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Brief Research Report ARTICLE

Front. Phys., 05 June 2020 | https://doi.org/10.3389/fphy.2020.00182

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

Ricai Luo1, Yanping Deng2* and Yuling Xie3
  • 1School of Mathematics and Statistics, Hechi University, Yizhou, China
  • 2School of Mathematics and Physics, Guangxi University for Nationalities, Nanning, China
  • 3Department of Trial Production, North Institute of Automatic Control Technology, Taiyuan, China

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 [110], 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 (1W), 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 [1318], 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 [1923]. 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 [2832]. 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]

{dωdt=σ(iq-ω)-T~L,diqdt=-iq-idω+δω+u~q,diddt=-id+iqω+u~d,    (1)

where ω, id, iq are system variables representing the angular velocity and shaft current of the motor, respectively, and T~L, ũq, and ũd represent load torque and shaft voltage. When there are no external inputs, denoting x = ω, y = iq, z = id, and putting an input u(t) to the third equation, the PMSM system (1) can be written as

{x·=σ(y-x),y·=-y-xz+δx,z·=-z+xy+u.    (2)

Let the tracking error be ϵ1=x-xc, with xcR 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:

x·=σy+Δg1(x),    (3)

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

Δg1=W1Tϑ1=W1*Tϑ1+ε1    (4)

with W1T being the adjustable parameter of the NN, W1*T being the optimal parameter, and ε1 being the optimal estimation error [33, 34]. Then, we can use the following virtual input

ρ1=-1σ[k1ϵ1+W1Tϑ1(x)+ε^1arctan(ϵ~1α1)-x·c]    (5)

with ε^1 being the estimation of ε1, k1, α1 > 0, and ϵ~1 being a compensated tracking error. Thus, (3), (4), and (5) imply

ϵ1=σy+Δg1-x·c      =-k1ϵ1+σ(y-ρ1)+W1*Tϑ1(x)+ε1-W1Tϑ1(x)      -ε^1arctan(ϵ~1α1)      =-k1ϵ1+σ(yc-ρ1)-W~1Tϑ1(x)+ε1      -ε^1arctan(ϵ~1α1)+σϵ2,    (6)

with W~1T=W1T-W1*T being the NN approximation error and ϵ2=y-yc being filtered error. We can define the following signal:

ϵ~1=ϵ1-ζ1    (7)

where ζ1 can be obtained by solving:

ζ1=-k1ζ1+σ(yc-ρ1)+σζ2    (8)

with ζ2 being defined in the next step and yc being the solution of the following equation:

y·c=-ϖ2(yc-ρ1)    (9)

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

W1=a11ϵ~1ϑ1(x)-a11a12W1    (10)

and

ε^1=a41ϵ~1tanh(ϵ~1α1)-a41a42ε^1    (11)

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

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

y·=-xz+Δg2    (12)

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

Δg2=W2Tϑ2=W2*Tϑ2+ε2.    (13)

Define

{ϵ2=y-yc,ϵ~2=ϵ2-ζ2,    (14)

with

ζ2=-k2ζ2+zc-ρ2+ζ3    (15)

with ζ2(0) = 0,

z·c=-ϖ3(zc-ρ2),    (16)
ρ2=-k2ϵ2-W2Tϑ2-ε^2arctan(ϵ~2α2)+y·c-σϵ~1,    (17)

where k2, α2 > 0. W2 and ε^2 are updated by

W2=a21ϵ~2ϑ2(x̄2)-a21a22W2    (18)

and

ε^2=a51ϵ~2tanh(ϵ~2α1)-a51a52ε^2    (19)

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

ϵ2=z+Δg2-y·c      =-k2ϵ2+z-ρ2+W2*Tϑ2+ε2-W2Tϑ2      -ε^2arctan(ϵ~2α2)-σϵ~1      =-k2ϵ2+zc-ρ2-W~2Tϑ2+ε2-ε^2arctan(ϵ~2α2)      -σϵ~1+ϵ3.    (20)

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

z·=Δg3+u    (21)

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

Δg3=W3Tϑ3=W3*Tϑ3+ε3.    (22)

We can implement the controller as

u=-k3ϵ3-W3Tϑ3-ε^2arctan(ϵ~3α3)+z·c-ϵ2    (23)

with k3, α3 > 0. Define

{ϵ3=z-zc,ϵ~3=ϵ3-ζ3,    (24)

with

ζ3=-k3ζ3-ζ2.    (25)

The parameters are updated by

W3=a31ϵ~3ϑ2-a31a32W3,    (26)
ε^3=a61ϵ~3tanh(ϵ~3α3)-a61a62ε^3    (27)

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

ϵ3=Δg3-z·c+u      =-k3ϵ3+W3*Tϑ3(x̄)+ε3-W3Tϑ3(x̄)-ε^3arctan     (ϵ~3α3)-ϵ2      =-k3ϵ3-W~3Tϑ3(x̄)+ε3-ε^3arctan(ϵ~3α3)-ϵ2.    (28)

Let us give the following reasonable assumption and lemma.

Assumption 1. The NN approximate error is bounded, i.e., there exists εi* such that εiεi*.

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

|y|-ytanh(yα)κα.

Then, we have

ϵ~1=-k1ϵ1+σ(yc-ρ1)-W~1Tϑ1+ε1-ε^1arctan(ϵ~1α1)      +σϵ2-ζ1      =-k1ϵ1-W~1Tϑ1+ε1-k1ζ1-ε^1arctan(ϵ~1α1)      +σϵ2-σζ2      =-k1ϵ~1+σϵ~2-W~1Tϑ1+ε1-ε^1arctan(ϵ~1α1),    (29)
ϵ~2=-k2ϵ2+zc-ρ2-W~2Tϑ2+ε2-ε^2arctan(ϵ~2α2)      -σϵ~1+ϵ3-ζ2      =-k2ϵ2-W~2Tϑ2+ε2-ζ3-ε^2arctan(ϵ~2α2)      -σϵ~1+ϵ3-k2ζ2      =-k2ϵ~2-W~2Tϑ2+ε2+ϵ~3-ε^2arctan(ϵ~2α2)-σϵ~1,    (30)
e~3=-k3ϵ3-W~3Tϑ3+ε3-ζ3-ϵ2-ε^3arctan(ϵ~3α3)     =-k3ϵ~3-W~3Tϑ3(x̄)+ε3-ϵ~2-ε^3arctan(ϵ~3α3).    (31)

Theorem 1. When |xic-zi|b with b > 0, then (8), (15), and (25) imply

ζc2k^(1-e-2k^t)    (32)

with ζ=[ζ1,ζ2,ζ3]TR3, k^=12 min{k1,k2,k3}, and c = b + σ.

Proof. Let V1=12ζ2. It follows from (8), (15), and (25) that

V1=-i=13kiζi2+σζ1(yc-ρ1)+ζ2(zc-ρ2)      -2k^ζ2+Aζ      -4k^V1+2aV1.    (33)

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 ϵ~1, ϵ~2 and ϵ~3 to a small region.

Proof. Define

V=12i=13ϵ~i2+i=1312ci1W~iTW~i+i=1312ci+3,1ε~i2    (34)

with ε~i=ε^i-εi*. Then, (29), (30), (31), Assumption 1, and Lemma 1 imply

i=13ϵ~iϵ~i=i=13e~i[εi-ε^iarctan(e~iαi)]-ϵ~1W~1Tϑ1-ϵ~2W~2Tϑ2                   -ϵ~3W~3Tϑ3-i=13kie~i2                   i=13[|e~i|εi*-e~iε^iarctan(e~iαi)]-ϵ~1W~1Tϑ1                   -ϵ~2W~2Tϑ2-ϵ~3W~3Tϑ3-i=13kie~i2                   =i=13[|e~i|εi*-e~iε^iarctan(e~iαi)-e~iεi*arctan(e~iαi)]                   -ϵ~2W~2Tϑ2-ϵ~1W~1Tϑ1                   +i=13e~iεi*arctan(e~iαi)-i=13kie~i2-ϵ~3d*arctan(ϵ~3α3)                   -ϵ~3W~3Tϑ3                   i=13[-e~iε^iarctan(e~iαi)+e~iεi*arctan(e~iαi)]                   -i=13kie~i2-ϵ~2W~2Tϑ2                    -ϵ~3W~3Tϑ3-ϵ~1W~1Tϑ1+κi=13αiεi*                   =-i=13e~iε~iarctan(e~iαi)-i=13kie~i2-ϵ~2W~2Tϑ2                   -ϵ~3W~3Tϑ3-ϵ~1W~1Tϑ1+κi=13αiεi*.    (35)

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

i=131ai1W~iTW~i=ϵ~2W~2Tϑ2+ϵ~3W~3Tϑ3+ϵ~1W~1Tϑ1                               -i=13ai2W~iTWi                               =ϵ~2W~2Tϑ2+ϵ~3W~3Tϑ3+ϵ~1W~1Tϑ1                               -i=13ai2W~iT(W~i+Wi*)                               ϵ~2W~2Tϑ2+ϵ~3W~3Tϑ3+ϵ~1W~1Tϑ1                               -i=13ai22W~iTW~i+i=13ai22Wi*TWi*,    (36)
i=1312ai+3,1ε~iε~i=i=13ε~i[e~itanh(e~iαi)-ai+3,2ε^i]                                    =i=13ε~ie~itanh(e~iαi)-i=13ai+3,2ε~i(ε~i+εi*)                                    i=13ε~ie~itanh(e~iαi)-i=13ai+3,22ε~i2                                    +i=13ai+3,22εi*2.    (37)

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

V-i=13kiϵ~i2+κi=13αiεi*-i=13ai+3,22ε~i2+i=13ai+3,22εi*2   -i=13ai22W~iTW~i+i=13ai22Wi*TWi*   =-i=13kie~i2-i=13ai+3,22ε~i2-i=13ai22W~iTW~i+κi=13αiεi*   +i=13ai+3,22εi*2+i=13ai22Wi*TWi*   -d12i=13e~i2-d2i=1312ai1W~iTW~i-d3i=1312ai+3,1ε~i2+d4    (38)

with d1 = 2 min{k1, k2, k3}, d2 = min{a11a12, a21a22, a31a32}, d3 = min{a41a42, a51a52, a61a62}, d4=κi=13αiεi*+i=13ai+3,22εi*2+i=13ai22Wi*TWi* 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 12i=13e~i2d4d1, i=1312ai1W~iTW~id4d2, i=1312ai+3,1ε~i2d4d3. As a result, all variables are indeed bounded, and ϵ~1, ϵ~2 and 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
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Figure 1. Chaotic phenomenon of (2) in (A) xyz space, (B) x(t), (C) y(t), and (D) z(t).

Let the desired signal be xc, defined by

xc={0t[0,8],2t>8.

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) = 01 × 2, W2(0) = 01 × 25, and W3(0) = 01 × 125. The design parameters are k1 = k2 = k3 = 1.5, ai1 = 6, ai2 = 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 xc(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
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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
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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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Keywords: adaptive neural network control, command filtered control, backstepping, permanent magnet synchronous motor, chaos control

Citation: Luo R, Deng Y and Xie Y (2020) Neural Network Backstepping Controller Design for Uncertain Permanent Magnet Synchronous Motor Drive Chaotic Systems via Command Filter. Front. Phys. 8:182. doi: 10.3389/fphy.2020.00182

Received: 11 April 2020; Accepted: 27 April 2020;
Published: 05 June 2020.

Edited by:

Jia-Bao Liu, Anhui Jianzhu University, China

Reviewed by:

Guangming Xue, Guangxi University of Finance and Economics, China
Shengda Zeng, Jagiellonian University, Poland
Ahmed Zarzour, British University in Egypt, Egypt

Copyright © 2020 Luo, Deng and Xie. This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

*Correspondence: Yanping Deng, ypdenggxun@163.com