T{\vartheta }_1+\kappa {\displaystyle \sum _{i=1}^3}{\alpha }_i{\epsilon }_i^{*}.\end{array}& (35)\end{array}$

It follows from (10), (11), (18), (19), (26), and (27) that

$\begin{array}{ll}\begin{array}{l}{\displaystyle \sum _{i=1}^3}\frac{1}{a_{i1}}{\tilde{W}}_i^T{\stackrel{\bullet }{\tilde{W}}}_i={\widetilde{ϵ}}_2{\tilde{W}}_2^T{\vartheta }_2+{\widetilde{ϵ}}_3{\tilde{W}}_3^T{\vartheta }_3+{\widetilde{ϵ}}_1{\tilde{W}}_1^T{\vartheta }_1\\ -{\displaystyle \sum _{i=1}^3}{a}_{i2}{\tilde{W}}_i^T{W}_i\\ ={\widetilde{ϵ}}_2{\tilde{W}}_2^T{\vartheta }_2+{\widetilde{ϵ}}_3{\tilde{W}}_3^T{\vartheta }_3+{\widetilde{ϵ}}_1{\tilde{W}}_1^T{\vartheta }_1\\ -{\displaystyle \sum _{i=1}^3}{a}_{i2}{\tilde{W}}_i^T({\tilde{W}}_i+W_i^{*})\\ \le {\widetilde{ϵ}}_2{\tilde{W}}_2^T{\vartheta }_2+{\widetilde{ϵ}}_3{\tilde{W}}_3^T{\vartheta }_3+{\widetilde{ϵ}}_1{\tilde{W}}_1^T{\vartheta }_1\\ -{\displaystyle \sum _{i=1}^3}\frac{a_{i2}}{2}{\tilde{W}}_i^T{\tilde{W}}_i+{\displaystyle \sum _{i=1}^3}\frac{a_{i2}}{2}{W}_i^{*T}{W}_i^{*},\end{array}& (36)\end{array}$

$\begin{array}{ll}\begin{array}{l}{\displaystyle \sum _{i=1}^3}\frac{1}{2a_{i+3,1}}{\tilde{\epsilon }}_i{\stackrel{\bullet }{\tilde{\epsilon }}}_i={\displaystyle \sum _{i=1}^3}{\tilde{\epsilon }}_i\left[{\tilde{e}}_{i}tanh\left(\frac{{\tilde{e}}_i}{{\alpha }_i}\right)-a_{i+3,2}{\widehat{\epsilon }}_i\right]\\ ={\displaystyle \sum _{i=1}^3}{\tilde{\epsilon }}_i{\tilde{e}}_{i}tanh\left(\frac{{\tilde{e}}_i}{{\alpha }_i}\right)-{\displaystyle \sum _{i=1}^3}{a}_{i+3,2}{\tilde{\epsilon }}_i({\tilde{\epsilon }}_i+{\epsilon }_i^{*})\\ \le {\displaystyle \sum _{i=1}^3}{\tilde{\epsilon }}_i{\tilde{e}}_{i}tanh\left(\frac{{\tilde{e}}_i}{{\alpha }_i}\right)-{\displaystyle \sum _{i=1}^3}\frac{a_{i+3,2}}{2}{\tilde{\epsilon }}_i^2\\ +{\displaystyle \sum _{i=1}^3}\frac{a_{i+3,2}}{2}{\epsilon }_i^{*2}.\end{array}& (37)\end{array}$

As a result, (35), (36), and (37) imply

$\begin{array}{ll}\begin{array}{l}\stackrel{\bullet }{V}\le -{\displaystyle \sum _{i=1}^3}{k}_i{\widetilde{ϵ}}_i^2+\kappa {\displaystyle \sum _{i=1}^3}{\alpha }_i{\epsilon }_i^{*}-{\displaystyle \sum _{i=1}^3}\frac{a_{i+3,2}}{2}{\tilde{\epsilon }}_i^2+{\displaystyle \sum _{i=1}^3}\frac{a_{i+3,2}}{2}{\epsilon }_i^{*2}\\ -{\displaystyle \sum _{i=1}^3}\frac{a_{i2}}{2}{\tilde{W}}_i^T{\tilde{W}}_i+{\displaystyle \sum _{i=1}^3}\frac{a_{i2}}{2}{W}_i^{*T}{W}_i^{*}\\ =-{\displaystyle \sum _{i=1}^3}{k}_i{\tilde{e}}_i^2-{\displaystyle \sum _{i=1}^3}\frac{a_{i+3,2}}{2}{\tilde{\epsilon }}_i^2-{\displaystyle \sum _{i=1}^3}\frac{a_{i2}}{2}{\tilde{W}}_i^T{\tilde{W}}_i+\kappa {\displaystyle \sum _{i=1}^3}{\alpha }_i{\epsilon }_i^{*}\\ +{\displaystyle \sum _{i=1}^3}\frac{a_{i+3,2}}{2}{\epsilon }_i^{*2}+{\displaystyle \sum _{i=1}^3}\frac{a_{i2}}{2}{W}_i^{*T}{W}_i^{*}\\ \le -\frac{d_1}{2}{\displaystyle \sum _{i=1}^3}{\tilde{e}}_i^2-d_2{\displaystyle \sum _{i=1}^3}\frac{1}{2a_{i1}}{\tilde{W}}_i^T{\tilde{W}}_i-d_3{\displaystyle \sum _{i=1}^3}\frac{1}{2a_{i+3,1}}{\tilde{\epsilon }}_i^2+d_4\end{array}& (38)\end{array}$

with d₁ = 2 min{k₁, k₂, k₃}, d₂ = min{a₁₁a₁₂, a₂₁a₂₂, a₃₁a₃₂}, d₃ = min{a₄₁a₄₂, a₅₁a₅₂, a₆₁a₆₂}, $d_4=\kappa \sum _{i=1}^3{\alpha }_i{\epsilon }_i^{*}+\sum _{i=1}^3\frac{a_{i+3,2}}{2}{\epsilon }_i^{*2}+\sum _{i=1}^3\frac{a_{i2}}{2}{W}_i^{*T}{W}_i^{*}$ being non-negative constants. Apparently, the constants d₁, d₂, d₃, d₄ are determined by design parameters and some unknown constant variable (see, the optimal NN parameter). Thus, (38) implies $\frac{1}{2}\sum _{i=1}^3{\tilde{e}}_i^2\le \frac{d_4}{d_1}$, $\sum _{i=1}^3\frac{1}{2a_{i1}}{\tilde{W}}_i^T{\tilde{W}}_i\le \frac{d_4}{d_2}$, $\sum _{i=1}^3\frac{1}{2a_{i+3,1}}{\tilde{\epsilon }}_i^2\le \frac{d_4}{d_3}$. As a result, all variables are indeed bounded, and ${\widetilde{ϵ}}_1$, ${\widetilde{ϵ}}_2$ and ${\widetilde{ϵ}}_3$ tend to a small region determined by design parameters.     ■

Remark 1. It should be emphasized that the conclusion of Theorem 1 is very representative, and it can be widely used in the field of automatic control, finite-time control, and backstepping control. However, the proposed method can only guarantee the that tracking error tends to a very small region.

Remark 2. In the controller design, the proposed method is different from some related methods, for example, those detailed in references [4, 33, 34]. We introduce an auxiliary signal to approximate the virtual input, and the approximation error can be made as small as possible.

3. Simulation Study

In system (2), let σ = 5.45, δ = 20.0, x(0) = 0.5, y(0) = −1, z(0) = 0. When u(t) ≡ 0, the chaotic behavior of (2) can be seen in Figure 1.

FIGURE 1

Figure 1. Chaotic phenomenon of (2) in (A) x − y − z space, (B) x(t), (C) y(t), and (D) z(t).

Let the desired signal be x^c, defined by

$\begin{array}{l}{x}^c=\{\begin{array}{ll}0& t\in \left[0,8\right],\\ 2& \hfill t>8.\end{array}\end{array}$

With respect to the NNs, the basic functions are chosen on interval [-8 8], and five functions are used for each state. Their initial conditions are W₁(0) = 0_{1 × 2}, W₂(0) = 0_{1 × 25}, and W₃(0) = 0_{1 × 125}. The design parameters are k₁ = k₂ = k₃ = 1.5, a_i1 = 6, a_i2 = 0.05, i = 1, 2, 3, 4, 5, 6, α₁ = α₂ = α₃ = 1.

The simulation results are presented in Figure 2. It can be seen in Figure 2 that the tracking error has rapid convergence and the signal x(t) tracks x^c(t) tightly; the control input u(t) has a small amplitude, and it fluctuates very gently; the compensated error ϵ₃ converges to zero very quickly, but ϵ₂ does not converge to zero (in fact, in our method, it is not necessary for the compensated error to converge to zero); the parameters of the NNs also fluctuate gently.

FIGURE 2

Figure 2. Simulation results in (A) tracking performance, (B) control input, (C) compensated errors, (D) W₁(t), (E) W₂(t), and (F) W₃(t).

To show the robustness of the proposed method, let us add an external term 0.95 sin t into the third equation of the system (2). The simulation results are presented in Figure 3. Comparing Figures 2, 3, we can see that under the external disturbance, the proposed method has very good robustness.

FIGURE 3

Figure 3. Simulation results with external disturbance in (A) tracking performance, (B) control input, (C) compensated errors, (D) ||W₁(t)||, ||W₂(t)|| and ||W₃(t)||.

4. Conclusions

This paper presents an NN CFC method for PMSMs with fully unknown system models. To avoid the “explosion of complexity” problem, we propose a one-order command filter. It has been proven that the virtual input and its derivative can together be approximated by the proposed filter, and the approximation error can be made as small as possible. The proposed method is performed by using a backstepping technique. It is also shown that the proposed NN CFC can guarantee the boundedness of all signals. Investigating CFC for PMSMs with input constraints will be our future research direction.

Data Availability Statement

All datasets generated for this study are included in the article/supplementary material.

Author Contributions

RL contributed the conception of the study. YD wrote the study and organized the literature analysis. YX wrote the simulation programs. RL and YX revised the manuscript. All authors contributed to manuscript revision and read and approved the submitted version.

Funding

This work was supported by the Natural Science Foundation of Guangxi University for Nationalities (2019KJYB001), and the Guangxi Natural Science Foundation (2019GXNSFAA185007).

Conflict of Interest

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

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