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Abstract

The permanent-magnet synchronous motor system will display a variety of chaotic phenomenon when its parameters or external inputs satisfy certain condition, and thus its performance would be deteriorated. Therefore, chaos should be suppressed or eliminated. In this article, a practical method which combines adaptive robust control with a single-layer neural network–based disturbance observer is proposed for elimination of the chaos and high-performance motion control of permanent-magnet synchronous motor. The proposed controller not only accounts for the load torque disturbance but also takes the parametric uncertainties into account. A single-layer neural network–based disturbance observer is designed to estimate the disturbance while an adaptive control law is designed to estimate the parameters respectively. Then, all the estimated values are used in the feedforward cancelation item in the controller via a backstepping technique. Lyapunov’s method is used to prove the stability of the novel control scheme. Sufficient comparative simulation results are obtained to validate the effectiveness of the proposed control strategy.

Keywords

Adaptive robust control chaos neural network observer permanent-magnet synchronous motor

Introduction

Nowadays, permanent-magnet synchronous motors (PMSM) have been adopted as the primary actuating components in many industries due to its high power density, high efficiency, eco-friendly, low cost, and large torque-to-inertia ratio^1–3 compared with its hydraulic part.⁴ However, it is still a challenge to realize a high-accuracy motion control of this kind of motor servo system since it is often a multi-variable nonlinear system with parametric uncertainty and external disturbances, which may lead to undesired behavior of PMSM servo system and even make the system unstable. Most studies just focus on modeling and controller designing, while ignoring some special nonlinear dynamic characteristics in the system, such as the irregular oscillation and low-frequency electro-magnetic noise. Recently, more and more research results show that the motion systems driven by PMSM can make chaotic movements when their parameters reach some certain values.^5–7 These nonlinear chaotic behaviors may explain why the aforementioned irregular oscillation and low-frequency electro-magnetic noise often occur in the motion systems driven by PMSM. This would deteriorate the performance of the systems and even make them instable.^8–11

The Ott–Grebogi–Yorke (OGY) method is a classic method for chaos control, which is proposed by Ott et al.¹² and could stabilize a sequence of Poincare section crossing points by adjusting a control parameter. However, it costs too much time to stabilize the system. Then, the time-delay feedback control (TDFC) was proposed by Zhang et al.¹³ It could make the system move along a desired periodic trajectory when the system moves close to the periodic trajectory. However, the desired trajectory must be one of the unstable periodic trajectories of the system or at the equilibrium, which could not satisfy the requirement in the reality. Feedback linearization technique was used for the chaos control in motor servo systems in the literature.^14,15 It enabled the controlled system to evolve to any desired value according to the need in reality. However, it is not easy to realize a high-performance motion control for motor servo systems with this method since it is based on the system’s mathematical model while we could not get accurate mathematical models for the systems in reality. Then, a kind of adaptive feedback control strategy was proposed by Hu et al.¹⁶ A parametric adaptive law was designed to estimate the parameters of PMSM servo systems which are unknown and then the estimated values are used in the feedforward cancelation item in the controller via a backstepping technique.^17–19 Thus, the control performance was improved dramatically. However, in Hu et al.,¹⁶ the load torque was just regarded as a known variable, while in reality, the load torque usually varies with time and it is also unknown. Thus, the hypothesis in Hu et al.¹⁶ is not appropriate. In fact, the load torque could be regarded as a kind of disturbance. In this case, the adaptive control cannot improve the control performance of the system and may also make the system unstable when the disturbance becomes larger and larger. Then, in Hu et al.,²⁰ a kind of adaptive robust control strategy was proposed to suppress the chaos and improve the control performance. A parametric adaption law and a high-gain robust feedback item were designed respectively to handle parameter uncertainties and time-varying disturbance together in one controller. However, the high gain often tends to make systems unstable. Thus, it is better to design an observer to observe the disturbance which can be canceled in the controller designed later. In this way, the robust gain could be reduced and thus the system could be stable.

With the development of artificial intelligence technology, the intelligent control underwent a rapid development. Neural networks have attracted much attention and have been applied in many ways.^21,22 In Feng,²³ the parametric uncertainty and uncertain nonlinearity of nonlinear systems are generally considered as a kind of system model uncertainty. Neural networks are used to approximate the model uncertainty of the whole system, and feedforward compensation technology is used to compensate the uncertainty to improve the control performance of the system. Although this method can realize the integrated design of nonlinear controller, it increases the modeling error and the approximation burden of neural network. When the system has only parametric uncertainty, this kind of controller design method is obviously inferior to the parameter adaptive control strategy based on the system model. Considering this point, we just design an intelligent compound controller which combines the advantage of neural network and the advantage of adaptive control, and it is different from the existing methods and novel. In virtue of the universal approximation of the neural network,²⁴ we would like to use it as an observer of load torque disturbance to estimate the load torque. Adaptive control method is used to estimate the uncertain parameters. Then, both the parametric uncertainty and the disturbance (uncertain nonlinearity) could be compensated with feedforward cancelation technique using their estimation values. At this time, the residual model uncertainty is very small and a robust item with low gain could be designed to suppress it. Thus, the system could be more stable and more accurate. In a word, an integrated controller design idea of fusing parameter adaption with neural network–based load torque disturbance estimation is proposed, in order to suppress the chaos and improve the tracking performance of the PMSM system by combining the advantages of these two methods and overcome their drawbacks. A single-layer neural network is designed to approximate and compensate the load torque disturbance, and at the same time, a model-based parameter adaption control strategy is designed to estimate the uncertain parameter to reduce the parametric uncertainty. In addition, a low-gain robust item is also designed to suppress the observation error and other compensation errors. The extensive comparative simulation results are presented in the article and they prove the validity of this method.

This article is arranged as follows. The chaotic dynamics of the PMSM servo system is analyzed in section “Problem description,” and the proposed controller is designed in detail in section “Design of the controller with neural network–based disturbance observer.” The simulation results are illustrated in section “Results of the simulation,” and finally, the conclusion is given in section “Conclusion.”

Problem description

According to Li et al.,⁶ the electromechanical behavior of PMSM servo system can be described by the following first-order differential equations

{\begin{matrix} \overset{\cdot}{x} = σ (y - x) - {\tilde{T}}_{L} \\ \overset{\cdot}{y} = (μ - z) x - y + {\tilde{u}}_{q} \\ \overset{\cdot}{z} = - z + xy + {\tilde{u}}_{d} \end{matrix}

(1)

where $x = \frac{R}{L_{q}} \tilde{x}$ , $y = \frac{p_{n} L_{q} ψ_{fd}}{R B_{equ}} \tilde{y}$ , $z = \frac{p_{n} L_{q} ψ_{fd}}{R B_{equ}} \tilde{z}$ , $\tilde{x} = ω, \tilde{y} = i_{q}, \tilde{z} = i_{d}$ , $σ = \frac{B_{equ} L_{q}}{J_{equ} R}$ , $μ = \frac{p_{n} ψ_{fd}^{2}}{B_{equ} R}$ , ${\tilde{u}}_{d} = \frac{p_{n} L_{q} ψ_{fd}}{R^{2} B_{equ}} u_{d}$ , ${\tilde{u}}_{q} = \frac{p_{n} L_{q} ψ_{fd}}{R^{2} B_{equ}} u_{q}$ , and ${\tilde{T}}_{L} = \frac{L_{q}^{2}}{R^{2} J_{equ}} T_{L}$ , where u_q and u_d represent the quadrature- and direct-axis voltage of the stator, respectively; i_q and i_d represent the quadrature- and direct-axis current of the stator, respectively; L_q and L_d represent the quadrature- and direct-axis inductance of the stator respectively; $ψ_{fd} = 1.5 ψ_{f}$ is the direct-axis flux; R is the resistance of the stator; p_n is the number of pole-pairs; ω is the frequency of the motor angle; T_L is the load torque, B_equ is the equivalent friction coefficient; and J_equ is the equivalent polar moment of inertia.

According to equation (1), some conclusions can be reached that both Hopf bifurcation and chaos would appear when some parameters of the PMSM servo system drop into a certain range.¹⁶ For example, when the systematic parameters are as follows: σ = 10.5, ${\tilde{T}}_{L} = 6$ , and the control values are set as ${\tilde{u}}_{q} = - x$ and ${\tilde{u}}_{d} = 0$ which indicates linear feedback control, we could obtain the bifurcation figure with μ changing from 1 to 500, which is shown in Figure 1. It can be seen in Figure 2 that the phase trajectory falls into the chaotic movement when μ = 24.8.

Figure 1.

Systematic bifurcation figure.

Figure 2.

Systematic phase figure when σ = 10.5, ${\tilde{T}}_{L} = 6$ , and μ = 24.8.

In this article, our aim is to suppress the chaos since it would degrade the system’s performance and to make the system motion speed track a desired value or trajectory. Thus, we need to find some accessible manipulated variables first. From system dynamical equation (1), it can be seen that ${\tilde{u}}_{q}, {\tilde{u}}_{d}$ could be regarded as two manipulated control inputs and thus could be properly designed to drive the systematic trajectory to a desired one. Now, the control inputs should be designed. The general feedback linearization control method could achieve the aforementioned goal effectively when there is no parametric uncertainty or external disturbance in the system. However, there always exist all these uncertainties in PMSM which would degrade the control effect of the general feedback linearization control method. To make up for its deficiency, a single-layer neural network–based disturbance observer and a parametric adaption law would be deigned to estimate the disturbance and the parameters. All the estimated values could be used in the controller designed later.

To realize the aforementioned aim, the system state equation is built first as follows

{\begin{matrix} {\overset{\cdot}{x}}_{1} = σ (x_{2} - x_{1}) - {\tilde{T}}_{L} \\ {\overset{\cdot}{x}}_{2} = - x_{2} - x_{1} x_{3} + μ x_{1} + {\tilde{u}}_{q} \\ {\overset{\cdot}{x}}_{3} = - x_{3} + x_{1} x_{2} + {\tilde{u}}_{d} \end{matrix}

(2)

where x₁ = x, x₂ = y, and x₃ = z are the system states. The specific control strategy for system (2) is given in the next part.

Design of the controller with neural network–based disturbance observer

Parameter adaption

Denote $\hat{θ}$ as the estimate of θ and $\tilde{θ}$ as the error of the estimation (i.e. $\tilde{θ} = \hat{θ} - θ$ ). Define a discontinuous projection as

\Pr o j_{\hat{θ}} (•) = {\begin{matrix} 0, {\hat{θ}}_{i} = θ_{i max} and •_{i} > 0 \\ 0, {\hat{θ}}_{i} = θ_{i min} and •_{i} < 0 \\ •_{i}, otherwise \end{matrix}

(3)

In equation (3), •i represents the ith component of the vector •.

An adaptation law is given as follows

\overset{\overset{\cdot}{^}}{θ} = \Pr o j_{\hat{θ}} {Γ τ}, with θ_{min} \leq \hat{θ} \leq θ_{max}

(4)

where Γ > 0 is a diagonal matrix of adaptation rate and τ is a parameter regression law to be deduced later. For any τ, the projection mapping used in equation (4) guarantees²¹

(P 1) \hat{θ} \in Ω_{\hat{θ}} \overset{Δ}{=} {\hat{θ} : θ_{min} \leq \hat{θ} \leq θ_{max}}

(5)

(P 2) \tilde{θ} [Γ^{- 1} Pro j_{\hat{θ}} (Γ τ) - τ] \leq 0 \begin{matrix} \forall τ \end{matrix}

(6)

Radial basis function–based load torque disturbance observer design

Step 1. Design the structure of the radial basis function (RBF) network. RBF networks are universal approximators in that they can approximate any smooth nonlinear function within arbitrary accuracy, given a sufficient number of hidden layer neurons and input information. Here, an RBF neural network is trained online to approximate load torque disturbance ${\tilde{T}}_{L}$ (Figure 3).

Figure 3.

Structure of RBF network.

The input–output map of RBF network can be expressed as

f (x) = W^{*^{T}} h (x) + ε_{approx} = {\tilde{T}}_{L}

(7)

h_{j} = \exp (\frac{{‖ x - c_{j} ‖}^{2}}{2 b_{j}^{2}})

(8)

where x is the input of the network, j is the jth node of the hidden layer, h = [h_j]^T is the output of network’s Guassian radial function, W * is the ideal weight value of the network, and ε_approx is the approximation error of the network and ε_approx ≤ ε_N.

The input of the network is taken as x = [x₁, x₂]^T, and the actual output of the network is

\hat{f} (x) = {\hat{W}}^{T} h (x) = \overset{\hat{~}}{T_{L}}

(9)

where $\hat{W}$ is the estimation of $W^{*}$ . Then, the estimation error of ${\tilde{T}}_{L}$ can be expressed as

\begin{matrix} \tilde{f} (x) = \overset{\hat{~}}{T_{L}} - {\tilde{T}}_{L} = {\hat{W}}^{T} h (x) - (W^{* T} h (x) + ε_{approx}) \\ = - {\tilde{W}}^{T} h (x) - ε_{approx} \end{matrix}

(10)

where $\tilde{W} = W^{*} - \hat{W}$ is the estimation error of the ideal weight value.

Step 2. In this step, we need to design the weights adaption law. Let x_1d be the desired velocity and denote z₁ = x₁ − x_1d. Then, we could design the weights adaption law as follows

\overset{\overset{\cdot}{^}}{W} = \Pr o j_{\hat{W}} {Γ_{1} h (x) z_{1}}

(11)

where $Γ_{1}$ is the weights adaption rate matrix.

Adaptive robust backstepping controller design

Step 1. In this step, x₂ is regarded as a virtual manipulated control variable. Then, a control function x_2eq needs to be designed for the virtual control variable x₂ so that the output tracking performance can be guaranteed.

According to the definition of z₁, we have

{\overset{\cdot}{z}}_{1} = {\overset{\cdot}{x}}_{1} - {\overset{\cdot}{x}}_{1 d} = σ (x_{2} - x_{1}) - {\tilde{T}}_{L} - {\overset{\cdot}{x}}_{1 d}

(12)

Denote z₂ = x₂ − x_2eq as the input error, and then we rearrange equation (12) as follows

{\overset{\cdot}{z}}_{1} = σ z_{2} + σ x_{2 eq} - σ x_{1} - {\tilde{T}}_{L} - {\overset{\cdot}{x}}_{1 d}

(13)

According to equations (9) and (13), the resulting virtual control variable x_2eq can be given as follows

\begin{matrix} x_{2 eq} = x_{2 eq 1} + x_{2 eq 2} \\ x_{2 eq 1} = \frac{1}{σ} [σ x_{1} + \overset{\hat{~}}{T_{L}} + {\overset{\cdot}{x}}_{1 d}], x_{2 eq 2} = u_{s 1} + u_{s 2} \\ u_{s 1} = - k_{1} z_{1}, u_{s 2} = - \frac{ε_{N}^{2}}{4 ε} z_{1} \end{matrix}

(14)

where k₁ > 0 is a feedback gain, and ε can be regarded as a kind of control accuracy, which is bounded and positive.

In equation (14), x_2eq1 functions as a model-based compensation law through online disturbance estimation, and x_2eq2 is composed of u_s₁ and u_s₂. u_s₁ is a linear feedback stabilizing item which is used to stabilize the PMSM system. u_s₂ is a nonlinear robust item which can be used to cancel the approximation error of the neural network and it satisfies the following conditions

\begin{matrix} 1 . z_{1} [u_{s 2} - ε_{approx}] \leq ε \\ 2 . z_{1} u_{s 2} \leq 0 \end{matrix}

(15)

Substituting equation (14) into equation (13), we have

\begin{matrix} {\overset{\cdot}{z}}_{1} = σ z_{2} - k_{1} z_{1} + \overset{\hat{~}}{T_{L}} - {\tilde{T}}_{L} + u_{s 2} = σ z_{2} - k_{1} z_{1} \\ + u_{s 2} - {\tilde{W}}^{T} h (x) - ε_{approx} \end{matrix}

(16)

Step 2. In step 1, a robust virtual control variable x_2eq has been designed. In this step, an actual control law for u_q needs to be designed. According to equation (2), the time derivative of z₂ can be obtained as

{\overset{\cdot}{z}}_{2} = {\overset{\cdot}{x}}_{2} - {\overset{\cdot}{x}}_{2 eq} = - x_{2} - x_{1} x_{3} + μ x_{1} + {\tilde{u}}_{q} - {\overset{\cdot}{x}}_{2 eq}

(17)

Then, the following control law u_q is proposed

{\tilde{u}}_{q} = {\overset{\cdot}{x}}_{2 eq} + x_{2} + x_{1} x_{3} - \hat{μ} x_{1} - k_{2} z_{2}

(18)

where k₂ > 0 is a feedback gain, and $\hat{μ}$ is the estimation of μ. Since μ is a critical value which determines the chaotic movement of PMSM and we could not know its true value, we need to estimate its value. Substituting equation (18) into equation (17), we have

{\overset{\cdot}{z}}_{2} = - k_{2} z_{2} - (\hat{μ} - μ) x_{1} = - k_{2} z_{2} - \tilde{μ} x_{1}

(19)

Step 3. In this step, the parametric adaption law is designed for μ as follows

\overset{\overset{\cdot}{^}}{μ} = \Pr o j_{\hat{μ}} {x_{1} z_{2}}

(20)

where γ is the parameter adaption rate.

Step 4. In this step, an actual control law should be designed for u_d. Similarly, let x_3d denote the desired output d-axis current and z₃ = x₃ − x_3d, then a feedback control law is designed as ${\tilde{u}}_{d} = x_{3} - x_{1} x_{2} - k_{3} z_{3} + {\overset{\cdot}{x}}_{3 d}$ , where k₃ > 0 is a feedback gain. When the control law is substituted into the third equation of system (2), the error dynamical equation could be obtained as follows

{\overset{\cdot}{z}}_{3} = {\overset{\cdot}{x}}_{3} - {\overset{\cdot}{x}}_{3 d} = - k_{3} z_{3}

(21)

which ensures that the tracking error dynamics of is globally uniformly stable.

Main results

Theorem 1

With neural network–based control laws (9), (14), and (18) and parameter adaptive laws (11) and (20), the output tracking error of the system is bounded and it converges to $\sqrt{\frac{ε_{s}}{k_{1}}}$ exponentially with a converging rate larger than k₁, which indicates that it is guaranteed that the tracking has a prescribed transient and steady-state performance.

Proof

For proof, see Appendix 1.

Remark

Theorem 1 indicates that the proposed controller could not only guarantee a prescribed transient output tracking performance but also guarantee the final tracking accuracy. After a finite time, the tracking error can be suppressed to a given range by adjusting the parameter ε_s. Thus, it can be made arbitrarily small in theory. In addition, a neural network is used to estimate the load torque and an adaptive control method is used to estimate the uncertain parameters. Both the parametric uncertainty and the disturbance are compensated with feedforward cancelation technique using their estimation values. Thus, the residual model uncertainty is very small and a robust item with low gain could be designed to suppress it. Thus, the robustness of the system is strong.

Results of the simulation

The parameters of PMSM used in the simulation are chosen as follows: the number of pole-pairs p_n = 4, inductance L_d = L_q = L_a = 0.10 H, stator resistance R = 2.5 Ω, rotor flux linkage amplitude ψ_f = 0.45 Wb, equivalent viscous friction coefficient B_equ = 1.27 × 10⁻² Ns, equivalent inertia J_equ = 2.512 × 10⁻⁵ kg·m². After a series of calculation, we have σ = 10.5 and μ = 25, which means the system is in the chaotic movement state. The control objective of x₁ is set to be x_1d = sin(t) and the control objective of x₂ is still set to be x₂ = 0.

The following three controllers are compared:

1. Neural network observer–based adaptive robust nonlinear feedback controller (NNARC). This is the control strategy proposed in this article, which combines adaptive robust controllers (14) and (18) with adaptive laws (11) and (19) and neural network-based disturbance observer (9).

First, we need to choose the number of nodes of hidden layer. Generally speaking, the more the number of hidden layer nodes, the higher the approximation accuracy of the neural network. However, with the increase in the number of nodes, the computational burden will increase correspondingly and the control cycle would be prolonged which would lead to a decreased control performance. Thus, it is necessary to select a suitable number of nodes. There are several empirical formula of how to choose the number of nodes such as m = log2n, in which m means the number of nodes of hidden layer and n means the number of the nodes of input layer. First, we choose the number of nodes of hidden layer according to this formula. Then, we rectify it through multiple simulation tests. Finally, we decide to choose five nodes for hidden layer.

Then we need to choose the parameters of the controller. We mainly need to tune the parameter k₁ in equation (14) and k₂ in equation (18). We just simply choose a large constant to implement the robust control law (14) without the control item of model feed-forward compensation x_2eq1 since it not only reduces the online computation time significantly but also facilitates the gain tuning process in the implementation. After we have chosen an appropriate value for k₁, we add x_2eq1 in the control law. It is the same process for k₂. Finally, the control gains are chosen as k₁ = 1.0 and k₂ = 0.2. The bounds of the parameters are given by μ_min = 1.0 and μ_max = 30.0. The control accuracy ε_s = 0.3. The initial value of μ’s estimation is chosen as 1.0. The adaptation rate matrix is set as Γ = diag{0.05, 0.05, 0.01, 0.01, 0.01} and γ = 0.15. Parameters of the neural network are chosen as c _i = 0.5 × [−2 −1 0 1 2]^T and b_i = 5.0 according to the range of neural network’s inputs x₁ and x₂.

2. General nonlinear feedback controller (GNFC). This is the controller discussed in the study of Xu and Yao¹ which includes x_2eq1 in equations (14) and (18), but just uses the nominal value of ${\tilde{T}}_{L} = 5.0$ in x_2eq1 which is different from the practical one. The controller’s parameters are the same as those of NNARC.

3. Robust nonlinear feedback controller (RNFC). This is the controller discussed in the study of Xu and Yao¹ which includes equations (14) and (18), but there is no estimation of T_L in x_2eq. The controller’s parameters are the same as those of NNARC.

Two cases are tested for these three controllers.

Case 1. Load torque disturbance is a constant value

In this case, a constant load torque disturbance is implemented by applying ${\tilde{T}}_{L} = 6.0$ to the physical motion system. The simulation results are obtained for tracking the desired output trajectory. The working time of the controller is set as t = 15 s to show the control effect clearly. The three systematic state variables are shown in Figure 4, which shows that all the systematic states have been driven to the desired motion trajectory with NNARC. The control inputs are shown in Figure 5. Figure 6 shows the compare curves of three controllers. As it can be seen, NNARC and RNFC controllers are better than GNFC controller in terms of both the transient performance and the final tracking accuracy since both the controllers employed robust item to cancel the influence of uncertain load torque disturbance. This confirms that the proposed robust control law is effective. In addition, the maximal steady tracking error of NNARC is about 0.02°, which is slightly better than that of RNFC, since the former controller could estimate the load torque disturbance and cancel it.

Figure 4.

Curves of system states x₁, x₂, and x₃ in which NNARC controller is applied when $t \geq 15 s$ .

Figure 5.

Curves of control inputs u_q and u_d of NNARC controller after $t = 15 s$ .

Figure 6.

Compare curves of tracking error of three controllers with constant load torque disturbance.

Case 2. Load torque disturbance is a time-varying value

In this case, a time-varying load torque disturbance ${\tilde{T}}_{L} = 0.5 x_{1} + x_{2}$ is applied to the motion system. The systematic dynamics is almost totally changed with this time-varying disturbance and the simulation results can be regarded as the most exhaustive verification. Figure 7 shows the compare curves of the three controllers. As it can be seen, all these three controllers can eliminate the chaos when the motion system tracks a sinusoidal desired output. However, NNARC and RNFC controllers are better than GNFC controller in terms of both the transient performance and the final tracking errors since both the controllers employed robust item to cancel the influence of uncertain load torque disturbance. This confirms that the proposed robust control law is effective further. However, the maximal steady tracking error of NNARC controller is about 0.01°, which means it is the best one in all the three controllers. The reason is that the NNARC controller could estimate and cancel the time-varying load torque disturbance effectively using neural network and the residual uncertainty which is very small and handled by the robust item which leads to a better control performance, meanwhile the NNARC could compensate the parametric uncertainty with adaption law (the adaptive parameters and the disturbance estimation are shown in Figures 8 and 9, respectively). Thus, the overall uncertainties are greatly reduced and the tracking performance has been greatly improved.

Figure 7.

Compare curves of tracking error of three controllers with time-varying load torque disturbance.

Figure 8.

Parameter estimation in RBFARC.

Figure 9.

Disturbance estimation in RBFARC.

Conclusion

In this article, an adaptive robust control with a neural network–based load torque disturbance observer has been proposed to suppress the chaos in PMSM and realize the high-accuracy motion control of PMSM system, which not only considers the load torque disturbance but also takes the systematic parameter uncertainties into account. Lyapunov’s method is used to prove the stability of the novel control strategy, which shows that the controller could control the chaos effectively and a prescribed tracking performance could be guaranteed theoretically in the presence of various uncertainties. Sufficient comparative simulation has been done and the results show that the proposed strategy is effective, that is, a wonderful tracking accuracy can be achieved via introducing neural network–based disturbance observer in adaptive robust control compared to the other nonlinear feedback control methods. The proposed scheme can suppress the chaos and improve the tracking accuracy of PMSM system by combining the advantages of the two approaches fundamentally.

Footnotes

Appendix 1

Handling Editor: Muhammad Akhtar

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported in part by the National Natural Science Foundation of China under grant 51505224, in part by the Natural Science Foundation of Jiangsu Province in China under grant BK20150776, and the Fundamental Research Funds for the Central Universities (no. 30918011301).

References

Yao

Adaptive robust precision motion control of linear motors with negligible electrical dynamics: theory and experiments. IEEE/ASME T Mechatron 2001; 6: 444–452.

Qiu

Liu

High-order sliding-mode observer based output feedback adaptive robust control of a launching platform with backstepping. Int J Control 2016; 89: 2029–2039.

Yao

Wang

An orthogonal global task coordinate frame for contouring control of biaxial systems. IEEE/ASME T Mechatron 2012; 17: 622–634.

Qiu

Performance-oriented asymptotic tracking control of hydraulic systems with radial basis function network disturbance observer. Adv Mech Eng 2016; 8: 1–12.

Hemail

Strange attractors in brushless DC motors. IEEE T Circuits I 1994; 41: 40–45.

Park

Joo

et al . Bifurcations and chaos in a permanent-magnet synchronous motor. IEEE T Circuits I 2002; 49: 383–387.

Saberi Nik

Gorder

Gambino

. The chaotic Dadras–Momeni system: control and hyperchaotification. IMA J Math Control I 2016; 33: 497–518.

Chua

Chen

JH.

Modeling, analysis and experimentation of chaos in switched reluctance drive system. IEEE T Circuits I 2003; 50: 712–716.

Jing

Chen

Complex dynamics in a permanent-magnet synchronous motor model. Chaos Soliton Fract 2004; 22: 831–848.

10.

Wang

Chau

KT.

Design, analysis, and experimentation of chaotic permanent magnet DC motor drives for electric compaction. IEEE T Circuits II 2009; 56: 245–249.

11.

Liu

Robust nonlinear feedback control of chaotic permanent-magnet synchronous motor with a load torque disturbance. J Korean Phys Soc 2014; 65: 2132–2139.

12.

Ott

Grebogi

Yorke

JA.

Controlling chaos. Phys Rev Lett 1990; 64: 1196–1199.

13.

Zhang

Chau

Wang

Analysis and stabilization of chaos in the electric-vehicle steering system. IEEE T Veh Technol 2013; 62: 118–126.

14.

Ren

Ding

Nonlinear feedback control of chaos in permanent magnet synchronous motor. IEEE T Circuits II 2006; 53: 45–50.

15.

Babaei

Nazarzadeh

Faiz

. Nonlinear feedback control of chaos in synchronous reluctance motor drive systems. In: Proceedings of the international conference on industrial technology, Chengdu, China, 21–24 April 2008, pp.1–5. New York: IEEE.

16.

Liu

et al . Adaptive nonlinear feedback control of chaos in permanent-magnet synchronous motor system with parametric uncertainty. Proc IMechE, Part C: J Mechanical Engineering Science 2015; 229: 2314–2323.

17.

Yao

Jiao

et al . High-accuracy tracking control of hydraulic rotary actuators with modeling uncertainties. IEEE/ASME T Mechatron 2014; 19: 633–641.

18.

Yao

Reedy

et al . Adaptive robust motion control of single-rod hydraulic actuators: theory and experiments. IEEE/ASME T Mechatron 2000; 5: 79–91.

19.

Yao

Jiao

Yao

et al . Nonlinear adaptive robust force control of hydraulic load simulator. Chin J Aeronaut 2012; 25: 766–775.

20.

Qiu

Adaptive robust nonlinear feedback control of chaos in PMSM system with modeling uncertainty. Appl Math Model 2016; 40: 8265–8275.

21.

Lai

Lewis

Venkataramanan

et al . Disturbance and friction compensations in hard disk drives using neural networks. IEEE T Ind Electron 2010; 57: 784–792.

22.

Xia

Wang

Chai

Neural-network-friction compensation-based energy swing-up control of pendubot. IEEE T Ind Electron 2014; 61: 1411–1423.

23.

Feng

A compensating scheme for robot tracking based on neural networks. Robot Auton Syst 2005; 15: 100–206.

24.

Park

Sandberg

IW.

Approximation and radial-basis-function networks. Neur Comput 1993; 5: 305–316.

Motion control of chaotic permanent-magnet synchronous motor servo system with neural network–based disturbance observer

Abstract

Keywords

Introduction

Problem description

Design of the controller with neural network–based disturbance observer

Parameter adaption

Radial basis function–based load torque disturbance observer design

Adaptive robust backstepping controller design

Main results

Theorem 1

Proof

Remark

Results of the simulation

Conclusion

Footnotes

Appendix 1

Declaration of conflicting interests

Funding

References