Practical finite-time adaptive neural networks control for incommensurate fractional-order nonlinear systems

Abstract

The issue associated with the practical finite-time adaptive neural networks control is studied in this paper for a class of incommensurate fractional-order nonlinear systems with external disturbances. With the help of a practical finite-time stability criterion for an integer-order system, the criterion for a fractional-order system is first established, where only the first-order derivative is adopted instead of the fractional-order derivative of the Lyapunov function. This provides an avenue to address the practical finite-time adaptive control problem for incommensurate fractional-order systems. By using the property of fractional-order calculus, a practical finite-time adaptive control scheme is then designed, which is the first time that a practical finite-time adaptive control problem for an incommensurate fractional-order nonlinear system has been considered. In contrast with the control schemes in the literature, the control signals in this paper are generated by some filters, with the result that the fluctuation range of control signals is reduced (especially around the initial time). In addition, a compensated signal is introduced into the design processes of the control scheme, which can not only compensate for the difference caused by the filter of the control scheme, but also simplify the design processes of the control scheme and stability analysis of the controlled system. Finally, numerical simulations and some comparisons are presented to illustrate the effectiveness of the proposed practical finite-time control scheme.

1 Introduction

Fractional-order calculus, as a global concept, is very suitable to describe the memory and heredity properties of various processes and materials, which is completely neglected by the traditional integer-order derivative. In recent years, with the development of fractional-order calculus theory, an increasing number of practical phenomena have been modeled by fractional-order systems to improve modeling accuracy in many research fields, such as biology, engineering, electroanalytical chemistry and physics [1,2,3]. The stability of a system is one of the most important dynamic characteristics, which has attracted the attention of many scholars, and there are many excellent research results with regards to stability, synchronization and so on [4, 5]. Many practical systems are usually unstable or chaotic, and thus, it is necessary to introduce some intervention measures to aid systems in achieving the actual requirement. Thus, numerous control schemes have been proposed in recent years, such as synchronization control [4] and adaptive control [6,7,8,9,10].

Adaptive control, which is a classical and important control method, has been successfully applied in integer-order systems based on the backstepping method in the past few years [6, 7, 10,11,12,13,14,15]. For example, the adaptive control schemes were designed in [11, 12] for strict feedback systems, where the control direction was assumed to be unknown in [11, 12]. Furthermore, adaptive control schemes for nonstrict feedback nonlinear systems were designed in [13, 16] by using the property of radial basis function neural networks. Since then, many results about nonstrict feedback nonlinear systems that consider Markov jumps, input delays, actuator failures or input dead-zones [7, 14, 15, 17] have emerged. Recently, with the rapid development of research on fractional-order calculus, the adaptive control method has also been extended to fractional-order systems (FOSs) [8, 18,19,20,21,22]. For example, the observer-based adaptive control scheme was designed in [8] by constructing a fractional-order state observer. Liu et al. [20] designed a composite learning adaptive control scheme with the help of the dynamic surface technique. The authors in [22] proposed the notion of the Mittag–Leffler input-to-state practical stable Lyapunov function and designed an event-triggered adaptive control scheme for FOSs with unmodeled dynamics. The systems in the aforementioned literature are commensurate FOSs, which is a special case of incommensurate FOSs, i.e., the derivative orders of each equation in the system are not completely equal. Therefore, the adaptive control scheme for a commensurate FOS cannot be directly extended to that of an incommensurate FOS, which is mainly because the equations with unequal derivative order cannot be substituted into the fractional-order derivative of the Lyapunov function. For this reason, the authors in [23] introduced a disturbance-like assumption to solve the finite-time consensus tracking problem for incommensurate FOSs; the issue of practical fixed-time bipartite consensus was studied for incommensurate FOSs in [24] by introducing a sliding-mode manifold; some authors [9, 25, 26] also transformed the adaptive control problem for incommensurate FOSs into that of integer-order systems by using a frequency distributed model. However, as pointed out in [27], whether the same weighting function $\mu _{\alpha }(\omega ) = \frac{\sin \alpha \pi }{\pi \omega ^{\alpha }}$ can be adopted for different systems is still not strictly proved. In addition, the sign function was involved in the virtual and final controllers in [25, 26], which leads to not only the controllers being discontinuous but also the chatting phenomenon appearing in the applications. Furthermore, during the i-th step of the backstepping method, the $(i-1)$-th tracking error appeared in the i-th virtual controller, which makes the structure of the controller complex and difficult for practical use. Therefore, we hope some progress in the controller design will be made for incommensurate FOSs to overcome the above-mentioned shortcomings and avoid the use of the frequency distributed model, which is one of the motivations of this paper.

On the other hand, the convergence rate is also an important performance indicator for practical systems. The control schemes in most of the aforementioned literature were proposed based on the asymptotic stability theory [6, 7, 11,12,13,14,15]. That is, the preset control objective can be achieved, or the tracking error enters a given region with a long convergence time. In fact, regarding the actual requirement, the control goal is expected to be achieved in finite time, which promotes the development of finite-time stability theory [28, 29]. As a more general notion, practical finite-time stability was proposed by Zhu et al. in [30], and some significant results about practical finite-time stability have emerged [31,32,33]. For example, a criterion for practical finite-time stability was developed in [31]; then, the authors designed a finite-time adaptive control scheme for a class of nonlinear systems by using a command filter technique. In [33], an adaptive control scheme for a multi-agent system was designed based on the proposed fast finite-time stability criterion. In recent years, some finite-time control schemes have also been proposed for FOSs [3, 34,35,36]. The finite-time adaptive sliding mode control schemes were designed in [3, 34] for fractional-order nonlinear systems and fractional-order hydro-turbine governing systems. The finite-time synchronization was studied for fractional memristive neural networks in [35]. Among the many studies of finite-time control for FOSs, the formula ${\mathcal {D}}^{\alpha } g^{\iota }(t) = \frac{\varGamma (1+ \iota )}{\varGamma (1+ \iota +\alpha )} g^{\iota - \alpha }(t) {\mathcal {D}}^{\alpha } g(t)$ $(\iota \in {\mathbb {R}}, \alpha \in (0,1))$ plays a crucial role in obtaining the results of finite-time control. Unfortunately, as pointed out in [4], this crucial formula may be incorrect (a counterexample is given in [4]). To the best of our current knowledge, when $\iota \in (0, 1)$ and $\alpha \in (0,1)$, it is very difficult and even impossible to obtain the relationship ${\mathcal {D}}^{\alpha } g^{\iota }(t) \le c_1 g^{c_2}(t) {\mathcal {D}}^{\alpha } g(t)$, because the fractional-order derivative of the compound function is an infinite series of some complicated functions (see [1, Section 2.7.3]), where $c_1 > 0$ and $c_2$ are two constants. Therefore, bypassing this incorrect formula and solving the finite-time adaptive tracking control problem of FOSs are a challenging problem. In addition, even if an alternative criterion is obtained, designing a practical finite-time control scheme and proving the stability of the controlled system are also difficult. All these challenging problems are another motivation of this paper.

Motivated by the above discussions, in this paper, we design a practical finite-time adaptive neural networks control scheme for a class of incommensurate FOSs based on the backstepping method, where the external disturbance is considered. The main contributions of this paper can be summarized as follows:

• (i) Motivated by the criterion of finite-time control for an integer-order system in [31], the criterion of practical finite-time control for the FOS is established, where only the first-order derivative is adopted instead of the fractional-order derivative of the Lyapunov function. The merits of this criterion lie in two aspects: it provides a way to deal with the control problem for incommensurate FOSs, and the frequency distribution model adopted in [9, 25, 26] is avoided, which indicates that the shortcoming of the frequency distribution model is also overcome. In addition, it is worth pointing out that practical finite-time stability can be achieved for the FOS in this paper by using only the same conditions for the Lyapunov function as those for the integer-order system in [31].

• (ii) A novel practical finite-time adaptive neural networks control scheme is proposed in this paper for incommensurate FOSs. The proposed practical finite-time control scheme can also be degenerated into a non-finite time control scheme for commensurate/incommensurate FOSs, which, compared with the literature [9, 18, 20,21,22, 36, 37], is also novel and can be used to control both incommensurate and commensurate FOSs in [9, 18, 20,21,22, 36, 37].

• (iii) In contrast with the results in [18, 36, 37], in this paper, the virtual and final control signals are generated by some designed filters during the design processes of the control scheme, which can reduce the fluctuation range of control signals, especially around the initial time (see Remarks 9 and 11). Additionally, to compensate for the difference between the original signal (i.e., $h_i(t)$) and the control signal, a compensated signal is introduced, which simplifies the processes of control scheme design and stability analysis.

The rest of this paper is organized as follows: In Sect. 2, some preliminary results are presented, and the problems to be studied are also formulated; The practical finite-time adaptive neural networks control scheme is designed in Sect. 3; Two numerical examples are provided in Sect. 4 to illustrate the effectiveness of proposed adaptive control scheme; A summary is presented in Sect. 5.

Notations: ${\mathbb {N}} = \{0, 1, 2, \cdots \}$ is the set of natural numbers, ${\mathbb {N}}_+ = {\mathbb {N}}\setminus \{0\}$, and ${\mathbb {R}}^n$ is the Euclidean space with dimension n. ${\mathcal {C}}^{m}([0,+\infty ), {\mathbb {R}})$ consists of functions from $[0,+\infty )$ to ${\mathbb {R}}$ that have $m \in {\mathbb {N}}$ order continuous derivatives. $\Vert \cdot \Vert$ denotes the Euclidean norm of the vector. For vector $x = (x_1, \ldots , x_n)^\textrm{T} \in {\mathbb {R}}^n$, ${\bar{x}}_i {\mathop {=}\limits ^{\vartriangle }} (x_1, \ldots , x_i)^\textrm{T}$.

2 Preliminaries and problem formulation

2.1 Preliminaries

In this subsection, we first present the fractional-order calculus and some related results.

Definition 1

[1] For function $f(t) \in {\mathcal {C}}^m([0, +\infty ), {\mathbb {R}})$, its Caputo-type fractional-order derivative with order $\alpha > 0$ is defined as

$$\begin{aligned} {\mathcal {D}}^{\alpha } f(t) = \frac{1}{\varGamma (m-\alpha )} \int _{0}^t \frac{f^{(m)}(\tau )}{(t - \tau )^{\alpha + 1 - m}} \text {d}\tau , \end{aligned}$$

where m is a positive integer satisfying $m-1 < \alpha \le m$ and $\varGamma (\cdot )$ is the Gamma function.

Lemma 1

[38] For function $f(t) \in {\mathcal {C}}^m([0, +\infty ), {\mathbb {R}})$ with $m\in {\mathbb {N}}_+$ and $\alpha _1, \alpha _2 >0$, if there exists ${\bar{m}} \in {\mathbb {N}}_+$ with ${\bar{m}} \le m$ such that $\alpha _2, \alpha _1+\alpha _2 \in [{\bar{m}}-1, {\bar{m}}]$, then the following property holds

$$\begin{aligned} {\mathcal {D}}^{\alpha _1} {\mathcal {D}}^{\alpha _2} f(t) = {\mathcal {D}}^{\alpha _1 +\alpha _2} f(t). \end{aligned}$$

Definition 2

[1] The Mittag–Leffler function with one parameter $\alpha > 0$ is defined as

$$\begin{aligned} E_{\alpha } (\varsigma ) = \sum _{s = 0}^{\infty } \frac{\varsigma ^s}{\varGamma (\alpha s + 1)}, \end{aligned}$$

where $\varsigma$ is a complex number.

Lemma 2

[13] For any positive number $\zeta$ and $\omega \in {\mathbb {R}}$, the following inequality holds

$$\begin{aligned} 0 \le \vert \omega \vert - \omega \tanh \frac{\omega }{\zeta } \le \zeta \rho , \end{aligned}$$

where $\rho \approx 0.2785$ is positive root of the equation $\rho = e^{-(\rho +1)}$.

Lemma 3

[33] For any $x_1,x_2 \in {\mathbb {R}}$ and $r = \frac{r_1}{r_2} \in (0,1)$ with positive odd integers $r_1, r_2$, the inequality $x_1x_2^r \le - b_1 x_1^{1+r} + b_2 (x_1+x_2)^{1+r}$ holds, where $b_1 = \frac{1}{1+r}(2^{r-1} - 2^{r^2 -1})$ and $b_2 = \frac{1}{1+r} \big (1 - 2^{r-1} + \frac{r}{1+r} + \frac{1}{1+r}2^{r(1-r^2)}\big )$.

Lemma 4

[31] For $x_s \in {\mathbb {R}}, \, s = 1, \ldots , n$ and $0< r <1$, the following inequality holds

$$\begin{aligned} \left( \sum _{s=1}^n \vert x_s\vert \right) ^r \le \sum _{s=1}^n \vert x_s\vert ^r \le n^{1-r} \left( \sum _{s=1}^n \vert x_s\vert \right) ^r. \end{aligned}$$

Lemma 5

For FOS ${\mathcal {D}}^{\alpha } x(t) = f(x(t))$, if there exists a continuous and positive definite function V(x(t)) such that ${\dot{V}}(x(t)) \le -q_1 V(x(t)) - q_2 V^{\iota }(x(t)) + q_3$ with constants $q_1, q_2, q_3 > 0$ and $0< \iota < 1$, then the trajectory of FOS ${\mathcal {D}}^{\alpha } x(t) = f(x(t))$ is practical finite-time stable, and there exists a finite-time T satisfying

$$\begin{aligned} \begin{aligned} T = \max \bigg \{&\frac{1}{\varrho q_1 (1- \iota )} \ln \frac{\varrho q_1 V^{1- \iota }(x(0)) + q_2}{q_2}, \\&\quad \frac{1}{q_1 (1 - \iota )} \ln \frac{q_1 V^{1-\iota }(x(0)) + \varrho q_2}{\varrho q_2} \bigg \} \end{aligned} \end{aligned}$$

(1)

such that for any $t \ge T$, the following inequality holds

$$\begin{aligned} V(x(t)) \le \min \left\{ \frac{q_3}{(1 - \varrho )q_1}, \Big (\frac{q_3}{(1- \varrho ) q_2}\Big )^{1/\iota }\right\} , \end{aligned}$$

where $0< \varrho < 1$ is an arbitrary constant.

Proof

The results can be proven by using an argument similar to that in [31], so we omit it here. $\square$

Remark 1

In this paper, the definition of practical finite-time stability is adopted; namely, for any initial value $x(0)=x_0$, there exist $\varepsilon > 0$ and $T (\varepsilon , x_0)< \infty$, such that the solution of system satisfies $\Vert x(t)\Vert \le \varepsilon$ for all $t \ge T$. If we let $\varepsilon = 0$, then the above definition will degenerate to the definition of finite-time stability that was adopted in [3, 28, 29]. Besides, there is another definition of finite-time stability in [39]: for constants $0< \varepsilon _1 < \varepsilon _2$ and finite-time $T > 0$, $\Vert x_0\Vert \le \varepsilon _1$ implies $\Vert x(t) \Vert \le \varepsilon _2$ for $\forall t \in [0, T]$. Thus, we can see that the former definition of finite-time stability mainly focuses on whether the solution of the system converges to zero at T and is always equal to zero on the interval $(T, +\infty )$, whereas the latter definition of finite-time stability focuses on the boundedness of solution on the finite interval [0, T].

2.2 Problem formulation

Consider the following incommensurate fractional-order nonlinear system

$$\begin{aligned} {\left\{ \begin{array}{ll} {\mathcal {D}}^{\alpha _i} x_i = x_{i+1} + f_i(x) + \Delta _i(t), \\ {\mathcal {D}}^{\alpha _n} x_n = u + f_n(x) + \Delta _n(t), \\ y = x_1, \quad i = 1, \ldots , n-1, \end{array}\right. } \end{aligned}$$

(2)

where $\alpha _s \in (0,1)$ ($s = 1,\ldots , n$), $x_s$ is the state variable, u and y are the input and output of the system, respectively. $f_s(x)$ is an unknown smooth function with $1+ \partial f_s(x) /\partial x_{s+1} \ne 0$ ($s = 1,\ldots , n-1$), and $\Delta _s(t)$ is the external disturbance.

Remark 2

System (2) is a classical incommensurate fractional-order one with the nonstrict feedback form, which can characterize many practical systems, such as fractional-order Chua–Hartley’s system, fractional-order Rössler’s system, fractional-order Chen’s system, fractional-order Lorenz’s system, fractional-order Lotka-Volterra system [1, 2]. Meanwhile, (2) also contains numerous systems in the literature as special cases. For example, if $\alpha _1 = \cdots = \alpha _n \in (0,1)$, system (2) becomes the commensurate fractional-order one with the nonstrict feedback form, which was studied in [8, 19, 37], and if further letting $f_i(x) = f_i({\bar{x}}_i)$, system (2) degenerates into the one with the strict feedback form in [18, 21]. If $\alpha _1 = \cdots = \alpha _n = 1$, system (2) becomes an integer-order one, which was considered in [13, 14]. Besides, if we use the transformation method in [17], then the systems in [7, 20, 22, 40] can transformed into the special cases of the system (2). Therefore, the design of control scheme for the system (2) is of great significance from both the practical and theoretical points of view.

The control objective of this paper is to design a practical finite-time control scheme such that: (i) all signals in the closed-loop system are bounded; (ii) the system output can track the reference signal $y_d(t)$ with a small error and the tracking error can enter a small region in finite time.

To realize the control objective, the following assumptions are needed.

Assumption 1

The external disturbance $\Delta _i(t) \, (i = 1, \ldots , n)$ and its derivative are continuous and bounded.

Assumption 2

The reference signal $y_d$ and its derivative are continuous and bounded.

Assumption 3

The orders of fractional derivative in system (2) satisfy $\alpha _i + \alpha _{i+1} \ge 1 \, (i = 1, \ldots , n-1)$.

Remark 3

In some existing literature [19,20,21], the external disturbance is assumed to be bounded, which is standard. During the design process of the practical finite-time control scheme, it also requires that ${\mathcal {D}}^{1- \alpha _i} \Delta _i(t)$ is bounded, while it is difficult to verify. Therefore, we assume that ${\dot{\Delta }}_i(t)$ is bounded, which is an easily verifiable hypothesis and implies the boundedness of ${\mathcal {D}}^{1- \alpha _i} \Delta _i(t)$. In fact, since ${\mathcal {D}}^{\alpha } \Delta _i(t)$ is continuous with respect to $\alpha$, and ${\mathcal {D}}^1\Delta _i(t) = {\dot{\Delta }}_i(t)$ and ${\mathcal {D}}^0\Delta _i(t) = \Delta _i(t)$ are bounded, we can conclude that ${\mathcal {D}}^{1- \alpha _i} \Delta _i(t)$ is bounded. Assumption 2 is quite standard, which is adopted in many previous studies [7, 14, 15]. It should be pointed out that ${\mathcal {D}}^{\alpha } y_d \, (\alpha \in (0,1))$ is assumed to be bounded in [19, 20, 22], which is difficult to verify due to the complexity of fractional-order calculus, while Assumption 2, as an alternative, is easy to verify and implies the boundedness of ${\mathcal {D}}^{\alpha } y_d$ based on the continuity of fractional-order derivative with respect to order $\alpha$. We give Assumption 3 to ensure the continuity of ${\mathcal {D}}^{1 - \alpha _i} x_{i + 1}$ that will appear during the designing process of the adaptive control scheme. In Assumption 3, there are some constraints to the orders $\alpha _i \, (i = 1,\ldots , n)$, while it does not seriously influence the actual application, since we find that many practical systems mentioned in [1, 2] and references therein can satisfy Assumption 3.

2.3 Radial basis function neural networks

Since the function $f_s(x)$ is unknown and some terms during the design process of the control scheme are difficult to deal with, the radial basis function neural networks are introduced in this subsection to approximate them.

For continuous function ${\mathcal {H}}(Z): {\mathbb {R}}^{n_1} \rightarrow {\mathbb {R}}$ with $n_1 \in {\mathbb {N}}_+$, we construct the neural networks as ${\mathcal {H}}_{NN} = W^\textrm{T} \Phi (Z)$, where $Z \in \Omega \subset {\mathbb {R}}^{n_1}$ is the input of neural networks, $W \in {\mathbb {R}}^{n_2}$ is the weight vector, $n_2$ is the number of nodes in neural networks, and $\Phi (Z) = (\phi _1(Z), \ldots , \phi _{n_2}(Z))$ is the Gaussian-like radial basis function vector with

$$\begin{aligned} \phi _s(Z) = \exp \bigg [- \frac{(Z - \varpi _s)^\textrm{T} (Z - \varpi _s)}{\delta _s^2}\bigg ], \end{aligned}$$

$s= 1, \ldots , n_2$, $\varpi _s = (\varpi _{s1}, \ldots , \varpi _{sn_1})^\textrm{T}$ and $\delta _s$ are the center of receptive domain and the width of Gaussian function, respectively.

Lemma 6

[41] If the nodes number $n_2$ of neural networks is sufficiently large, then the radial basis function neural networks ${\mathcal {H}}_{NN} = W^\textrm{T} \Phi (Z)$ can approximate the continuous function ${\mathcal {H}}(Z)$ over compact set $\Omega \subset {\mathbb {R}}^{n_1}$ to arbitrary accuracy ${\tilde{\epsilon }}$ as

$$\begin{aligned} {\mathcal {H}}(Z) = W^{*T} \Phi (Z) + \epsilon (Z), \end{aligned}$$

where $W^* \in {\mathbb {R}}^{n_2}$ is the ideal weight vector of neural networks defined as

$$\begin{aligned} W^* = \arg \min _{W\in {\mathbb {R}}^{n_2}} \left\{ \sup _{Z \in \Omega } \vert {\mathcal {H}}(Z) - W^\textrm{T} \Phi (Z)\vert \right\} , \end{aligned}$$

and $\epsilon (Z)$ is the approximation error satisfying $\vert \epsilon (Z)\vert < {\tilde{\epsilon }}$.

Lemma 7

[15] Let $Z = (z_1,\ldots , z_{n_1})^\textrm{T}$, $\varpi _s$ and $\Phi (Z)$ are defined as in Lemma 6, ${\check{Z}} = (z_{i_1}, \ldots , z_{i_{\hbar }})^\textrm{T}$, ${\check{\varpi }}_s = (\varpi _{s,i_1}, \ldots , \varpi _{s, i_{\hbar }})^\textrm{T}$, and $\{i_1, \ldots , i_{\hbar }\}$ is a subsequence of $\{1, \ldots , n_1\}$. ${\check{\Phi }}({\check{Z}}) = ({\check{\phi }}_1({\check{Z}}), \ldots , {\check{\phi }}_{n_2}({\check{Z}}))$ with ${\check{\phi }}_s({\check{Z}}) = \exp \bigg [- \frac{({\check{Z}} - {\check{\varpi }}_s)^\textrm{T} ({\check{Z}} - {\check{\varpi }}_s)}{\delta _s^2}\bigg ]$, $s = 1, \ldots , n_2$ are radial basis function vectors. Then,

$$\begin{aligned} \Vert \Phi (Z)\Vert \le \Vert {\check{\Phi }}({\check{Z}})\Vert . \end{aligned}$$

For the sake of simplicity, ${\check{\Phi }}({\check{Z}})$ is still denoted by $\Phi ({\check{Z}})$ in the following if no confusion arises.

3 The design of control scheme and stability analysis

This section is devoted to designing the practical finite-time adaptive neural networks control scheme for system (2) and analyzing the stability of the controlled system. In order to do this, we first define the following coordinate transform

$$\begin{aligned} {\left\{ \begin{array}{ll} z_1 = x_1 - y_d, \\ z_i = x_i - \chi _{i-1}, \quad i = 2, \ldots , n, \\ \end{array}\right. } \end{aligned}$$

(3)

where $\chi _i, \, (i = 1, \ldots , n-1)$ is the virtual control signal that will be specified later. The control scheme is designed in the following by using the backstepping technique.

Step 1: According to (2) and (3), and using Lemma 1, we have

$$\begin{aligned} \begin{aligned} {\dot{z}}_1&= {\mathcal {D}}^{1- \alpha _1} {\mathcal {D}}^{\alpha _1} x_1 - {\dot{y}}_d \\&= {\mathcal {D}}^{1- \alpha _1}(z_2 + \chi _1 + f_1(x) + \Delta _1(t)) - {\dot{y}}_d. \end{aligned} \end{aligned}$$

(4)

Since the fractional-order derivative of $\chi _1$ with order ($1-\alpha _1$) is appeared in (4), the virtual control signal $\chi _1$ is difficult to design. Thus, motivated by the dynamic surface technique [20], we define the following fractional filter

$$\begin{aligned} {\mathcal {D}}^{1- \alpha _1} \chi _1(t) = - \kappa _1(\chi _1(t) - h_1(t)), \chi _1(0) = 0, \end{aligned}$$

(5)

where $h_1(t)$ is an intermediate signal that will be designed later and $\kappa _1 > 0$ is a designed parameter. From (5), we can only obtain that the error ${\tilde{h}}_1 = \chi _1 - h_1$ is bounded by a constant (see Lemma 8), but it cannot be guaranteed to converge to a small region of origin. Therefore, we design a compensated signal ${\tilde{\chi }}_1$ as follows

$$\begin{aligned} \dot{{\tilde{\chi }}}_1(t)= & {} - \lambda _1 {\tilde{\chi }}_1(t) - {\bar{\lambda }}_1 {\tilde{\chi }}_1^r(t) - \kappa _1 \chi _1(t), \nonumber \\ {\tilde{\chi }}_1(0)= & {} z_1(0), \end{aligned}$$

(6)

where $\lambda _1, {\bar{\lambda }}_1 > 0$ are the designed parameters and $r = \frac{r_1}{r_2} \in (0,1)$ with positive odd numbers $r_1,r_2$. The compensated tracking error is defined as

$$\begin{aligned} {\tilde{z}}_1 = z_1 - {\tilde{\chi }}_1. \end{aligned}$$

(7)

It then follows from (4) and (7) that

$$\begin{aligned} \begin{aligned} \dot{{\tilde{z}}}_1&= {\dot{z}}_1 - \dot{{\tilde{\chi }}}_1 \\&= {\mathcal {D}}^{1- \alpha _1}(z_2 + \chi _1 + f_1(x) + \Delta _1(t)) - {\dot{y}}_d - \dot{{\tilde{\chi }}}_1. \end{aligned} \end{aligned}$$

Based on Assumption 1, there exists a positive constant $d_1$ such that $\vert {\mathcal {D}}^{1- \alpha _1} \Delta _1(t)\vert \le d_1$, which, together with Lemma 2, implies that

$$\begin{aligned} {\tilde{z}}_1 {\mathcal {D}}^{1- \alpha _1} \Delta _1(t) \le d_1\vert {\tilde{z}}_1\vert \le d_1{\tilde{z}}_1 \tanh \left( \frac{d_1{\tilde{z}}_1}{\zeta _1}\right) + \zeta _1 \rho ,\nonumber \\ \end{aligned}$$

(8)

where $\zeta _1 > 0$ is a designed parameter. Construct the Lyapunov function as

$$\begin{aligned} V_1 = \frac{1}{2} {\tilde{z}}_1^2 + \frac{1}{2 \eta _1} {\tilde{\theta }}_1^2, \end{aligned}$$

where $\eta _1 > 0$ is a designed parameter, ${\tilde{\theta }}_1 = {\hat{\theta }}_1 - \theta _1$ is the estimation error of $\theta _1 = \Vert W_1^*\Vert ^2$, ${\hat{\theta }}_1$ is the estimate of $\theta _1$, and $W_1^*$ will be specified later. By using (8), the derivative of $V_1$ can be calculated as

$$\begin{aligned} \begin{aligned} {\dot{V}}_1 =&\, {\tilde{z}}_1 \big ({\mathcal {D}}^{1- \alpha _1} (z_2 + \chi _1 + f_1(x) + \Delta _1(t)) \\&- {\dot{y}}_d - \dot{{\tilde{\chi }}}_1\big ) + \frac{1}{\eta _1} {\tilde{\theta }}_1 \dot{{\hat{\theta }}}_1 \\ \le&{\tilde{z}}_1 {\mathcal {H}}_1(Z_1) + {\tilde{z}}_1 {\mathcal {D}}^{1- \alpha _1} \chi _1 - {\tilde{z}}_1 \dot{{\tilde{\chi }}}_1 + \frac{1}{\eta _1} {\tilde{\theta }}_1 \dot{{\hat{\theta }}}_1 \\&+ \zeta _1 \rho - \lambda _1 z_1 {\tilde{z}}_1 - {\bar{\lambda }}_1 {\tilde{z}}_1 {\tilde{\chi }}_1^r, \end{aligned} \end{aligned}$$

where ${\mathcal {H}}_1(Z_1) = {\mathcal {D}}^{1- \alpha _1} (z_2 + f_1(x)) + d_1 \tanh \left( \frac{d_1 {\tilde{z}}_1}{\zeta _1}\right) - {\dot{y}}_d + \lambda _1 z_1 + {\bar{\lambda }}_1 {\tilde{\chi }}_1^r$ with $Z_1 = (x^\textrm{T}, {\hat{\theta }}_1, {\tilde{\chi }}_1, y_d, {\dot{y}}_d)^\textrm{T} \in {\mathbb {R}}^{n+4}$. It should be pointed out that ${\mathcal {D}}^{1- \alpha _1} (z_2 + f_1(x))$ is a continuous function, which thus can be approximated subsequently by using neural networks. In fact, since ${\mathcal {D}}^{\alpha _2} x_2$ is continuous (from system (2)) and $\alpha _2 \ge 1 - \alpha _1$ (according to Assumption 3), we can conclude that ${\mathcal {D}}^{1- \alpha _1} x_2$ is continuous. Besides, from (5), we obtain that ${\mathcal {D}}^{1- \alpha _1} \chi _1$ is continuous. Thus, equation ${\mathcal {D}}^{1- \alpha _1} z_2 = {\mathcal {D}}^{1- \alpha _1} (x_2 - \chi _1)$ implies that ${\mathcal {D}}^{1- \alpha _1} z_2$ is continuous. In addition, since $f_1(x)$ is smooth, ${\mathcal {D}}^{1- \alpha _1} f_1(x)$ is continuous. Based on Lemma 6, the continuous function ${\mathcal {H}}_1(Z_1)$ can be approximated as

$$\begin{aligned} {\mathcal {H}}_1(Z_1) = W_1^{*T} \Phi _1(Z_1) + \epsilon _1(Z_1), \end{aligned}$$

where $W_1^{*}$ is the ideal weight vector and $\epsilon _1(Z_1)$ is the approximation error with $\vert \epsilon _1(Z_1)\vert < {\tilde{\epsilon }}_1$. It then follows from Lemma 7 that

$$\begin{aligned} \begin{aligned} {\tilde{z}}_1 {\mathcal {H}}_1(Z_1) \le&\, \,\vert {\tilde{z}}_1\vert (\Vert W_1^{*}\Vert \Vert \Phi _1(Z_1)\Vert + \vert \epsilon _1(Z_1))\vert \\ \le&\, \vert {\tilde{z}}_1\vert (\Vert W_1^{*}\Vert \Vert \Phi _1(X_1)\Vert + {\tilde{\epsilon }}_1) \\ \le&\, \,\frac{{\tilde{z}}_1^2}{2a_1^2} \theta _1 \Phi _1^\textrm{T}(X_1) \Phi _1(X_1) \\&\quad + \frac{a_1^2}{2} + \frac{{\tilde{z}}_1^2}{2} + \frac{{\tilde{\epsilon }}_1^2}{2} , \end{aligned} \end{aligned}$$

(9)

where $a_1 > 0$ is a designed parameter and $X_1 = (x_1, {\hat{\theta }}_1, {\tilde{\chi }}_1, y_d, {\dot{y}}_d)^\textrm{T} \in {\mathbb {R}}^5$. For the sake of simplicity, $\Phi _1(X_1)$ will be abbreviated as $\Phi _1$ in the following. Hence, ${\dot{V}}_1$ can be rewritten as

$$\begin{aligned} {\dot{V}}_1 =&\, \frac{1}{2a_1^2} {\tilde{z}}_1^2 \theta _1 \Phi _1^\textrm{T} \Phi _1 + \frac{1}{2} {\tilde{z}}_1^2 + {\tilde{z}}_1 {\mathcal {D}}^{1- \alpha _1} \chi _1 - {\tilde{z}}_1 \dot{{\tilde{\chi }}}_1 \\&+ \frac{1}{\eta _1} {\tilde{\theta }}_1 \dot{{\hat{\theta }}}_1 - \lambda _1 z_1 {\tilde{z}}_1 - {\bar{\lambda }}_1 {\tilde{z}}_1 {\tilde{\chi }}_1^r + \Theta _1, \end{aligned}$$

where $\Theta _1 = \frac{1}{2} a_1^2 + \frac{1}{2} {\tilde{\epsilon }}_1^2 + \zeta _1 \rho$. Design the intermediate signal $h_1(t)$ and the adaptive law of ${\hat{\theta }}_1$ as

$$\begin{aligned} h_1(t) =&\, -\left( \gamma _1 +\frac{1}{2}\right) \frac{{\tilde{z}}_1}{\kappa _1} - \frac{{\tilde{z}}_1 {\hat{\theta }}_1}{2a_1^2 \kappa _1} \Phi _1^\textrm{T} \Phi _1 - \frac{{\bar{\gamma }}_1}{\kappa _1} \vartheta _1({\tilde{z}}_1), \end{aligned}$$

(10)

$$\begin{aligned} \dot{{\hat{\theta }}}_1 =&\, \frac{\eta _1}{2a_1^2} {\tilde{z}}_1^2 \Phi _1^\textrm{T} \Phi _1 - \mu _1 {\hat{\theta }}_1 - \xi _1 {\hat{\theta }}_1^r, {\hat{\theta }}_1(0) \ge 0, \end{aligned}$$

(11)

where $\gamma _1, {\bar{\gamma }}_1, \mu _1, \xi _1 > 0$ are designed parameters, and

$$\begin{aligned} \vartheta _1({\tilde{z}}_1) = {\left\{ \begin{array}{ll} {\tilde{z}}_1^r, &{} \vert {\tilde{z}}_1\vert \ge \tau _1, \\ A_1 {\tilde{z}}_1 - B_1 {\tilde{z}}_1^3, &{} \vert {\tilde{z}}_1\vert < \tau _1, \end{array}\right. } \end{aligned}$$

with $A_1 = \frac{1}{2} (3 - r) \tau _1^{r-1}, B_1 = \frac{1}{2} \tau _1^{r-3} (1 -r)$, and a small constant $\tau _1>0$.

Remark 4

Motivated by [32], the term ${\tilde{z}}_1^r$ is modified to $A_1 {\tilde{z}}_1 - B_1 {\tilde{z}}_1^3$ when $\vert {\tilde{z}}_1\vert$ is less than the small positive constant $\tau _1$. This modification can ensure that: (i) the singularity phenomenon of ${\mathcal {D}}^{\alpha _1} h_1(t)$ can be avoided when ${\tilde{z}}_1 = 0$; (ii) ${\mathcal {D}}^{\alpha _1} h_1(t)$ is continuous function of $(x, {\hat{\theta }}_1, y_d)^\textrm{T}$.

It follows from (5)–(7), (10) and (11) that

$$\begin{aligned} {\dot{V}}_1 \le&\, \frac{1}{2a_1^2} {\tilde{z}}_1^2 \theta _1 \Phi _1^\textrm{T} \Phi _1 + \frac{1}{2} {\tilde{z}}_1^2 + \kappa _1 {\tilde{z}}_1 h_1(t) \\&+ \lambda _1 {\tilde{z}}_1 {\tilde{\chi }}_1 + \frac{1}{\eta _1} {\tilde{\theta }}_1 \dot{{\hat{\theta }}}_1 - \lambda _1 z_1 {\tilde{z}}_1 + \Theta _1 \\ =&\, -(\gamma _1 + \lambda _1) {\tilde{z}}_1^2 - {\bar{\gamma }}_1 {\tilde{z}}_1 \vartheta _1({\tilde{z}}_1) \\&- \frac{\mu _1}{\eta _1} {\tilde{\theta }}_1 {\hat{\theta }}_1 - \frac{\xi _1}{\eta _1} {\tilde{\theta }}_1 {\hat{\theta }}_1^r + \Theta _1. \end{aligned}$$

Step i ($2 \le i \le n-1$): From (2) and (3), and Lemma 1, one can obtain that

$$\begin{aligned} \begin{aligned} {\dot{z}}_i&= {\mathcal {D}}^{1- \alpha _i} {\mathcal {D}}^{\alpha _i} x_i - {\dot{\chi }}_{i-1} \\&= {\mathcal {D}}^{1- \alpha _i} \big (z_{i+1} + \chi _i + f_i(x) + \Delta _i(t)\big ) - {\dot{\chi }}_{i-1}. \end{aligned} \end{aligned}$$

(12)

Since ${\mathcal {D}}^{1- \alpha _i} \chi _i$ appears in the above equation, we design the following filter to approximate the virtual control signal $\chi _i$

$$\begin{aligned} {\mathcal {D}}^{1- \alpha _i} \chi _i(t) = - \kappa _i (\chi _i(t) - h_i(t)), \chi _i(0) = 0, \end{aligned}$$

(13)

where $\kappa _i > 0$ is a designed parameter and $h_i(t)$ is an intermediate signal that will be designed later. To compensate for the error between $\chi _i$ and $h_i$, we define a compensated signal ${\tilde{\chi }}_i$ that is determined by

$$\begin{aligned} \dot{{\tilde{\chi }}}_i(t) = - \lambda _i {\tilde{\chi }}_i(t) - {\bar{\lambda }}_i {\tilde{\chi }}_i^r(t) - \kappa _i \chi _i(t), {\tilde{\chi }}_i(0) = z_i(0),\nonumber \\ \end{aligned}$$

(14)

where $\lambda _i, {\bar{\lambda }}_i > 0$ are designed parameters. Let ${\tilde{z}}_i = z_i - {\tilde{\chi }}_i$, from (12), we have

$$\begin{aligned} \dot{{\tilde{z}}}_i = {\mathcal {D}}^{1- \alpha _i} \big (z_{i+1} + \chi _i + f_i(x) + \Delta _i(t)\big ) - {\dot{\chi }}_{i-1} - \dot{{\tilde{\chi }}}_i.\nonumber \\ \end{aligned}$$

(15)

Construct the following Lyapunov function

$$\begin{aligned} V_i = V_{i-1} + \frac{1}{2} {\tilde{z}}_i^2 + \frac{1}{2 \eta _i} {\tilde{\theta }}_i^2, \end{aligned}$$

where $\eta _i$ is a designed parameter, ${\tilde{\theta }}_i = {\hat{\theta }}_i - \theta _i$ is the estimation error of $\theta _i = \Vert W_i^*\Vert ^2$, ${\hat{\theta }}_i$ is the estimate of $\theta _i$, and $W_i^*$ is the weight vector that will be defined later. The derivative of $V_i$ can be computed as

$$\begin{aligned} {\dot{V}}_i =&\, {\dot{V}}_{i-1} + {\tilde{z}}_i \big ({\mathcal {D}}^{1 - \alpha _i} (z_{i+1} + \chi _i + f_i(x) + \Delta _i(t)) \\&- {\dot{\chi }}_{i-1} - \dot{{\tilde{\chi }}}_i\big ) + \frac{1}{\eta _i} {\tilde{\theta }}_i \dot{{\hat{\theta }}}_i. \end{aligned}$$

From Assumption 1, these exists a constant $d_i > 0$ such that $\vert {\mathcal {D}}^{1 - \alpha _i} \Delta _i(t)\vert \le d_i$, which, together with Lemma 2, implies that

$$\begin{aligned} {\tilde{z}}_i {\mathcal {D}}^{1 - \alpha _i} \Delta _i(t) \le d_i\vert {\tilde{z}}_i\vert \le d_i {\tilde{z}}_i \tanh \left( \frac{d_i {\tilde{z}}_i}{\zeta _i}\right) + \zeta _i \rho . \end{aligned}$$

Therefore, we have

$$\begin{aligned} \begin{aligned} {\dot{V}}_i \le&\, {\dot{V}}_{i-1} +{\tilde{z}}_i {\mathcal {H}}_i (Z_i) + {\tilde{z}}_i {\mathcal {D}}^{1 -\alpha _i} \chi _i - {\tilde{z}}_i \dot{{\tilde{\chi }}}_i\\&+ \frac{1}{\eta _i} {\tilde{\theta }}_i \dot{{\hat{\theta }}}_i + \zeta _i \rho - \lambda _i z_i {\tilde{z}}_i - {\bar{\lambda }}_i {\tilde{z}}_i {\tilde{\chi }}_i^r, \end{aligned} \end{aligned}$$

(16)

where ${\mathcal {H}}_i (Z_i) = {\mathcal {D}}^{1 - \alpha _i} (z_{i+1} + f_i(x)) + d_i \tanh \left( \frac{d_i {\tilde{z}}_i}{\zeta _i}\right) - {\dot{\chi }}_{i-1} + \lambda _i z_i + {\bar{\lambda }}_i {\tilde{\chi }}_i^r$ with $Z_i = (x^\textrm{T}, {\hat{\theta }}_1, \ldots , {\hat{\theta }}_i, {\tilde{\chi }}_i, y_d, {\dot{y}}_d)^\textrm{T} \in {\mathbb {R}}^{n+i+3}$. Similar to the statement in Step 1, the continuity of ${\mathcal {D}}^{1 - \alpha _i} (z_{i+1} + f_i(x))$ can be obtained. Besides, according to the fractional-order filter (5) and (13), ${\mathcal {D}}^{1 - \alpha _{i-1}} \chi _{i-1}$ must exist, which, together with Definition 1, implies that ${\dot{\chi }}_{i-1}$ is continuous. Based on Lemma 6, continuous function ${\mathcal {H}}_i (Z_i)$ can be approximated as

$$\begin{aligned} {\mathcal {H}}_i (Z_i) = W_i^{*T} \Phi _i(Z_i) + \epsilon _i(Z_i), \end{aligned}$$

where $W_i^*$ denotes the ideal weight and $\epsilon _i(Z_i)$ is the approximation error satisfying $\vert \epsilon _i(Z_i)\vert < {\tilde{\epsilon }}_i$ with positive constant ${\tilde{\epsilon }}_i$. According to Lemma 7 and using the same manner as in (9), we have

$$\begin{aligned} {\tilde{z}}_i {\mathcal {H}}_i(Z_i){} & {} \le \frac{1}{2a_i^2} {\tilde{z}}_i^2 \theta _i \Phi _i^\textrm{T}(X_i) \Phi _i(X_i) \\{} & {} + \frac{1}{2} a_i^2 + \frac{1}{2} {\tilde{z}}_i^2 + \frac{1}{2} {\tilde{\epsilon }}_i^2, \end{aligned}$$

where $a_i > 0$ is a designed parameter and $X_i = (x_1, \ldots , x_i, {\hat{\theta }}_1, \ldots , {\hat{\theta }}_i, {\tilde{\chi }}_i, y_d, {\dot{y}}_d)^\textrm{T} \in {\mathbb {R}}^{2i+3}$. For convenience, argument $X_i$ in $\Phi _i(X_i)$ will be omitted in the following. Thus, (16) can be rewritten as

$$\begin{aligned} {\dot{V}}_i \le&\, {\dot{V}}_{i-1} + \frac{1}{2a_i^2} {\tilde{z}}_i^2 \theta _i \Phi _i^\textrm{T} \Phi _i + \frac{1}{2} {\tilde{z}}_i^2 + {\tilde{z}}_i {\mathcal {D}}^{1 -\alpha _i} \chi _i - {\tilde{z}}_i \dot{{\tilde{\chi }}}_i \\&+ \frac{1}{\eta _i} {\tilde{\theta }}_i \dot{{\hat{\theta }}}_i- \lambda _i z_i {\tilde{z}}_i - {\bar{\lambda }}_i {\tilde{z}}_i {\tilde{\chi }}_i^r + \frac{1}{2} a_i^2 + \frac{1}{2} {\tilde{\epsilon }}_i^2 + \zeta _i \rho , \end{aligned}$$

The intermediate signal $h_i(t)$ and the adaptive law of ${\hat{\theta }}_i$ are designed as

$$\begin{aligned} h_i(t) =&\, -\left( \gamma _i +\frac{1}{2}\right) \frac{{\tilde{z}}_i}{\kappa _i} - \frac{{\tilde{z}}_i {\hat{\theta }}_i}{2a_i^2 \kappa _i} \Phi _i^\textrm{T} \Phi _i - \frac{{\bar{\gamma }}_i}{\kappa _i} \vartheta _i({\tilde{z}}_i), \end{aligned}$$

(17)

$$\begin{aligned} \dot{{\hat{\theta }}}_i =&\, \frac{\eta _i}{2a_i^2} {\tilde{z}}_i^2 \Phi _i^\textrm{T} \Phi _i - \mu _i {\hat{\theta }}_i - \xi _i {\hat{\theta }}_i^r, {\hat{\theta }}_i(0) \ge 0, \end{aligned}$$

(18)

where $\gamma _i, {\bar{\gamma }}_i, \mu _i, \xi _i > 0$ are designed parameters, and

$$\begin{aligned} \vartheta _i({\tilde{z}}_i) = {\left\{ \begin{array}{ll} {\tilde{z}}_i^r, &{} \vert {\tilde{z}}_i\vert \ge \tau _i, \\ A_i {\tilde{z}}_i - B_i {\tilde{z}}_i^3, &{} \vert {\tilde{z}}_i\vert < \tau _i, \end{array}\right. } \end{aligned}$$

with $A_i = \frac{1}{2} (3 - r) \tau _i^{r-1}, B_i = \frac{1}{2} \tau _i^{r-3} (1 -r)$, and a small constant $\tau _i>0$. Substituting (13), (14), (17) and (18) into ${\dot{V}}_i$, we have

$$\begin{aligned} {\dot{V}}_i \le&\, {\dot{V}}_{i-1} + \frac{1}{2a_i^2} {\tilde{z}}_i^2 \theta _i \Phi _i^\textrm{T} \Phi _i + \frac{1}{2} {\tilde{z}}_i^2 + \kappa _i {\tilde{z}}_i h_i(t) \\&+ \lambda _i {\tilde{z}}_i {\tilde{\chi }}_i+ \frac{1}{\eta _i} {\tilde{\theta }}_i \dot{{\hat{\theta }}}_i - \lambda _i z_i {\tilde{z}}_i + \frac{1}{2} a_i^2 + \frac{1}{2} {\tilde{\epsilon }}_i^2 + \zeta _i \rho \\ =&\, {\dot{V}}_{i-1} - (\gamma _i + \lambda _i) {\tilde{z}}_i^2 - {\bar{\gamma }}_i {\tilde{z}}_i \vartheta _i({\tilde{z}}_i) - \frac{\mu _i}{\eta _i} {\tilde{\theta }}_i {\hat{\theta }}_i \\&- \frac{\xi _i}{\eta _i} {\tilde{\theta }}_i {\hat{\theta }}_i^r + \frac{1}{2} a_i^2 + \frac{1}{2} {\tilde{\epsilon }}_i^2 + \zeta _i \rho \\ \le&\, - \sum _{s = 1}^i \bigg ((\gamma _s + \lambda _s) {\tilde{z}}_s^2 + {\bar{\gamma }}_s {\tilde{z}}_s \vartheta _s({\tilde{z}}_s) \\&+ \frac{\mu _s}{\eta _s} {\tilde{\theta }}_s {\hat{\theta }}_s + \frac{\xi _s}{\eta _s} {\tilde{\theta }}_s {\hat{\theta }}_s^r\bigg ) + \Theta _i, \end{aligned}$$

where $\Theta _i = \Theta _{i-1} + \frac{1}{2} a_i^2 + \frac{1}{2} {\tilde{\epsilon }}_i^2 + \zeta _i \rho$.

Step n: Let the final control signal u(t) be given by the filter

$$\begin{aligned} {\mathcal {D}}^{1 - \alpha _n} u(t) = -\kappa _n (u(t) - h_n(t)), \quad u(0) = 0, \end{aligned}$$

(19)

where $\kappa _n > 0$ is a designed parameter and $h_n(t)$ is a intermediate signal that will be specified later. As pointed out in 1-th step, the error between u(t) and $h_n(t)$ cannot be overlooked. Therefore, we design a compensated signal ${\tilde{\chi }}_n$ to eliminate this error, which is given by

$$\begin{aligned} \dot{{\tilde{\chi }}}_n(t) = - \lambda _n {\tilde{\chi }}_n(t) - {\bar{\lambda }}_n {\tilde{\chi }}_n^r(t) - \kappa _n u(t), \, {\tilde{\chi }}_n(0) = z_n(0), \nonumber \\ \end{aligned}$$

(20)

where $\lambda _n, {\bar{\lambda }}_n > 0$ are designed parameters. Let ${\tilde{z}}_n = z_n - {\tilde{\chi }}_n$. Then, by using Lemma 1, we have

$$\begin{aligned} \begin{aligned} \dot{{\tilde{z}}}_n =&\, {\mathcal {D}}^{1-\alpha _n} {\mathcal {D}}^{\alpha _n}_n x_n - {\dot{\chi }}_{n-1} - \dot{{\tilde{\chi }}}_n \\ =&\, {\mathcal {D}}^{1- \alpha _n} \big (u + f_n(x) + \Delta _n(t)\big ) - {\dot{\chi }}_{n-1} - \dot{{\tilde{\chi }}}_n. \end{aligned} \end{aligned}$$

(21)

Define the Lyapunov function $V_n$ as follows

$$\begin{aligned} V_n = V_{n-1} + \frac{1}{2} {\tilde{z}}_n^2 + \frac{1}{2\eta _n} {\tilde{\theta }}_n^2, \end{aligned}$$

where $\eta _n > 0$ is a designed parameter, ${\tilde{\theta }}_n = {\hat{\theta }}_n - \theta _n$ is the estimation error of $\theta _n = \Vert W_n^*\Vert ^2$, ${\hat{\theta }}_n$ is the estimate of $\theta _n$, and $W_n^*$ will be defined later. From (21), we obtain

$$\begin{aligned} {\dot{V}}_n =&\, {\dot{V}}_{n-1} + {\tilde{z}}_n \big ({\mathcal {D}}^{1- \alpha _n} \big (u + f_n(x) + \Delta _n(t)\big ) \\&- {\dot{\chi }}_{n-1} - \dot{{\tilde{\chi }}}_n\big ) + \frac{1}{\eta _n} {\tilde{\theta }}_n \dot{{\hat{\theta }}}_n. \end{aligned}$$

Based on Assumption 1, there exists a constant $d_n > 0$ such that $\vert {\mathcal {D}}^{1- \alpha _n} \Delta _n(t)\vert \le d_n$. So, by using Lemma 2, we have

$$\begin{aligned} {\tilde{z}}_n {\mathcal {D}}^{1- \alpha _n} \Delta _n(t) \le d_n \vert {\tilde{z}}_n\vert \le d_n {\tilde{z}}_n \tanh \left( \frac{d_n {\tilde{z}}_n}{\zeta _n}\right) + \zeta _n \rho . \end{aligned}$$

Therefore, ${\dot{V}}_n$ becomes

$$\begin{aligned} \begin{aligned} {\dot{V}}_n \le&\, {\dot{V}}_{n-1} + {\tilde{z}}_n {\mathcal {H}}_n(Z_n) + {\tilde{z}}_n {\mathcal {D}}^{1 - \alpha _n} u - {\tilde{z}}_n \dot{{\tilde{\chi }}}_n \\&+ \frac{1}{\eta _n} {\tilde{\theta }}_n \dot{{\hat{\theta }}}_n+ \zeta _n \rho - \lambda _n z_n {\tilde{z}}_n - {\bar{\lambda }}_n {\tilde{z}}_n {\tilde{\chi }}_n^r, \end{aligned} \end{aligned}$$

(22)

where ${\mathcal {H}}_n(Z_n) = {\mathcal {D}}^{1- \alpha _n} f_n(x) + d_n \tanh \left( \frac{d_n {\tilde{z}}_n}{\zeta _n}\right) + {\dot{\chi }}_{n-1} + \lambda _n z_n + {\bar{\lambda }}_n {\tilde{\chi }}_n^r$ with $Z_n = (x^\textrm{T}, {\hat{\theta }}_1, \ldots , {\hat{\theta }}_n, {\tilde{\chi }}_n, y_d, {\dot{y}}_d)^\textrm{T} \in {\mathbb {R}}^{2n+3}$. From Lemma 6, function ${\mathcal {H}}_n(Z_n)$ is approximated as

$$\begin{aligned} {\mathcal {H}}_n(Z_n) = W_n^{*T} \Phi _n(Z_n) + \epsilon _n(Z_n), \end{aligned}$$

where $W_n^*$ is the ideal weight vector and $\epsilon _n(Z_n)$ is the approximation error satisfying $\vert \epsilon _n(Z_n)\vert < {\tilde{\epsilon }}_n$. According to Lemma 7 and using the same manner as in (9), we have

$$\begin{aligned} {\tilde{z}}_n {\mathcal {H}}_n(Z_n) \le&\, \frac{1}{2a_n^2} {\tilde{z}}_n^2 \theta _n \Phi _n^\textrm{T}(X_n) \Phi _n(X_n) \\&+ \frac{1}{2} a_n^2 + \frac{1}{2} {\tilde{z}}_n^2 + \frac{1}{2} {\tilde{\epsilon }}_n^2, \end{aligned}$$

where $a_n > 0$ is a designed parameter and $X_n = (x^\textrm{T}, {\hat{\theta }}_1, \ldots , {\hat{\theta }}_n, {\tilde{\chi }}_n, y_d, {\dot{y}}_d)^\textrm{T} \in {\mathbb {R}}^{2n+3}$. For simplicity, $\Phi _n(X_n)$ will be abbreviated as $\Phi _n$ in the following. Substituting the above inequality into (22), we get

$$\begin{aligned} {\dot{V}}_n \le&\, {\dot{V}}_{n-1} + \frac{1}{2a_n^2} {\tilde{z}}_n^2 \theta _n \Phi _n^\textrm{T} \Phi _n + \frac{1}{2} {\tilde{z}}_n^2 + {\tilde{z}}_n {\mathcal {D}}^{1 - \alpha _n} u \\&- {\tilde{z}}_n \dot{{\tilde{\chi }}}_n+ \frac{1}{\eta _n} {\tilde{\theta }}_n \dot{{\hat{\theta }}}_n - \lambda _n z_n {\tilde{z}}_n - {\bar{\lambda }}_n {\tilde{z}}_n {\tilde{\chi }}_n^r \\&+ \frac{1}{2} a_n^2 + \frac{1}{2} {\tilde{\epsilon }}_n^2 + \zeta _n \rho . \end{aligned}$$

The intermediate signal $h_n(t)$ and the adaptive laws of ${\hat{\theta }}_n$ are designed as

$$\begin{aligned} h_n(t) =&\, -(\gamma _n +\frac{1}{2}) \frac{{\tilde{z}}_n}{\kappa _n} - \frac{ {\tilde{z}}_n {\hat{\theta }}_n}{2a_n^2 \kappa _n} \Phi _n^\textrm{T} \Phi _n - \frac{{\bar{\gamma }}_n}{\kappa _n} \vartheta _n({\tilde{z}}_n), \end{aligned}$$

(23)

$$\begin{aligned} \dot{{\hat{\theta }}}_n =&\, \frac{\eta _n}{2a_n^2} {\tilde{z}}_n^2 \Phi _n^\textrm{T} \Phi _n - \mu _n {\hat{\theta }}_n - \xi _n {\hat{\theta }}_n^r, {\hat{\theta }}_n(0) \ge 0, \end{aligned}$$

(24)

where $\gamma _n, {\bar{\gamma }}_n, \mu _n, \xi _n > 0$ are designed parameters and

$$\begin{aligned} \vartheta _n({\tilde{z}}_n) = {\left\{ \begin{array}{ll} {\tilde{z}}_n^r, &{} \vert {\tilde{z}}_n\vert \ge \tau _n, \\ A_n {\tilde{z}}_n - B_n {\tilde{z}}_n^3, &{} \vert {\tilde{z}}_n\vert < \tau _n, \end{array}\right. } \end{aligned}$$

with $A_n = \frac{1}{2} (3 - r) \tau _n^{r-1}, B_n = \frac{1}{2} \tau _n^{r-3} (1 -r)$, and a small constant $\tau _n>0$. Taking (19), (20), (23) and (24) into ${\dot{V}}_n$, one has

$$\begin{aligned} {\dot{V}}_n \le&\, {\dot{V}}_{n-1} + \frac{1}{2a_n^2} {\tilde{z}}_n^2 \theta _n \Phi _n^\textrm{T} \Phi _n + \frac{1}{2} {\tilde{z}}_n^2 + \kappa _n {\tilde{z}}_n h_n(t) \nonumber \\&+ \lambda _n {\tilde{z}}_n {\tilde{\chi }}_n + \frac{1}{\eta _n} {\tilde{\theta }}_n \dot{{\hat{\theta }}}_n - \lambda _n z_n {\tilde{z}}_n \nonumber \\&+ \frac{1}{2} a_n^2 + \frac{1}{2} {\tilde{\epsilon }}_n^2 + \zeta _n \rho \nonumber \\ \le&\, - \sum _{s=1}^n \bigg ((\gamma _s + \lambda _s) {\tilde{z}}_s^2 + {\bar{\gamma }}_s {\tilde{z}}_s\vartheta _s({\tilde{z}}_s) + \frac{\mu _s}{\eta _s} {\tilde{\theta }}_s {\hat{\theta }}_s \nonumber \\&+ \frac{\xi _s}{\eta _s} {\tilde{\theta }}_s {\hat{\theta }}_s^r\bigg ) + \Theta _n, \end{aligned}$$

(25)

where $\Theta _n = \Theta _{n-1} + \frac{1}{2} a_n^2 + \frac{1}{2} {\tilde{\epsilon }}_n^2 + \zeta _n \rho = \sum _{s = 1}^n (\frac{1}{2} a_s^2 + \frac{1}{2} {\tilde{\epsilon }}_s^2 + \zeta _s \rho )$.

Before giving the main result of this paper, we give two useful lemmas.

Lemma 8

For the filters (5), (13) and (19), if the signal $h_i(t) \, (1 \le i \le n)$ is bounded, then signals ${\tilde{h}}_i(t) = \chi _i(t) - h_i(t) \, (1 \le i \le n-1)$, ${\tilde{h}}_n(t) = u(t) - h_n(t)$, $\chi _i(t) \, (1 \le i \le n-1)$ and u(t) are bounded for $t \ge 0$.

Proof

For the sake of simplicity, u(t) is denoted by $\chi _n(t)$, so ${\tilde{h}}_n(t) = \chi _n(t) - h_n(t)$. Since $h_i(t) \, (1 \le i \le n)$ is bounded, there exists a positive constant ${\bar{h}}_i$ such that $\vert h_i(t)\vert \le {\bar{h}}_i$. Considering the Lyapunov function $V_{\chi _i}(t) = \frac{1}{2} \chi _i^2(t)$, we have

$$\begin{aligned} {\mathcal {D}}^{1- \alpha _i} V_{\chi _i}(t)&\le \chi _i(t) {\mathcal {D}}^{1- \alpha _i} \chi _i(t) \\&= -\kappa _i \chi _i^2(t) + \kappa _i \chi _i(t) h_i(t). \end{aligned}$$

It follows from $\vert h_i(t)\vert \le {\bar{h}}_i$ that $\chi _i(t) h_i(t) \le \frac{1}{2} \chi _i^2(t) + \frac{1}{2} h_i^2(t) \le \frac{1}{2} \chi _i^2(t) + \frac{1}{2} {\bar{h}}_i^2$, which means

$$\begin{aligned} {\mathcal {D}}^{1- \alpha _i} V_{\chi _i}(t){} & {} \le -\frac{1}{2} \kappa _i \chi _i^2(t) + \frac{1}{2} \kappa _i {\bar{h}}_i^2 \\ {}{} & {} = -\kappa _i V_{\chi _i}(t) + \frac{1}{2} \kappa _i {\bar{h}}_i^2. \end{aligned}$$

Therefore, from $0 \le E_{1- \alpha _i} (-\kappa _i t^{1-\alpha _i}) < 1$ for $t \ge 0$ and Lemma 5 in [42], we have

$$\begin{aligned} V_{\chi _i}(t)&\le \left( V_{\chi _i}(0) - \frac{1}{2}{\bar{h}}_i^2\right) E_{1- \alpha _i} (-\kappa _i t^{1-\alpha _i}) + \frac{1}{2}{\bar{h}}_i^2 \\&\le V_{\chi _i}(0) + \frac{1}{2}{\bar{h}}_i^2, \end{aligned}$$

from which we can obtain that $\chi _i^2(t) \le \chi _i^2(0) + {\bar{h}}_i^2$ for $t \ge 0$. That is, $\chi _i(t) \, (1 \le i \le n)$ is bounded. From the boundedness of $\chi _i(t)$ and $h_i(t)$, it is obvious that ${\tilde{h}}_i(t)$ is bounded. This completes the proof. $\square$

Lemma 9

For the compensated filters (6), (14) and (20), if signals $\chi _i \, (1 \le i \le n-1)$ and u(t) are bounded, then signal ${\tilde{\chi }} = ({\tilde{\chi }}_1, \ldots , {\tilde{\chi }}_n)^\textrm{T}$ converges to a small region $\Omega _1 = \left\{ {\tilde{\chi }} \vert \Vert {\tilde{\chi }}\Vert ^2 \le 2 \omega _1 \right\}$ in finite-time $T_1$, where $T_1 = T_1({\bar{q}}_1, {\bar{q}}_2, {\bar{q}}_3)$ is determined by (1), ${\bar{q}}_1, {\bar{q}}_2, {\bar{q}}_3 >0$ will be given in the following, and

$$\begin{aligned} \omega _1 = \min \left\{ \frac{{\bar{q}}_3}{(1 - \varrho ){\bar{q}}_1}, \Big (\frac{{\bar{q}}_3}{(1 - \varrho ) {\bar{q}}_2}\Big )^{\frac{2}{1 + r}}\right\} . \end{aligned}$$

Proof

For convenience, u(t) is denoted by $\chi _n(t)$. Defining the Lyapunov function $V_{{\tilde{\chi }}}(t) = \frac{1}{2} \sum _{s = 1}^n {\tilde{\chi }}_s^2$, and noting (6), (14) and (20), we have

$$\begin{aligned} {\dot{V}}_{{\tilde{\chi }}}(t)= & {} \sum _{s = 1}^n{\tilde{\chi }}_s\dot{{\tilde{\chi }}}_s = - \sum _{s = 1}^n \big (\lambda _s{\tilde{\chi }}_s^2 + {\bar{\lambda }}_s {\tilde{\chi }}_s^{1+ r}\big ) \\{} & {} - \sum _{s = 1}^n\kappa _s {\tilde{\chi }}_s \chi _s. \end{aligned}$$

By using the Young’s inequality, we obtain

$$\begin{aligned} -{\tilde{\chi }}_s \chi _s \le \vert {\tilde{\chi }}_s \chi _s\vert \le \frac{\lambda _s}{2 \kappa _s} {\tilde{\chi }}_s^2 + \frac{\kappa _s}{2 \lambda _s} \chi _s^2. \end{aligned}$$

Because $\chi _s$ is bounded, there is a constant $\bar{{\bar{\Theta }}}_s > 0$ such that $\frac{\kappa _s^2}{2\lambda _s} \chi _s^2 \le \bar{{\bar{\Theta }}}_s$. Thus, $-\kappa _s {\tilde{\chi }}_s \chi _s \le \frac{1}{2} \lambda _s {\tilde{\chi }}_s^2 + \bar{{\bar{\Theta }}}_s$. By using Lemma 4, ${\dot{V}}_{{\tilde{\chi }}}(t)$ can be rewritten as

$$\begin{aligned} \begin{aligned} {\dot{V}}_{{\tilde{\chi }}}(t) \le&\, - \sum _{s = 1}^n \big (\frac{1}{2} \lambda _s{\tilde{\chi }}_s^2 + {\bar{\lambda }}_s {\tilde{\chi }}_s^{1+ r}\big ) + \sum _{s = 1}^n \bar{{\bar{\Theta }}}_s \\ \le&\, - {\bar{q}}_1 V_{{\tilde{\chi }}}(t) - {\bar{q}}_2 V_{{\tilde{\chi }}}^{\frac{1 + r}{2}}(t) + {\bar{q}}_3, \end{aligned} \end{aligned}$$

(26)

where ${\bar{q}}_1 = \min \{\lambda _s, s = 1, \ldots , n\}$, ${\bar{q}}_2 = 2^{\frac{1 + r}{2}} \min \{{\bar{\lambda }}_s, s = 1, \ldots , n\}$ and ${\bar{q}}_3 = \sum _{s = 1}^n \bar{{\bar{\Theta }}}_s$. It then follows from Lemma 5 that ${\tilde{\chi }} = ({\tilde{\chi }}_1, \ldots , {\tilde{\chi }}_n)^\textrm{T}$ converges to $\Omega _1$ in finite-time $T_1$. The proof is completed. $\square$

Eventually, the practical finite-time adaptive neural networks control schemes and the parameter adaptive laws are designed, from which we establish the main result of this paper.

Theorem 1

For incommensurate fractional-order nonlinear system (2) under Assumptions 1–3, the designed virtual control signals determined by (5), (13), final control signal determined by (19), adaptive laws (11), (18), (24), intermediate signals (10), (17), (23), and compensated signals determined by (6), (14), (20) can ensure that:

• (i) all signals in the closed-loop system are bounded;

• (ii) the signal ${\tilde{z}} = ({\tilde{z}}_1, \ldots , {\tilde{z}}_n)^\textrm{T}$ converges to a small region $\Omega _2 = \left\{ {\tilde{z}} \vert \Vert {\tilde{z}}\Vert ^2 \le 2 \omega _2 \right\}$ in finite-time $T_2$, and the signal $z = (z_1, \ldots , z_n)^\textrm{T}$ converges to $\Omega _3 = \left\{ z \vert \Vert z\Vert ^2 \le 4 \omega _3 \right\}$ in finite-time T,

where $T = \max \{T_1, T_2\}$, $T_1 = T_1({\bar{q}}_1, {\bar{q}}_2, {\bar{q}}_3)$ and $T_2 = T_2(q_1, q_2, q_3)$ are determined by (1), $q_1, q_2, q_3$ will be specified later, ${\bar{q}}_1, {\bar{q}}_2, {\bar{q}}_3 >0$ are given in Lemma 9, and $\omega _3 = \omega _1 + \omega _2$,

$$\begin{aligned} \omega _2 = \min \left\{ \frac{q_3}{(1 - \varrho ) q_1}, \Big (\frac{q_3}{(1 - \varrho ) q_2}\Big )^{\frac{2}{1 + r}}\right\} . \end{aligned}$$

Proof

For convenience, let ${\mathbb {S}} = \{s\vert s=1, \ldots , n\}$, ${\mathbb {S}}_1 = \{s \in {\mathbb {S}}\vert \vert {\tilde{z}}_s\vert < \tau _s\}$ and ${\mathbb {S}}_2 = \{s \in {\mathbb {S}}\vert \vert {\tilde{z}}_s\vert \ge \tau _s\}$. By using Lemma 3 and noting that $r = \frac{r_1}{r_2} \in (0,1)$ with positive odd numbers $r_1,r_2$, for $s \in {\mathbb {S}}$, we have

$$\begin{aligned} -{\tilde{\theta }}_s {\hat{\theta }}_s^r = {\tilde{\theta }}_s (- {\hat{\theta }}_s)^r \le -b_1 {\tilde{\theta }}_s^{1 +r} + b_2 \theta _s^{1+r}, \end{aligned}$$

(27)

where $b_1, b_2 > 0$ are given in Lemma 3. Consider the Lyapunov function

$$\begin{aligned} V_n(\Upsilon (t)) = \sum _{s \in {\mathbb {S}}} \left( \frac{1}{2} {\tilde{z}}_s^2 + \frac{1}{2 \eta _s} {\tilde{\theta }}_s^2\right) , \end{aligned}$$

(28)

where $\Upsilon (t) = ({\tilde{z}}_1, \ldots , {\tilde{z}}_n, {\tilde{\theta }}_1, \ldots , {\tilde{\theta }}_n)^\textrm{T}$. Then, from (25), (27) and the inequality $-{\tilde{\theta }}_s {\hat{\theta }}_s = -{\tilde{\theta }}_s^2 - {\tilde{\theta }}_s \theta _s \le - \frac{1}{2} {\tilde{\theta }}_s^2 + \frac{1}{2} \theta _s^2$, we have

$$\begin{aligned} \begin{aligned} {\dot{V}}_n \le&\, -\sum _{s \in {\mathbb {S}}} \bigg ((\gamma _s + \lambda _s) {\tilde{z}}_s^2 + {\bar{\gamma }}_s {\tilde{z}}_s\vartheta _s({\tilde{z}}_s) + \frac{\mu _s}{2\eta _s} {\tilde{\theta }}_s^2 \\&+ \frac{b_1 \xi _s}{\eta _s} {\tilde{\theta }}_s^{1+r}\bigg ) + {\bar{\Theta }}_1, \end{aligned} \end{aligned}$$

(29)

where ${\bar{\Theta }}_1 = \Theta _n + \sum _{s \in {\mathbb {S}}} \left( \frac{\mu _s}{2\eta _s} \theta _s^2 + \frac{b_2 \xi _s}{\eta _s} \theta _s^{1+r}\right)$. According to the form of $\vartheta _s({\tilde{z}}_s)$, we can obtain that

$$\begin{aligned} \begin{aligned}&-\sum _{s \in {\mathbb {S}}} {\bar{\gamma }}_s {\tilde{z}}_s\vartheta _s({\tilde{z}}_s) \\&\,\quad = -\sum _{s \in {\mathbb {S}}_1} {\bar{\gamma }}_s A_s {\tilde{z}}_s^2 - \sum _{s \in {\mathbb {S}}_2} {\bar{\gamma }}_s {\tilde{z}}_s^{1 + r} + \sum _{s \in {\mathbb {S}}_1} {\bar{\gamma }}_s B_s {\tilde{z}}_s^4 \\&\,\quad \le -\sum _{s \in {\mathbb {S}}_1} {\bar{\gamma }}_s A_s {\tilde{z}}_s^2 - \sum _{s \in {\mathbb {S}}} {\bar{\gamma }}_s {\tilde{z}}_s^{1 + r} + {\bar{\Theta }}_2, \end{aligned} \end{aligned}$$

where ${\bar{\Theta }}_2 = \sum _{s \in {\mathbb {S}}_1} {\bar{\gamma }}_s (B_s \tau _s^4 + \tau _s^{1+r})$. Therefore, by using Lemma 4, (29) becomes

$$\begin{aligned} {\dot{V}}_n \le{} & {} \, -\sum _{s \in {\mathbb {S}}_1} ({\bar{\gamma }}_s A_s + \gamma _s + \lambda _s) {\tilde{z}}_s^2 - \sum _{s \in {\mathbb {S}}_2} (\gamma _s + \lambda _s) {\tilde{z}}_s^2 \nonumber \\{} & {} - \sum _{s \in {\mathbb {S}}} \left( {\bar{\gamma }}_s {\tilde{z}}_s^{1+ r} + \frac{\mu _s}{2 \eta _s} {\tilde{\theta }}_s^2 + \frac{b_1 \xi _s}{\eta _s} {\tilde{\theta }}_s^{1+r} \right) + {\bar{\Theta }}_1 + {\bar{\Theta }}_2 \nonumber \\ \le{} & {} \, - q_1 V_n - q_2 \sum _{s \in {\mathbb {S}}} \bigg (\Big (\frac{1}{2} {\tilde{z}}_s^2\Big )^{\frac{1+ r}{2}} + \Big (\frac{1}{2 \eta _s} {\tilde{\theta }}_s^2 \Big )^{\frac{1+ r}{2}}\bigg ) + q_3 \nonumber \\ \le{} & {} \, - q_1 V_n -q_2\bigg (\sum _{s \in {\mathbb {S}}} \Big (\frac{1}{2} {\tilde{z}}_s^2 + \frac{1}{2 \eta _s} {\tilde{\theta }}_s^2 \Big )\bigg )^{\frac{1+ r}{2}} + q_3 \nonumber \\ \le{} & {} \, - q_1 V_n -q_2V_n^{\frac{1+ r}{2}} + q_3, \end{aligned}$$

(30)

where $q_1 = \min \{2 \min _{s \in {\mathbb {S}}_1} \{{\bar{\gamma }}_s A_s + \gamma _s + \lambda _s\}, 2 \min _{s \in {\mathbb {S}}_2} \{\gamma _s + \lambda _s\}, \min _{s \in {\mathbb {S}}} \{\mu _s\}\}$, $q_2 = 2^{\frac{1+ r}{2}} \min _{s \in {\mathbb {S}}} \{{\bar{\gamma }}_s, b_1\xi _s \eta _s^{\frac{1+ r}{2}}\}$ and $q_3 = {\bar{\Theta }}_1 + {\bar{\Theta }}_2$.

• (i) Since $V_n$ is positive definite, from (28) and (30), we have ${\dot{V}}_n \le - q_1 V_n + q_3$, which means that

  $$\begin{aligned} V_n = V_n(\Upsilon (t))&\le (V_n(\Upsilon (0)) - \frac{q_3}{q_1}) e^{-q_1 t} + \frac{q_3}{q_1} \\&\le V_n(\Upsilon (0)) + \frac{q_3}{q_1}. \end{aligned}$$

  Thus, ${\tilde{z}}_i, {\tilde{\theta }}_i \, (i=1, \ldots , n)$ are bounded. Since $\theta _i$ is a constant, ${\hat{\theta }}_i = {\tilde{\theta }}_i + \theta _i$ is bounded, which means $h_i$ in (10), (17) and (23) is bounded. Based on the boundedness of $h_i$ and Lemma 8, $\chi _i \, (i = 1, \ldots , n-1)$ and u are bounded. Further, the boundedness of ${\tilde{\chi }}_i$ can be obtained by using Lemma 9, from which we get that $z_i = {\tilde{z}}_i + {\tilde{\chi }}_i$ is bounded. According to coordinate transform (3) and Assumption 2, we obtain the boundedness of $x_i$. Therefore, all signals in the closed-loop system are bounded.

• (ii) It follows from Lemma 5 and (30) that there exists a finite-time $T_2 = T_2(q_1, q_2, q_3) > 0$ such that ${\tilde{z}} = ({\tilde{z}}_1, \ldots , {\tilde{z}}_n)^\textrm{T}$ enters $\Omega _2$ for $t \ge T_2$. Noting that $z_i = {\tilde{z}}_i + {\tilde{\chi }}_i$ and using Lemma 9, $\Vert z\Vert ^2 \le 2\Vert {\tilde{z}}\Vert ^2 + 2\Vert {\tilde{\chi }}\Vert ^2 \le 4(\omega _1 + \omega _2) = 4 \omega _3$, i.e., $z \in \Omega _3$ for $t \ge T = \max \{T_1, T_2\}$, where $T_1$ is given in Lemma 9.

This completes the proof. $\square$

Remark 5

In some previous literature concerning finite-time control of FOS, the formula ${\mathcal {D}}^{\alpha } g^{\iota }(t) = \frac{\varGamma (1+ \iota )}{\varGamma (1+ \iota +\alpha )} g^{\iota - \alpha }(t) {\mathcal {D}}^{\alpha } g(t)$ $(\iota \in {\mathbb {R}}, \alpha \in (0,1))$ plays a key role in stability analysis. Unfortunately, as pointed out in [4], this formula may not feasible (a counterexample was given in [4]), with the result that the design of a (practical) finite-time adaptive control scheme is quite difficult, because the condition of the fractional-order version (i.e., ${\mathcal {D}}^{\alpha } V \le -q_1 V(x) - q_2 V^{\iota } + q_3$) does not imply the practical finite-time stability of FOS. However, in this paper, with the help of Lemma 1, the first-order derivative of the Lyapunov function is calculated instead of its fractional-order derivative, and thus, the practical finite-time stability of FOS is obtained based on the integer-order condition ${\dot{V}} \le -q_1 V(x) - q_2 V^{\iota } + q_3$ (see Lemma 5). According to Lemma 5, a novel and practical finite-time adaptive control scheme is successfully designed for FOS and the stability analysis is also derived.

Remark 6

In order to highlight the proposed novel practical finite-time adaptive control scheme, only a classical incommensurate FOS is considered in this paper. In fact, it is easy to extend the proposed control scheme to cases with more general assumptions, such as systems with unmodeled dynamics, input quantization, unknown control direction, or actuator faults. These systems were considered in [8, 19, 20, 22, 37], while the practical finite-time control for these systems remains open.

Remark 7

It can be found that the backstepping method has been successfully applied to the control problem of FOSs in [9, 18, 19, 21, 22, 37], while these control schemes can only achieve the goal of non-finite time control. In this paper, the practical finite-time control scheme is proposed for incommensurate FOS, which can not only be used for practical finite-time control of commensurate FOS but also be degenerated to the non-finite time control scheme for incommensurate FOS. In addition, the reference signal $y_d$ in this paper is a time-varying function. If $y_d = 0$, then the tracking problem in this paper becomes a stabilization problem.

The following non-finite time control scheme can be obtained from Theorem 1 for an incommensurate fractional-order nonlinear system.

Corollary 1

Consider the incommensurate fractional-order nonlinear system (2) under Assumptions 1–3, the designed virtual control signals determined by (5), (13), final control signal determined by (19), adaptive laws, intermediate signals and compensated signals determined by ($i = 1, \ldots , n$)

$$\begin{aligned}&\dot{{\hat{\theta }}}_i = \frac{\eta _i}{2a_i^2} {\tilde{z}}_i^2 \Phi _i^\textrm{T} \Phi _i - \mu _i {\hat{\theta }}_i, \end{aligned}$$

(31)

$$\begin{aligned}&h_i(t) = -\left( \gamma _i +\frac{1}{2}\right) \frac{{\tilde{z}}_i}{\kappa _i} - \frac{1}{2a_i^2 \kappa _i} {\tilde{z}}_i {\hat{\theta }}_i \Phi _i^\textrm{T} \Phi _i, \end{aligned}$$

(32)

$$\begin{aligned}&\dot{{\tilde{\chi }}}_i(t) = - \lambda _i {\tilde{\chi }}_i(t) - \kappa _i \chi _i(t), \quad {\tilde{\chi }}_i(0) = z_i(0), \end{aligned}$$

(33)

$$\begin{aligned}&\chi _n (t) = u(t), \end{aligned}$$

(34)

can ensure that: (i) all signals in the closed-loop system are bounded; (ii) the signals ${\tilde{z}} = ({\tilde{z}}_1, \ldots , {\tilde{z}}_n)^\textrm{T}$ and $z = (z_1, \ldots , z_n)^\textrm{T}$ converge to a small regions of origin.

Proof

Let parameters ${\bar{\gamma }}_i = {\bar{\lambda }}_i = \xi _i = 0$, then from (6), (10), (11), (14), (17), (18), (20), (23) and (24), we can obtain the forms of adaptive laws, intermediate signals and compensated signals shown in (31)–(34). In addition, consider the same Lyapunov functions $V_n (\Upsilon (t))$ as in Theorem 1 and $V_{{\tilde{\chi }}}(t)$ as in Lemma 9, the inequalities (26) and 30 will degenerate into

$$\begin{aligned}{} & {} {\dot{V}}_{{\tilde{\chi }}}(t) \le - {\bar{q}}_1 V_{{\tilde{\chi }}}(t) + {\bar{q}}_3, \end{aligned}$$

(35)

$$\begin{aligned}{} & {} {\dot{V}}_n(\Upsilon (t)) \le - q_1 V_n(\Upsilon (t)) + q_3, \end{aligned}$$

(36)

where ${\bar{q}}_1, {\bar{q}}_3, q_1$ and $q_3$ are the same as the ones in Lemma 9 and Theorem 1. Then, we can obtain the boundedness of all signals in the closed-loop system by using a similar argument as in the proof of Theorem 1. It follows from (35) and (36) that

$$\begin{aligned}&V_{{\tilde{\chi }}}(t) \le \Big (V_{{\tilde{\chi }}}(0) - \frac{{\bar{q}}_3}{{\bar{q}}_1} \Big ) e^{-{\bar{q}}_1 t} + \frac{{\bar{q}}_3}{{\bar{q}}_1}, \\&V_n(\Upsilon (t)) (t) \le \Big (V_n(\Upsilon (0)) - \frac{q_3}{q_1} \Big ) e^{-q_1 t} + \frac{q_3}{q_1}, \end{aligned}$$

from which we can only get that signals ${\tilde{z}}$ and z converge to small regions of origin, and cannot derive the results of finite-time convergence. $\square$

Remark 8

The practical finite-time control scheme proposed in Theorem 1 for FOS (2) contains an adjustable parameter r, which provides a greater degree of freedom in the choice of parameters. Therefore, compared with the non-finite time control scheme in Corollary 1, Theorem 1 gives us a possible option to achieve better control performance.

Remark 9

Compared with the literature [9, 18, 20,21,22, 36, 37], the control scheme in Corollary 1 is also novel and can be used to control both incommensurate and commensurate FOSs in [9, 18, 20,21,22, 36, 37]. The merits of the control scheme in Corollary 1 can be summarized as follows: (i) in contrast to the adaptive laws in [9, 20,21,22], the derivative order of adaptive laws in Corollary 1 is 1, which is easier to implement in practice; (ii) it can be seen from the numerical simulations in some previous studies [18, 36, 37] that the fluctuation range of control input around the initial time is very large, while in this paper, the control input is generated by the designed filter (19) and such fluctuation of control input can be reduced, which is also illustrated by the numerical simulations in the next section (see Remark 11 for details).

In addition, the form of intermediate signal $h_1(t)$ is similar to the virtual controllers in many literature, but the virtual control signal $\chi _1$ in this paper is generated by fractional filter (5), which is quite different from the form of the virtual control signal in literature. Besides, if system (2) degenerates to integer-order one, then fractional filter (5) becomes $\chi _1(t) = \frac{\kappa _1}{1 + \kappa _1} h_1(t)$, and thus, there is only a constant coefficient between $\chi _1(t)$ and control virtual control signal in the literature, which indicates that the virtual control signal is more comprehensive and cover the ones in literature as special cases.

Remark 10

The form of intermediate signals $h_i(t)\, (i =1, \ldots , n)$ are similar to the virtual controllers in many literature, but the virtual control signal $\chi _i$ in this paper is generated by fractional filters (5), (13) and (19), which are quite different from the form of the virtual control signal in literature. It is worth pointing out that if $\alpha _i = 1$, then the practical finite-time and non-finite time control schemes for FOSs presented in Theorem 1 and Corollary 1 will degenerate to the ones of integer-order nonlinear systems. More precisely, the filters (5), (13) and (19) become ${\mathcal {D}}^0 \chi _i(t) = \chi _i(t) = - \kappa _i (\chi _i(t) - h_i(t))$, i.e., $\chi _i(t) = \frac{\kappa _i}{1 + \kappa _i} h_i(t)$. Then substituting $h_i(t)$ into this equation, we have

$$\begin{aligned} \chi _i(t) = -\frac{1}{1 + \kappa _i}&\bigg (\left( \gamma _i +\frac{1}{2}\right) {\tilde{z}}_i \\&+ \frac{1}{2a_i^2} {\tilde{z}}_i {\hat{\theta }}_i \Phi _i^\textrm{T} \Phi _i + {\bar{\gamma }}_i \vartheta _i({\tilde{z}}_i)\bigg ). \end{aligned}$$

Further, letting $\kappa _i = 0$ in the above equation, one can obtain

$$\begin{aligned} \chi _i(t) = -\left( \gamma _i +\frac{1}{2}\right) {\tilde{z}}_i - \frac{1}{2a_i^2} {\tilde{z}}_i {\hat{\theta }}_i \Phi _i^\textrm{T} \Phi _i - {\bar{\gamma }}_i \vartheta _i({\tilde{z}}_i), \end{aligned}$$

which coincides with the finite-time control scheme in [32, 33]. This indicates that the control signals proposed in this paper are more comprehensive and cover the ones in literature as special cases. If we further let ${\bar{\gamma }}_i = {\bar{\lambda }}_i = \xi _i = 0$, then the non-finite time control scheme for an integer-order system can be obtained, which was extensively discussed in the existing literature, such as [7, 13, 15]. Therefore, the proposed control scheme in this paper is more general than the above-mentioned studies.

Fig. 1

Block diagram of the closed-loop system

4 Numerical simulations

In this section, two numerical examples are presented to illustrate the effectiveness of the control scheme proposed in the previous section. In order to enhance the readability of this article, a block diagram of the closed-loop system is presented in Fig. 1.

Example 1

Consider the following incommensurate fractional-order nonlinear system

$$\begin{aligned} {\left\{ \begin{array}{ll} {\mathcal {D}}^{\alpha _1}x_1 = 2x_2 +1 -\cos (x_1x_3) + x_3 + \Delta _1(t), \\ {\mathcal {D}}^{\alpha _2}x_2 = x_3 + x_2^2 x_3 e^{-x_2^2} + \Delta _2(t), \\ {\mathcal {D}}^{\alpha _3}x_3 = u + x_1 x_2 e^{-x_3^2} + x_3 \sin (x_1 x_2) + \Delta _3(t), \\ y = x_1, \end{array}\right. } \end{aligned}$$

(37)

where $\alpha _1 = 0.95$, $\alpha _2 = 0.85$, $\alpha _3 = 0.92$, $\Delta _1(t) = 0.3\cos (0.7t), \Delta _2(t) = - 0.1\sin (0.6t), \Delta _3(t) = 0.2\cos (0.8t)$. The reference signal is $y_d = \sin (1.4t) + 0.8 \cos (0.7t)$.

It is obvious that $y_d, \Delta _1, \Delta _2, \Delta _3$ and their derivatives are continuous and bounded. Thus, Assumptions 1 and 2 are satisfied. The designed parameters are chosen as: $r = \frac{5}{9}$, $\gamma _1 = 11.8, \gamma _2 = 6.4, \gamma _3 = 3.2$, ${\bar{\gamma }}_1 = 10.76, {\bar{\gamma }}_2 = 4.55, {\bar{\gamma }}_3 = 2.86$, $\lambda _1 = 4.62, \lambda _2 = 1.96, \lambda _3 = 2.1$, ${\bar{\lambda }}_1 = 2.9, {\bar{\lambda }}_2 = 1.4, {\bar{\lambda }}_3 = 1.5$, $\kappa _1 = 0.9, \kappa _2 = 1.7, \kappa _3 = 1.2$, $\eta _1 = 0.2, \eta _2 = 0.1, \eta _3 = 0.5$, $\mu _1 = 0.003, \mu _2 = 0.005, \mu _3 = 0.01$, $\xi _1 = 0.001, \xi _2 = 0.004, \xi _3 = 0.02$, $a_1 = a_2 = 1, a_3 = 1.2$, $\tau _1 = \tau _2 = \tau _3 = 0.02$, $\varpi _s = (-4.5, -3, -1.5, 0, 1.5, 3, 4.5)^\textrm{T}$ and $\delta _s = 0.8$. The initial values of the system (37) and the adaptive laws are $x(0) = (-0.1, -0.1, 0.9)^\textrm{T}$, ${\hat{\theta }}_1(0) = 10^{-6}$, ${\hat{\theta }}_2(0) = 10^{-4}$ and ${\hat{\theta }}_3(0) = 10^{-3}$.

Fig. 2

Trajectories of reference signal $y_d$ and system output y

Fig. 3

Trajectories of tracking error $z_1 = y_d - y$

The simulation results are shown in Figs. 2–7. The system output $y = x_1$ and the tracking error $z_1$ under the proposed practical finite-time control scheme are given in Figs. 2 and 3, respectively, from which we see that y can track the reference signal $y_d$ with a small tracking error. To compare with the non-finite time control scheme (i.e., the results in Corollary 1), we also consider the following two cases for Corollary 1: Case I, all parameters are the same as in case of practical finite-time control, except ${\bar{\gamma }}_i = {\bar{\lambda }}_i = \xi _i = 0$; Case II, $\gamma _1 = 19.6, \gamma _2 = 11, \gamma _3 = 11.2$, $\lambda _1 = 11.62, \lambda _2 = 7.56, \lambda _3 = 7.7$, ${\bar{\gamma }}_i = {\bar{\lambda }}_i = \xi _i = 0$ and other parameters are the same as in case of practical finite-time control. We present the trajectories of y and $z_1$ under the non-finite time control scheme in Figs. 2 and 3, respectively. From Fig. 3, compared with the tracking error under the non-finite time control scheme, the proposed finite-time control scheme has a smaller tracking error and has a slighter oscillation. In addition, from the trajectories of system input u under practical finite-time and non-finite time control schemes shown in Fig. 4, we see that: (i) the tracking performance is very bad in Case I from Fig. 3, although the control force is smaller than that of practical finite-time control; (ii) the fluctuation range of u for practical finite-time control is smaller than that for non-finite time control in Case II, which, together with Fig. 3, means that the proposed practical finite-time control scheme can achieve better tracking performance under a smaller control force compared with the non-finite time control scheme in Case II. The trajectories of state variables $x_2$ and $x_3$ are given in Fig. 5. Figure 6 shows the trajectories of adaptive parameters ${\hat{\theta }}_1, {\hat{\theta }}_2$ and ${\hat{\theta }}_3$. Figure 7 gives the trajectories of virtual control signals $\chi _1, \chi _2$ and compensated signals ${\tilde{\chi }}_1, {\tilde{\chi }}_2, {\tilde{\chi }}_3$. All figures indicate the boundedness of signals in the proposed practical finite-time control scheme.

Fig. 4

Trajectories of system input u

Fig. 5

Trajectories of state variables $x_2$ and $x_3$

Fig. 6

Trajectories of estimated parameter ${\hat{\theta }}_i$

Fig. 7

Trajectories of virtual control signals $\chi _1, \chi _2$ and compensated signals ${\tilde{\chi }}_i$

Example 2

We consider the following fractional-order Chua–Hartley’s system [2]

$$\begin{aligned} {\left\{ \begin{array}{ll} {\mathcal {D}}^{\alpha _1} x_1 = 10x_2 + \frac{10}{7} (x_1 - x_1^3) + \Delta _1(t),\\ {\mathcal {D}}^{\alpha _2} x_2 = x_3 + x_1 - x_2 + \Delta _2(t),\\ {\mathcal {D}}^{\alpha _3} x_3 = u -\frac{100}{7}x_2 + \Delta _3(t),\\ y=x_1, \end{array}\right. } \end{aligned}$$

(38)

where $\alpha _1 = \alpha _2 = 0.98, \alpha _3 = 0.94$, $\Delta _1(t) = 0.5\sin (0.8t), \Delta _2(t) = 0.4\cos (0.8t), \Delta _3(t) = -0.5\sin (0.8t)$. The reference signal is $y_d = \sin (0.8t) + \cos (0.8t)$.

It follows from the forms of the reference signal and external disturbances that Assumptions 1 and 2 are satisfied. The designed parameters in the practical finite-time control scheme are chosen as: $r = 5/11$, $\gamma _1 = 1.16, \gamma _2 = 13.88, \gamma _3 = 12.84$, ${\bar{\gamma }}_1 = 0.66, {\bar{\gamma }}_2 = 13.7, {\bar{\gamma }}_3 = 4.36$, $\lambda _1 = 3.02, \lambda _2 = 5.78, \lambda _3 = 2.42$, ${\bar{\lambda }}_1 = 1.66, {\bar{\lambda }}_2 = 0.68, {\bar{\lambda }}_3 = 0.5$, $\kappa _1 = 0.8, \kappa _2 = 0.6, \kappa _3 = 0.4$, $\eta _1 = 0.8, \eta _2 = 0.13, \eta _3 = 0.12$, $\mu _1 = 0.22, \mu _2 = 0.25, \mu _3 = 0.11$, $\xi _1 = 0.33, \xi _2 = \xi _3 = 0.22$, $a_1 = 24, a_2 = a_3 = 17.6$, $\tau _1 = 0.06, \tau _2 = 0.02, \tau _3 = 0.04$, and the parameters of radial basis function neural networks are the same as ones in Example 1. The initial values are chosen as $x(0) = (-0.1, 0.4, -0.6)$ and ${\hat{\theta }}_1(0) = 2.2, {\hat{\theta }}_2(0) = {\hat{\theta }}_3(0) = 5.6$.

Fig. 8

Trajectories of reference signal $y_d$ and output y

Fig. 9

Trajectory of tracking error $z_1 = y_d - y$

Fig. 10

Trajectories of system input u

Fig. 11

Trajectories of state variables $x_2$ and $x_3$

Fig. 12

Trajectories of estimated parameter ${\hat{\theta }}_i$

The simulation results are presented in Figs. 8, 9,  10, 11, 12 and 13. Figures 8 and 9 give the system output y and the corresponding tracking error $z_1$ ($r = \frac{5}{11}$), from which we see that the proposed practical finite-time control scheme can effectively control system (38) to achieve the tracking objective. As pointed out in Remark 8, r is an important parameter in the proposed practical finite-time control scheme. Thus, we choose the different r ($r = \frac{7}{11}$ and $\frac{9}{11}$) and compare the tracking error $z_1$ and the system input u. From Figs. 9 and 10, we see that a small r implies a small tracking error, while the fluctuation range of the control input u becomes large with the reduction of r, i.e., the large control force is needed. Thus, in practice, it is necessary to make a trade-off between the tracking performance and control force according to the actual engineering requirements. The state variables $x_2, x_3$ and the estimated parameters ${\hat{\theta }}_1, {\hat{\theta }}_2, {\hat{\theta }}_3$ are presented in Figs. 11 and 12, respectively. Figure 13 gives the virtual control signals $\chi _1, \chi _2$ and the compensated signals ${\tilde{\chi }}_1, {\tilde{\chi }}_2, {\tilde{\chi }}_3$. Based on Figs. 8, 9,  10, 11, 12 and 13, we also know that all signals in the closed-loop system (38) are bounded, which indicates the effectiveness of the proposed control scheme.

Remark 11

In some existing literature, such as [18, 36, 37], the input signal u has a large fluctuation around the initial time, which may be unreasonable and unacceptable in practice. Unlike the aforementioned literature, the control input u in this paper is generated by a filter and thus the fluctuation range of u can be reduced, which is also illustrated by two numerical examples. More precisely, the signal $h_3$ in this paper has a similar form with the control input in [18, 36, 37], so we compare the difference between $h_3$ and u, and the results are shown in Fig. 14 for Examples 1 and 2. From Fig. 14, we see that the fluctuation range of u is significantly lower than that of $h_3$ for both practical finite-time and non-finite time control schemes, especially around the initial time. Therefore, the proposed control scheme has an apparent effect on reducing the fluctuation range of the input signal. In addition, the difference between $h_3$ and u is large, and should not be neglected, which indicates that the introduction of compensated signal ${\tilde{\chi }}_i$ is needed and reasonable.

Fig. 13

Trajectories of virtual control signals $\chi _1, \chi _2$ and compensated signals ${\tilde{\chi }}_i$

Fig. 14

Trajectories of system input u and intermediate signal $h_3$ in Examples 1 and 2

5 Conclusion

In this paper, the issue associated with the practical finite-time adaptive neural networks control has been considered for a class of incommensurate FOS with external disturbances. With the help of the established practical finite-time stability criterion and the property of fractional-order calculus, a practical finite-time adaptive control scheme has been designed for an incommensurate FOS, where the control signals are generated by some designed filters. During the design processes of the control scheme, a compensated signal has been introduced to compensate for the difference from the filters of control signals, and thus, the design processes of the control scheme and stability analysis have been simplified. The proposed control scheme provides a new way for the practical finite-time adaptive control of FOSs, while the constraint on the derivative-order $\alpha _i$ (i.e., Assumption 3) brings some limitations on the practical applications of the proposed control scheme, which inspires us to study how to remove this assumption in the future. In addition, to highlight the novelty of the proposed practical finite-time control scheme, only a classical incommensurate fractional-order nonlinear system has been considered in this paper. Thus, we expect to extend the practical finite-time control scheme to a more general and practical system in the future.