In late decades, fractional-order non-linear systems (FONSs) [1] have been widely studied, not only owing to their accurate performance in modeling physical phenomena (e.g., chaos, oscillations, impulses, diffusions, see [2–5]), but also owing to their successful applications in a variety of fields, such as chemistry, medicine, biology, electronics, robotics, fuel cells, and so on [6–11]. Stability analysis [12] is regarded as a fundamental and crucial task in the development of cybernetics. Recently, more and more scholars have paid attention to stability analysis of fractional-order non-linear systems [13–16]. It is not exaggerated to say that stability analysis of FONSs along with their robust control have become a hot and promising research topic.

The researches on the control of chaotic systems are widely concerned due to its valuable significance in both theoretical and practical aspects [17, 18]. Since Kuroe and Hayashi [19] originally discovered chaotic phenomenon from the motor drive system in the late 1980's, chaos control has been one of the most popular research topics in cybernetics. There are several types of chaotic motor drivers that capture widespread interests. For instance, DC motor drivers [20], step motor drivers [21], single-phase induction motor drivers [22], synchronous reluctance motor drivers [23], switched reluctance motor drivers [24] and so on. The extensive utilization of permanent magnet synchronous motors (PMSMs) in industries mainly benefits from their merits of high speed, high efficiency, high power, low loss and low temperature rise. Chaotic non-linear systems are very complex due to the irregular and unpredictable behaviors. A remarkable feature of chaotic systems is that they are very sensitive to the initial conditions. The small change of initial state will lead to great distinction. On the other hand, they have many other desired properties, such as information processing, secure communication and mechanical system. However, it may cause unexpected oscillations and even destroy the system stability. Therefore, such oscillations should be effectively suppressed. For this reason, various methods have been developed to stabilize non-linear chaotic systems, in which fractional order chaos control has also been focused, such as OGY type [25], feedback type [26–28], dynamic surface type [29], sliding mode type [30–33], backstepping type [34, 35], etc.

Neural network (NN) control technique [36, 37] is an intelligent method for controlling non-linear systems with uncertainties. Analogizing to fuzzy control approach [38, 39], the idea of NN control technique is to approximate unknown non-linear functions by using radial basis function neural networks (RBFNNs), which is a type of neuron-modeled structure formed by the computation of some adjustable parameter vectors and some specific continuous functions. As one of the most powerful tools to realizing approximation of functions, NN control technique is popular because it facilitates to control most of many non-linear systems in which the data are too imprecise or too complex to construct mathematical modeling. It provides an available way for the control designs, and it is considerably applicable in the field of control engineering.

Backstepping technique has engaged much attention due to its efficient performance in handling mismatched uncertainties of integer-order non-linear systems [40, 41]. Unfortunately, this control method has an inherent drawback, namely “explosion of complexity,” triggered by iteratively differentiating virtual control inputs [42]. Additionally, it requires complicated analysis to compute a so-called “regression matrix” [43]. Dawson et al. [44] pointed out that the size of the regression matrix displays too large when backstepping technique was applied to manipulate DC motors in a conventional manner. Such complexities might be augmented remarkably for fractional-order non-linear systems.

It is well-known that the design of NN control is rarely systematic, which is difficult to work for the control of complex systems. It is also challenging to establish a systematic NN control theory to solve a series of problems, such as the mechanism of NN control, stability analysis, systematic design, etc. Backstepping control usually leads to the problem of “complexity explosion” when it is applied in the processing of unknown functions, so the methods of adaptive NN control [36], adaptive fuzzy control [45] and adaptive NN backstepping control [35] are put forward to address such a problem. These techniques enable systems to be greatly adaptive and robust obeying the required performance criteria for the control. However, the control performance is not desired for the non-linear systems with triangular structures, and the problem of “complexity explosion” will occur during the control proceeding. Based on the above discussion, this paper proposes an adaptive neural network control method of chaotic fractional-order permanent magnet synchronous motors using backstepping technique, which can improve the control performance of non-linear systems.

To deal with the synchronization issue of fractional-order permanent magnet synchronous motor (FOPMSM) with triangular structure, we expect to construct an adaptive NN controller combined with backstepping technique. This enables every uncertain complex non-linear functions being approximated by a radial basis function neural network (RBFNN) during each control step. The main contributions of this work can be summarized as follows:

The synchronization control scheme design and the stability analysis of FOPMSMs are investigated. In order to analyze the stability of the controlled systems, firstly, some basic results related to fractional calculus and RBFNN are recalled, including a fractional differential inequality, which lays the foundation for the application of the fractional Lyapunov function method. Meanwhile, it lays a foundation for the stability analysis of other types of FONSs. Secondly, an adaptive NN backstepping recursive control method is proposed for a class of uncertain FOPMSMs. The stability of FOPMSMs is analyzed based on fractional Lyapunov criterion. NN control technique is employed when dealing with the approximation of uncertain functions of FOPMSMs, and the fractional adaptive law is designed to update the parameters of NNs. The relevant properties of Mittag-Leffler function and Laplace transform are applied when the fractional Lyapunov function is defined to implement the system control. Our proposed control method fully averts the superfluous terms which are aroused by repeated derivation on virtual control inputs, and facilitates to overcome the so-called “complexity explosion” inherent drawback of the traditional backstepping technique. Finally, we present a numerical example to verify the main results. The simulation results show that our method embodies a perfect control effect. This also reveals the effectiveness of our control algorithm in another way.

The remainder of this work is arranged as below: In section 2, we recall several fundamental preliminaries of fractional calculus and RBFNN. Then, a brief overview of a class of FOPMSMs is provided. In section 3, we propose a RBFNN-based control scheme in three steps and present stability analysis. In section 4, we illustrate the effectiveness of the proposed synchronous controller via a simulation example. Finally, in section 5, we summarize the results of this work and put forward the prospect for our further investigation.

Some basic concepts, notations and lemmas, involved with fractional calculus and radial basis function neural network (RBFNN), need to be stated in this section before used. For convenience, we adopt the symbol ℝ (resp. ℝⁿ, ℂ) to represent the collection of all real numbers (resp. n-dimensional real vectors, complex numbers). Ω ⊆ ℝⁿ is always assumed to be compact. T = [0, +∞) means the time-variable domain. The notation C¹(T, Ω) stands for the collection of all continuous functions from T to Ω with continuous derivatives. Given a vector x ∈ ℝⁿ, x^T denotes its transpose, ∥x∥ denotes its Euclidean norm.

Definition 1 ([8]). Let α ≥ 0. For a given function f :[0, ∞) → ℝ, its α-th order integral is written as

where Γ(α)=∫0+∞sα-1e-sds.

Definition 2 ([8]). Let α ≥ 0. For a given function f :[0, ∞) → ℝ, its α-th order Caputo derivative is expressed by

where α ∈ [n − 1, n), n = 1, 2, ⋯ .

Definition 3 ([8]). Let α, γ > 0. The Mittag-Leffler function E_α,γ on ℂ is expressed as

Moreover, taking the Laplace transform on E_α,γ generates

Lemma 1 ([1]). Let 0 < α < 1, γ ∈ ℂ and ν ∈ ℝ fulfilling the following:

If |ζ| → ∞, ν ≤ |arg(ζ)| ≤ π, then the following statement holds:

where n is a non-zero natural number.

Lemma 2 ([1]). Let α ∈ (0, 2), β ∈ ℝ. If μ is a constant fulfilling

then there exists C > 0 such that

with |arg(ζ)| ∈ [μ, π].

Lemma 3 ([35]). Let z(t) be a smooth function. Then

Lemma 4 ([34, 46]). Let z = 0 be the equilibrium point of a FONS, which is given by

where f : T × Ω → ℝ is a function with the Lipschitz condition. Suppose there exist a Lyapunov function V(t, z(t)) and a family of class-K functions¹ ĝ_i (i = 1, 2, 3) satisfying

Then system (10) is asymptotical stable, i.e., limt→∞z(t)=0.

Next, let us introduce some basic notions and notations about the radial basis function NN (RBFNN) [43, 47]. The goal of the control procedure is to establish a adaptive NN control scheme, which enables the tracking signal x₁(t) and the given reference signal x_d(t) are synchronized.

A RBFNN can be formed as

where z(t)=(z1(t),z2(t),⋯,zn(t))T∈C1(T,Ω) and f^ are the input-variable and the output-variable, respectively, θ(t)=(θ1(t),θ2(t),⋯,θm(t))T is an adjustable parameter vector, ϑ(z(t))=(ϑ1(z(t)),ϑ2(z(t)),⋯,ϑm(z(t)))T with ϑj(z(t))=(ϑj(z1(t)),ϑj(z2(t)),⋯,ϑj(zn(t)))T (j = 1, 2, ⋯ , m) being a continuous function, called the regressor variable. To illustrate its structure, we refer to Figure 1.

Suppose that all of the continuous functions ϑ₁(z(t)), ϑ₂(z(t)), ⋯ , ϑ_m(z(t)) in the above RBFNN are chosen as Gaussian functions, that is, for j = 1, 2, ⋯ , m,

where cj=[cj1,cj2,…,cjn]T is the center vector and σ_j > 0 is the width of the Gaussian function ϑ_j(z(t)). Then the next lemma is obtained.

Lemma 5 ([35]). Let f :Ω → ℝ be a Lipschitz function. For each z ∈ C¹(T, Ω) and for each ε > 0, there is a RBFNN fulfilling Equation (13) and the following property:

It is well-known that non-linear theory has yet been widely applied in the stability analysis of integer-order non-linear PMSMs. Yu et al. [48] investigated a type of classical PMSMs, which are described as follows:

Yu et al. [48] also studied that when the parameters σ, γ of a PMSM decrease in a certain range, chaos will appear in the PMSM. To eliminate chaos in PMSM drive systems, they treated u_d as an adjustable variable, and proposed an adaptive NN control method based on backstepping control technique. It is well-known that backstepping technique usually makes great efforts to the effective control of integer-order triangular non-linear systems. Nevertheless, it is difficult to incorporate backstepping control technique into FONSs because of the complexities of fractional derivatives of composite functions. Moreover, the applications of FONSs broadly cover a great deal of fields, such as physics, chemistry, mathematics, etc., which suggests that the mathematical structures modeled by FONSs are more accurate and more practical.

Based on the aforementioned facts, this paper concerns a class of FOPMSMs. For simplicity, denote ω = x₁, i_q = x₂, i_d = x₃ in system (15), and extend system (15) into the next fractional-order form:

where 0 < α < 1, x(t)=[x1(t),x2(t),x3(t)]T∈ℝ3 is a measurable state-variable, x₁(t) ∈ ℝ is an output-variable, u_d(t) ∈ ℝ is an input-variable, σ and γ are positive constants, both of them represent system operating parameters.

In this section, we will improve the conventional control method combined with backstepping technique, by which the chaos of FOPMSMs realizes to be eliminated or restrained in a high effective manner. The design process includes three steps. Each of them will construct a virtual control variable based on a proper Lyapunov function. At the end, a controller in real sense will be produced to manipulate FOPMSM. Assume that x_d(t) is a given reference signal. Our goal is to establish an appropriate controller u_d(t), ensuring that the tracking error e(t): = x₁(t) − x_d(t) will ultimately converge to an arbitrarily small neighborhood of the origin. Next, we present a recursive backstepping procedure to reach our goal in three steps:

Step 1: From (16), we obtain

The virtual control input α₁(e(t), x₁(t), x_d(t)) is adopted as

where k₁₁ > 0, k₂₁ > 0 are design parameters, sign(·) denotes a signum function.

Denote α₁(t) = α₁(e(t), x₁(t), x_d(t)). Let

Introduce Equations (18) and (19) into Equation (17) yields

Multiplying e(t) with Equation (20) generates

Let the Lyapunov function candidate V₁(t) be taken as

By Lemma 3 and Equation (21), one obtains

where κ₁ = 2k₁₁ is a positive constant.

Step 2: From Equations (16) and (19), we have

where F1(x1(t)=D0Ctαα1(t)) is an unknown function. To approximate F₁(x₁(t)), we adopt a RBFNN formulated by

Suppose θ1* is the optimal parameter, which is represented as

Here, θ1* is presented for the purpose of analysis, in other words, it is not required in the controller design procedure.

Define the parameter estimation error θ~1(t) as

Also, formulate the optimal approximate error ϵ₁(x₁(t)) by

According to Tong and Li [49], we know that ϵ₁(x₁(t)) is bounded. Therefore,

where ϵ̄1 is a known constant. Consequently,

Define the virtual control input by

where k₁₂ > 0 and k22>ϵ̄1 are design parameters.

Implement the following fractional-order adaptation law:

where ρ₁ is a positive design parameters. Noting that the α-th order derivatives of constants are equal to 0, by Equation (27), we immediately get

Put

Then Equations (24), (30), and (31) lead to

By multiplying e₁(t) with (35), we obtain

Adopt the next Lyapunov function candidate V₂(t):

Applying Lemma 3 and Equation (23), ont gets

Substituting Equations (36) and (32) into Equation (38) derives

where κ₂ = min{κ₁, 2k₁₂, ρ₁} and H1=ρ12θ1*Tθ1* are positive constants.

Step 3: Using Equation (34), one has

where F2(x1(t),x2(t))=D0Ctαα2(t) is unknown. We approximate F₂x₁(t), x₂(t) via RBFNN as follows:

Furthermore, Equation (40) can be reformulated by

Let the virtual control input be expressed by

Design the fractional-order adaptation law as

where k₁₃ > 0, k23>ϵ̄2 (ϵ̄2 are design parameters with ||ϵ2(x1(t),x2(t))||≤ϵ̄2), ρ₂ > 0. Substitute it into Equation (43). By multiplying e₂(t) with Equation (42), we get

Choose the Lyapunov function V₃(t) as

Employing Lemma 3 with Equations (39), (44), and (45) together gives

where κ₃ = min{κ₂, 2k₁₃, ρ₂} and H2=H1+12ρ1θ2*Tθ2* are positive constants.

Theorem 1. In system (16), if the control outputs are formulated by Equations (18), (31), and (43), and the adaptation law is designed as Equations (32) and (44), then the tracking error e(t) must tend to a sufficiently small neighborhood of the equilibrium point.

Proof. Applying (47), one gets

where Q̄(t)≥0. By the implementation of the Laplace transform on Equation (48), we obtain

where V₃(s) and M(s) are given by the Laplace transform on V₃(t) and Q̄(t), respectively.

By Equations (4), (49), V₃(t) can be rearranged as