Neural network control of fractional-order chaotic systems with unknown control direction

Abstract

In this paper, neural network control of fractional order chaotic systems (FOCSs) with input saturation and unknown sign of the controller gain is addressed by employing the Nussbaum function, where neural networks are utilized to model system uncertainties. To get rid of the limitation that reaching phase should be active before sliding motion in the traditional sliding mode control, a stable sliding surface is constructed. Then, by using the integer order Nussbaum gain control method, a novel controller with neural network sliding mode variable structure is designed. Finally, the practicality of the designed method is confirmed by a simulation experiment.

1. Introduction

The unknown control direction is also an uncertainty. The so-called control direction, that is, the sign of control gain, usually needs to be known in advance in general control design [1], [2], [3], [4], [5], [6]. However, when the control direction is uncertain, the control problem of chaotic systems will become very difficult, especially when the control coefficient changes with time. If the change of gain sign is involved, the originally stable system may become unstable due to the change of control direction. Nussbaum gain control method is usually used when the control direction is unknown. Since Nussbaum proposed the concept of Nussbaum gain control, this method has been widely used in control design. For different systems and control objectives, designers usually combined Nussbaum gain control with other different control technologies to design controllers to achieve control purposes [7], [8].

Nussbaum gain based controller is introduced into the chaos control, which has been studied in relevant literatures, and the limitation added in the control gain is also a factor that should be considered. The Nussbaum gain method is introduced to study the synchronization issue of integer order chaotic systems in [9]. Ref. [10] gave an adaptive backstepping control strategy by using Nussbaum function, which realized the system stable control. In Ref. [11], the Nussbaum function and a recursive technique were utilized to realize tracking control for intrinsically nonlinear systems and time-varying uncertainty, combined with adaptive control. For multivariable systems with unknown actuator nonlinearity and control direction, a fuzzy adaptive controller was provided by using Nussbaum gain control technology in [12]. The adaptive control problem of discrete-time systems with unknown control direction was studied in [13]. The control difficulty caused by the change of control direction is overcome by using discrete Nussbaum gain. Some other interesting results about the control of nonlinear systems with unknown control direction can be referred to [14], [15], [16], [17], [18] and the references therein.

It is worth emphasizing that the aforementioned research works are confined to integer order chaotic systems. In recent decades, due to the characteristics of heredity and memorability, scholars have found that using fractional calculus to describe various systems and processes can obtain more accurate results [19], [20], [21], [22]. Meanwhile, because the control for fractional order nonlinear systems has its particularity and is quite different from that of integer order systems, it is still a novel and difficult topic to introduce Nussbaum gain control to address the synchronization issue of fractional order chaotic systems (FOCSs) when the control direction is uncertain. On the other hand, although there are many literatures about the synchronization control of FOCSs, none of them apply Nussbaum gain control method to solve the synchronization control of FOCSs when the control direction is unknown. Up to now, only few literatures have studied the control issue of FOCS with unknown control direction. Ref. [23] used Nussbaum to investigate the control question of fractional order systems with unknown control directions, and some valuable fruits on how to tackle the unknown control gain were also given. In [24], the quantized control strategy was employed to study the control topic of FOCSs, and the unknown control direction was also considered. In [25], by using a Nussbaum function, the unknown control gain parameter was approximated by fuzzy logic systems. It should be noted that in above literatures, the Nussbaum function is used as the same in the integer-order systems, i.e., the characteristics of fractional order systems are not well utilized. Therefore, how to solve this problem is worthy to being explored further.

Inspired by the above analysis, neural network (NN) control of FOCSs with unknown control direction is considered in this paper. The main contributions are concluded as follows. (1) A useful lemma that can be used to apply Nussbaum functions in FOCSs is given, and consequently, one can check the stability of fractional-order systems with unknown direction easily. (2) A sliding surface that can avoid the chattering phenomena is proposed for FOCSs, and an NN sliding mode controller is implemented. (3) Input saturation is also considered, and it is estimated by a continuous function.

The rest parts of this article are arranged as follows. Some fundamental knowledge about the fractional calculus and the Nussbaum function is given in Section 2. The problem description, the sliding surface design, the controller design and stability analysis are designed in Section 3. A numerical example on fractional L-V system is shown in Section 4, and the conclusion is arranged in Section 5.

2. Preliminaries

2.1. Fractional calculus

There are three kinds of fractional derivatives that usually used in literature, namely Riemann-Liouville, Caputo, and Grünwald Letnikov fractional derivative [26], [27]. Since the initial condition of Caputo fractional derivative is the same as that of traditional calculus and has good physical meaning, this definition will be used in this paper.

Definition 1

[28] Let $f\in \mathbb{R}$ be a smooth function, then its ν-th Caputo fractional derivative can be given by

$${\mathfrak{D}}^{\nu }f(t)=\frac{1}{\mathrm{\Gamma }(m-\nu )}\underset{0}{\overset{t}{\int }}\frac{f^{(m)}(\tau )}{{(t-\tau )}^{\nu }}d\tau ,$$

with $n\in \mathbb{N},m-1<\nu \le m$, and $\mathrm{\Gamma }(\cdot )$ being the Gamma function.

For convenience, it is assumed that the order $\nu \in (0,1)$ is studied in this work. For the controller design, one need the following results.

Lemma 1

[29]Consider the following system:

$${\mathfrak{D}}^{\nu }y=f(y).$$

Then, one has

$$\{\begin{array}{cc}\hfill & \frac{\partial Z(\mathrm{\Xi },t)}{\partial t}=-\mathrm{\Xi }Z(\mathrm{\Xi },t)+f(y),\hfill \\ \hfill & y=\underset{0}{\overset{\infty }{\int }}{\mu }_{\nu }(\mathrm{\Xi })Z(\mathrm{\Xi },t)d\mathrm{\Xi },\hfill \end{array}$$

with Ξ being the frequency, $Z(\mathrm{\Xi },t)$ being the state, and ${\mu }_{\nu }(\mathrm{\Xi })\triangleq \frac{\sin (\nu \pi )}{\pi }{\mathrm{\Xi }}^{-\nu }$ being a frequency weighting function.

Definition 2

[30], [31] Let $N(t)$ be a continuous function. If it satisfies

$$\underset{\ell \to \infty }{\lim }\sup \frac{1}{\ell }\underset{0}{\overset{\ell }{\int }}N(\wp )d\wp =\infty ,\underset{\ell \to -\infty }{\lim }\inf \frac{1}{\ell }\underset{0}{\overset{\ell }{\int }}N(\wp )d\wp =-\infty ,$$

one says that N is a Nussbuam-type function.

Lemma 2

[3], [32]Consider the following Nussbuam-type function:

$$N(\wp )=\vartheta \sqrt{p^2+q^2}{e}^{p|\wp |}\sin (q\wp ),$$

where ℘ is a real variable, $\eta ,c,r,p,\in {\mathbb{R}}^{+}$, and $q=r{c}^{\frac{1}{m}}$. Denote

$$p>\max \{\frac{\overline{b}}{\ell }\ln (\frac{1}{\varpi }(m-1)),q\cot \frac{1}{c}\},$$

with $\varpi <1$, $\frac{\pi }{2r}<\ell <\frac{\pi }{r}$ and $\varpi ,\ell ,\overline{b}>0$. If one can find a Lyapunov function $V(t)$ satisfying that for all $ϵ>0$, it holds

$$V(t)\le \underset{0}{\overset{t}{\int }}(bN(\wp )-1)\dot{\wp }(\tau )d\tau +ϵ,$$

then one has that $V(t)$, ℘ and ${\int }_0^{t}gN(\wp )\dot{\wp }(\tau )d\tau$ are bounded for all $t>0$.

Lemma 3

[32], [33] Suppose that $h:[0,\infty )\to {\mathbb{R}}^{+}$ is a uniform continuous function satisfying ${\int }_0^{+\infty }h(s)ds<\infty$ . Then, for any $z\in \mathbb{R}$ , it holds

$$0\le |z|-\frac{z^2}{\sqrt{z^2+h^2(t)}}<h(t).$$

Remark 1

In the conventional controller design, the discontinuous sign function is often used, which will leads to chattering phenomenon. In this paper, we will use the following property to avoid using the sign function:

$$\mathrm{rel}(z,h)\triangleq \frac{z}{|z|+h},\mathrm{Rel}(z,h)\triangleq \frac{z}{\sqrt{z^2+h^2}},$$

with $h>0$ being a constant or a bounded continuous positive function. When $x>0$, one has

$$\mathrm{sgn}(z)=\frac{z}{|z|}>\underset{\mathrm{Rel}(z,h)}{\underbrace{\frac{z}{\sqrt{z^2+h^2}}}}\ge \underset{\mathrm{rel}(z,h)}{\underbrace{\frac{z}{|z|+h}}}.$$

2.2. Description of an NN

A three-layer NN can be described by

$$y_a(b,w_a)=\sum _{j=1}^{ħ}{\omega }_{aj}{\phi }_{aj}(\sum _{i=1}^c{v}_{ji}{b}_i+{\theta }_j)=w_a^T{\psi }_a(\cdot ),$$

where $a=1,\cdots ,m$ and $w_a=\left[\begin{array}{c}\hfill {\omega }_{a1}\hfill \\ \hfill ⋮\hfill \\ \hfill {\omega }_{ah}\hfill \end{array}\right]$, ${\psi }_a=\left[\begin{array}{c}\hfill {\phi }_{a1}({\sum }_{i=1}^c{v}_{1i}{b}_i+{\theta }_1)\hfill \\ \hfill ⋮\hfill \\ \hfill {\phi }_{ah}({\sum }_{i=1}^c{v}_{hi}{b}_i+{\theta }_h)\hfill \end{array}\right]$, with $c,ħ$ and m being the number of neurons in input layer, hidden layer and output layer, respectively, $y_a$ representing the output. With the utilization of the NN, $v_{ji}$ for $j=1,2,\cdots ħ$ and $i=1,2,\cdots ,c$ can be selected arbitrarily on $[-1,1]$, which is usually chosen as

$$\phi (d)=\frac{e^d-e^{-d}}{e^d+e^{-d}}.$$

Then, one has

$$y={\theta }^T\psi (\cdot ),$$ (1)

where $\theta =\left[\begin{array}{c}\hfill w_1^T\hfill \\ \hfill w_2^T\hfill \\ \hfill ⋮\hfill \\ \hfill w_m^T\hfill \end{array}\right]$, $\psi (\cdot )=\left[\begin{array}{cccc}\hfill {\psi }_1(\cdot )\hfill & \hfill 0\hfill & \hfill \cdots \hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill {\psi }_2(\cdot )\hfill & \hfill \cdots \hfill & \hfill 0\hfill \\ \hfill ⋮\hfill & \hfill ⋮\hfill & \hfill \ddots \hfill & \hfill ⋮\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill \cdots \hfill & \hfill {\psi }_m(\cdot )\hfill \end{array}\right]$.

To approximate an unknown continuous nonlinear function f, the NN (1) can be used as

$$f(d)={\theta }^{⁎T}\psi (d)+\epsilon (d),$$

where $\epsilon (d)$ signifies the optimal approximation error, and ${\theta }^{⁎}$ holds that

$${\theta }^{⁎}=\mathrm{\text{arg}}\underset{\theta }{\min }[\sup |\hat{f}(d)-f(d)|].$$

3. Main results

Consider a class of FOCSs that can be described by

$$\begin{array}{cc}\hfill & {\mathfrak{D}}^{\nu }{x}_i=\sum _{i=1}^n{a}_i{x}_i+f_i(x)+g_i{u}_i,i=1,\cdots ,n\hfill \end{array}$$ (2)

in which $x={[x_1,x_2,\cdots ,x_n]}^T\in {\mathbb{R}}^n$ denotes the system state, $0<\nu <1$ is the fractional order, $f_i:{\mathbb{R}}^n\to \mathbb{R}$ is an unknown nonlinear function, $a_i\in {\mathbb{R}}^n$ is a known system parameter, $g_i\in {\mathbb{R}}^n$ is a positive or negative control gain satisfying $|g_i|>g_{i0}$ with $g_{i0}$ being a positive constant, and $u=[u_1,\cdots ,u_n]\in {\mathbb{R}}^n$ is the control with input saturation, which is given as

$$u_i=\mathrm{sat}({\overline{v}}_i)=\{\begin{array}{c}{u}_{iM},{\overline{v}}_i\ge u_{iM}\hfill \\ {\overline{v}}_i,-u_{im}<v<u_{iM}\hfill \\ u_{im},\overline{v}\le -u_{im}\hfill \end{array}$$ (3)

with ${\overline{v}}_i\in \mathbb{R}$ being the actual input, and $u_{im},u_{iM}\in {\mathbb{R}}^{+}$ being two constants.

It is easy to know from (3) that $u_i$ is not smooth at ${\overline{v}}_i=u_{iM}$ and ${\overline{v}}_i=u_{im}$. Let the following smooth function

$${\rho }_i({\overline{v}}_i)=\{\begin{array}{cc}{u}_{iM}\times \tanh \left(\frac{{\overline{v}}_i}{u_M}\right),\hfill & {\overline{v}}_i>0\hfill \\ u_{im}\times \tanh \left(\frac{{\overline{v}}_i}{u_{im}}\right),\hfill & {\overline{v}}_i\le 0,\hfill \end{array}$$

be the estimation of $u_i$. Thus, one has

$$u_i=\mathrm{sat}({\overline{v}}_i)=\rho ({\overline{v}}_i)+\mathrm{\Delta }({\overline{v}}_i),$$ (4)

with $\mathrm{\Delta }({\overline{v}}_i)=\mathrm{sat}({\overline{v}}_i)-\rho ({\overline{v}}_i)$ being the estimation error satisfying

$$|\mathrm{\Delta }({\overline{v}}_i)|\le {\delta }_i,$$

where ${\delta }_i\triangleq \max \{u_{iM}(1-\tanh (1)),u_{im}(\tanh (1)-1)\}$. Let ${\beta }_i({\overline{v}}_i)=\frac{\partial \rho ({\overline{v}}_i)}{\partial {\overline{v}}_i}$. Thus, (4) implies that

$$u_i={\beta }_i({\overline{v}}_i){\overline{v}}_i+{\mathrm{\Delta }}_i({\overline{v}}_i).$$

The main purpose is to implement an NN controller such that x tracks a known smooth enough signal $x_r={[x_{r1},x_{r2},\cdots ,x_{rn}]}^T\in {\mathbb{R}}^n$. Let the tracking error be $e_i=x_i-x_{di}$. Define the following sliding surface

$$s_i=h_i(e_i(t)-e_i(0)\exp (-{\lambda }_{i}t)),$$ (5)

with $h_i$ and ${\lambda }_i$ being two positive design parameters.

Remark 2

It should be noted that in the conventional SMC method, one usually requires reaching phase should be active before sliding motion, which may increase control system's overshoot. In this paper, the above alternative switching function $s_i(t)$ is used to overcome this problem. In (5), $h_i$ and ${\lambda }_i$ are chosen to regulate initial value and raise decay ratio, respectively. In addition, the sliding surface can be arrived without any reaching phase.

Noting that $f_i(x)$ in (2) is unknown, by the NN (1), it can be estimated as ${\hat{f}}_i(x,{\theta }_i)={\theta }_i^T{\psi }_i(x)$. Thus, one has

$$f_i(x)={\theta }_i^{⁎T}{\psi }_i(x)+{\epsilon }_i(x),$$ (6)

in which ${\theta }_i^{⁎}$ is the optimal parameter vector. Define the optimal parameter error as ${\tilde{\theta }}_i={\theta }_i^{⁎}-{\theta }_i$. According to the universal approximation theorem, it is reasonable to suppose that the approximation error satisfies $|{\epsilon }_i(x)|\le {\overline{\epsilon }}_i$ with ${\overline{\epsilon }}_i$ being an arbitrary small constant.

It follows from (2) and (5) that

$$\begin{array}{cc}\hfill {\mathfrak{D}}^{\nu }{s}_i& =r_i({\mathfrak{D}}^{\nu }{x}_i-{\mathfrak{D}}^{\nu }{x}_{ri}-e_i(0){\mathfrak{D}}^{\nu }\exp (-{\lambda }_{i}t))\hfill \\ \hfill & =r_i[\sum _{i=1}^n{a}_i{x}_i+f_i(x)+g_i{u}_i-{\mathfrak{D}}^{\nu }{x}_{ri}-e_i(0){\mathfrak{D}}^{\nu }\exp (-{\lambda }_{i}t)]\hfill \\ \hfill & =r_i[\sum _{i=1}^n{a}_i{x}_i+{\theta }_i^{⁎T}{\psi }_i(x)+{\epsilon }_i(x)+g_i{u}_i-{\mathfrak{D}}^{\nu }{x}_{ri}-e_i(0){\mathfrak{D}}^{\nu }\exp (-{\lambda }_{1}t)].\hfill \end{array}$$ (7)

According to Lemma 1, (7) can be rewritten by

$$\{\begin{array}{cc}\hfill \frac{\partial Z_{s_i}}{\partial t}& =r_i[\sum _{i=1}^n{a}_i{x}_i+{\theta }_i^{⁎T}{\psi }_i(x)+{\epsilon }_i(x)+g_i{u}_i-e_i(0){\mathfrak{D}}^{\nu }\exp (-{\lambda }_{1}t)]\hfill \\ \hfill & -\mathrm{\Xi }{Z}_{s_i}(\mathrm{\Xi },t)-r_i{\mathfrak{D}}^{\nu }{x}_{ri},\hfill \\ \hfill s_i=& \underset{0}{\overset{\infty }{\int }}{\mu }_{\nu }(\mathrm{\Xi })Z_{s_i}(\mathrm{\Xi },t)d\mathrm{\Xi },\hfill \end{array}$$ (8)

where ${\mu }_{\nu }(\mathrm{\Xi })\triangleq \frac{\sin (\nu \pi )}{\pi }{\mathrm{\Xi }}^{-\nu }$.

To proceed, one needs to analyze the following two Lyapunov functions, namely the monochromatic Lyapunov function of Ξ:

$${\overline{V}}_{s_i}(\mathrm{\Xi },t)=\frac{1}{2r_i}{Z}_{s_i}^2(\mathrm{\Xi },t),$$ (9)

and the Lyapunov function to weight (9):

$$\begin{array}{cc}\hfill V_{s_i}& =\underset{0}{\overset{\infty }{\int }}{\mu }_{\nu }(\mathrm{\Xi }){\overline{V}}_{s_i}(\mathrm{\Xi },t)d\mathrm{\Xi }\hfill \\ \hfill & =\frac{1}{2r_i}\underset{0}{\overset{\infty }{\int }}{\mu }_{\nu }(\mathrm{\Xi })Z_{s_i}^2(\mathrm{\Xi },t)d\mathrm{\Xi }.\hfill \end{array}$$ (10)

Substituting (8) into (10) yields

$$\begin{array}{cc}\hfill {\dot{V}}_{s_i}& =\frac{1}{r_i}\underset{0}{\overset{\infty }{\int }}{\mu }_{\nu }(\mathrm{\Xi })Z_{s_i}(\mathrm{\Xi },t)\frac{\partial Z_{s_i}}{\partial t}d\mathrm{\Xi }\hfill \\ \hfill & =-\frac{1}{r_i}\underset{0}{\overset{\infty }{\int }}\mathrm{\Xi }{\mu }_{\nu }(\mathrm{\Xi })Z_{s_i}^2(\mathrm{\Xi },t)d\mathrm{\Xi }+\underset{0}{\overset{\infty }{\int }}{\mu }_{{\nu }_1}(\mathrm{\Xi })Z_{s_1}(\mathrm{\Xi },t)d\mathrm{\Xi }\hfill \\ \hfill & \times [\sum _{i=1}^n{a}_i{x}_i+{\theta }_i^{⁎T}{\psi }_i(x)+{\epsilon }_i(x)+g_i{u}_i-{\mathfrak{D}}^{\nu }{x}_{ri}-e_i(0){\mathfrak{D}}^{\nu }\exp (-{\lambda }_{1}t)]\hfill \\ \hfill & =-\underset{0}{\overset{\infty }{\int }}\frac{\sin (\nu \pi )}{r_i\pi }{\mathrm{\Xi }}^{1-\nu }{Z}_{s_i}^2(\mathrm{\Xi },t)d\mathrm{\Xi }\hfill \\ \hfill & +s_i[\sum _{i=1}^n{a}_i{x}_i+{\theta }_i^{⁎T}{\psi }_i(x)+{\epsilon }_i(x)+g_i{u}_i-{\mathfrak{D}}^{\nu }{x}_{ri}-e_i(0){\mathfrak{D}}^{\nu }\exp (-{\lambda }_{1}t)].\hfill \end{array}$$ (11)

Note that $0<{\nu }_1<1$ and $\mathrm{\Xi }>0$, one has ${\int }_0^{\infty }\frac{\sin ({\nu }_1\pi )}{r_1\pi }{\mathrm{\Xi }}^{1-{\nu }_1}{Z}_{s_1}^2(\mathrm{\Xi },t)d\mathrm{\Xi }\ge 0$. Thus, by (10), (11), one has

$$\begin{array}{cc}\hfill {\dot{V}}_{s_1}& \le s_i{\theta }_i^{⁎T}{\psi }_i(x)+s_i{\epsilon }_i(x)+s_i{g}_i{u}_i-s_i{\mathfrak{D}}^{\nu }{x}_{ri}-s_i{e}_i(0){\mathfrak{D}}^{\nu }\exp (-{\lambda }_{i}t)\hfill \\ \hfill & \le s_i{\theta }_i^{⁎T}{\psi }_i(x)+|s_i|{\overline{\epsilon }}_i+s_i{g}_i{u}_i-s_i{\mathfrak{D}}^{\nu }{x}_{ri}-s_i{e}_i(0){\mathfrak{D}}^{\nu }\exp (-{\lambda }_{i}t).\hfill \end{array}$$ (12)

Then, it follows from Lemma 3 and Remark 1 that

$$\begin{array}{c}\hfill |s_i|\le s_i\mathrm{Rel}(s_i,{\sigma }_i)+{\sigma }_i,\end{array}$$ (13)

with ${\sigma }_i$ denoting the descending positive function $l_i{e}^{-l_{i}t}$ with $l_i>0$. Thus, it follows from (13) that

$$\begin{array}{c}\hfill |s_i|{\overline{\epsilon }}_i\le s_i\mathrm{Rel}(s_1,\sigma ){\overline{\epsilon }}_1+\sigma {\overline{\epsilon }}_1.\end{array}$$ (14)

Since ${\psi }_i^T(x){\psi }_i(x)\le 1$, (13) and Young's inequality imply that

$$\begin{array}{cc}\hfill s_i{\theta }_i^T{\psi }_i& \le \frac{1}{2}|s_i|({\Vert {\theta }_i\Vert }^2{\psi }_i^T{\psi }_i+1)\hfill \\ \hfill & \le \frac{1}{2}(s_i\mathrm{Rel}(s_i,{\sigma }_i)+{\sigma }_i)({\Vert {\theta }_i\Vert }^2{\psi }_i^T{\psi }_i+1)\hfill \\ \hfill & \le \frac{1}{2}{s}_i\mathrm{Rel}(s_i,{\sigma }_i)({\Vert {\theta }_i\Vert }^2{\psi }_i^T{\psi }_i+1)+\frac{1}{2}{\sigma }_i({\Vert {\theta }_i\Vert }^2+1).\hfill \end{array}$$ (15)

Then, according to (12), (14) and (15), one has

$$\begin{array}{cc}\hfill {\dot{V}}_{s_i}& \le \frac{1}{2}{s}_i\mathrm{Rel}(s_i,{\sigma }_i)({\phi }_i^2{\psi }_i^T{\psi }_i+1)+{\sigma }_i(\frac{1}{2}{\phi }_i^2+\frac{1}{2})+s_i{g}_i{u}_i\hfill \\ \hfill & -s_i{\mathfrak{D}}^{\nu }{x}_{ri}-s_i{e}_i(0){\mathfrak{D}}^{\nu }\exp (-{\lambda }_{1}t)+s_i\sum _{i=1}^n{a}_i{x}_i\hfill \\ \hfill & =\frac{1}{2}{s}_i\mathrm{Rel}(s_i,{\sigma }_i)({\phi }_i^2{\psi }_i^T{\psi }_i+1)+{\sigma }_i(\frac{1}{2}{\phi }_i^2+\frac{1}{2})+s_i{g}_i{\beta }_i{\overline{v}}_i\hfill \\ \hfill & -s_i{\mathfrak{D}}^{\nu }{x}_{ri}-s_i{e}_i(0){\mathfrak{D}}^{\nu }\exp (-{\lambda }_{1}t)+s_i\sum _{i=1}^n{a}_i{x}_i+s_i{\mathrm{\Delta }}_i,\hfill \end{array}$$ (16)

with ${\phi }_1={\Vert {\theta }_1\Vert }^2$. Thus, the actual controller can be constructed as

$$\{\begin{array}{cc}\hfill & {\overline{v}}_i=N_i\left({\xi }_i\right){\overline{\alpha }}_i,\hfill \\ \hfill & {\overline{\alpha }}_i=e_i(0){\mathfrak{D}}^{\nu }\exp (-{\lambda }_{n}t)-k_i{s}_i-s_i\mathrm{Rel}(s_i,{\sigma }_i)[\frac{1}{2}{\hat{\phi }}_i{\psi }_i^T{\psi }_i+\frac{1}{2}]-\sum _{i=1}^n{a}_i{x}_i,\hfill \\ \hfill & {\dot{\xi }}_i=s_i{\overline{\alpha }}_i,\hfill \end{array}$$ (17)

with $k_i>1$ being a design parameter, ${\hat{\phi }}_i$ is the estimation of ${\phi }_i$. Denote ${\tilde{\phi }}_n={\hat{\phi }}_n-{\phi }_n$ as the estimation error of the optimal parameter of the NN, and then, one can use the following law to update the control parameter:

$$\begin{array}{c}\hfill {\mathfrak{D}}^{\nu }{\hat{\phi }}_i=\frac{{\eta }_i}{2}{s}_i\mathrm{Rel}(s_i,{\sigma }_i){\psi }_i^T{\psi }_i-{\eta }_i^{⁎}{\hat{\phi }}_i,\end{array}$$ (18)

where ${\eta }_i,{\eta }_i^{⁎}\in {\mathbb{R}}^{+}$ are design parameters. Consequently, by Lemma 1, one has

$$\{\begin{array}{cc}\hfill & \frac{\partial Z_{{\tilde{\phi }}_i}}{\partial t}=-\mathrm{\Xi }{Z}_{{\tilde{\phi }}_i}+\frac{{\eta }_i}{2}{s}_n\mathrm{Rel}(s_i,{\sigma }_i){\psi }_i^T{\psi }_i-{\eta }_i^{⁎}{\hat{\phi }}_i\hfill \\ \hfill & {\tilde{\phi }}_i=\underset{0}{\overset{\infty }{\int }}{\mu }_{\nu }(\mathrm{\Xi })Z_{{\tilde{\phi }}_i}(\mathrm{\Xi },t)d\mathrm{\Xi }.\hfill \end{array}$$ (19)

Thus, one can choose the following Lyapunov function:

$$V_{i1}=V_{s_i}+V_{{\tilde{\phi }}_n}+\frac{1}{2{\eta }_n}\underset{0}{\overset{\infty }{\int }}{\mu }_{\nu }(\mathrm{\Xi })Z_{{\tilde{\phi }}_n}^2(\mathrm{\Xi },t)d\mathrm{\Xi }.$$ (20)

Then, it follows from (16), (19) and (20) that

$$\begin{array}{cc}\hfill {\dot{V}}_{i1}& \le -k_i{s}_i^2-\frac{{\eta }_i^{⁎}}{2{\eta }_i}{\tilde{\phi }}_i^2+\frac{{\eta }_i^{⁎}}{2{\eta }_i}{\phi }_i^2+[g_i{\beta }_i({\overline{v}}_i)N_i\left({\xi }_i\right)-1]{\dot{\xi }}_i+{\sigma }_i.\hfill \end{array}$$ (21)

Thereby, the major result can be summarized as the following theorem.

Theorem 1

Consider the FOCS (2) . If the sliding surface is provided as (5) , the controller is devised as (17) , the adaptation law is chosen as (18) , then, if the control parameters are selected suitably, the tracking error signal $e_i$ will remain sufficiently small eventually.

Proof

Let $V={\sum }_{i=1}^n{V}_i$, and from (21), one has

$$\begin{array}{cc}\hfill \dot{V}& \le -\sum _{i=1}^n(k_i^{⁎}{s}_i^2+\frac{{\eta }_i^{⁎}}{2{\eta }_i}{\tilde{\phi }}_i^2)+\sum _{i=1}^n\frac{1}{2}\frac{{\eta }_i^{⁎}}{{\eta }_i}{\phi }_i^2+\sum _{i=1}^n(g_i{\beta }_i{N}_i-1){\dot{\xi }}_i\hfill \\ \hfill & \le -(d_1-\frac{d_2}{{\Vert \kappa \Vert }^2}){\kappa }^T\kappa +\sum _{i=1}^n{l}_i{e}^{-l_{i}t}+\sum _{i=1}^n({\overline{c}}_i{N}_i-1){\dot{\xi }}_i,\hfill \end{array}$$ (22)

in which $\kappa ={[s_1,\cdots ,s_n,{\tilde{\phi }}_1,\cdots ,{\tilde{\phi }}_n]}^T\in {\mathbb{R}}^{2n}$, $d_1=\underset{1\le i\le n}{\min }\{k_i^{⁎},\frac{{\eta }_i^{⁎}}{2{\eta }_i}\}$, ${\overline{c}}_i=g_i{\beta }_i$, and $d_2=\sum _{i=1}^n\frac{1}{2}\frac{{\eta }_i^{⁎}}{{\eta }_i}{\phi }_i^2$.

Integrating (22) on $[0,t]$ yields

$$\begin{array}{cc}\hfill V& \le \underset{0}{\overset{t}{\int }}\left[\sum _{i=1}^n({\overline{c}}_i{N}_i-1){\dot{\xi }}_i\right]d\tau -\underset{0}{\overset{t}{\int }}\left[(d_1-\frac{d_2}{{\Vert \kappa \Vert }^2}){\kappa }^T\kappa \right]d\tau +\varsigma ,\hfill \end{array}$$ (23)

with $\varsigma ={\int }_0^{\infty }{\sum }_{i=1}^n{l}_i{e}^{-l_{i}t}d\tau +V(0)$.

If the control parameters are chosen such that $d_1-\frac{d_2}{{\Vert \kappa \Vert }^2}>0$, using (23), one has

$$\begin{array}{cc}\hfill V& \le \underset{0}{\overset{t}{\int }}\left[\sum _{i=1}^n({\overline{c}}_i{N}_i-1){\dot{\xi }}_i\right]d\tau +\varsigma ,\hfill \end{array}$$ (24)

Therefore, it is known from (24) and Lemma 2 that ${\int }_0^t\left[{\sum }_{i=1}^n(b_i{N}_i\left({\xi }_i\right)-1){\dot{\xi }}_i\right]d\tau$ and V are all bounded.

Then, taking a simple transposition of (23) yields

$$\begin{array}{cc}\hfill 0& \le \underset{0}{\overset{t}{\int }}\left[(d_1-\frac{d_2}{{\Vert \kappa \Vert }^2}){\kappa }^T\kappa \right]d\tau \le \underset{0}{\overset{t}{\int }}\left[\sum _{i=1}^n({\overline{c}}_i{N}_i-1){\dot{\xi }}_i\right]d\tau -V+\varsigma ,\hfill \end{array}$$ (25)

Thus, (25) implies that ${\int }_0^t\left[(d_1-\frac{d_2}{{\Vert \kappa \Vert }^2}){\kappa }^T\kappa \right]d\tau$ is bounded. Nothing that $(d_1-\frac{d_2}{{\Vert \kappa \Vert }^2}){\kappa }^T\kappa$ is monotonically increasing and uniformly continuous, one knows that the integral ${\int }_0^{\infty }\left[(d_1-\frac{d_2}{{\Vert \kappa \Vert }^2}){\kappa }^T\kappa \right]d\tau$ is convergent, and as a result, ${\lim }_{t\to \infty }(d_1-\frac{d_2}{{\Vert \kappa \Vert }^2})=0$, which means that $s_i,{\tilde{\phi }}_i$ converge into the set $\mathrm{\Omega }=\{\kappa |\Vert \kappa \Vert \le \sqrt{\frac{d_1}{d_2}}\}$ eventually. The whole proof ends.

4. Simulation results

To verify the feasibility of the proposed scheme, a simulation example was analyzed. Consider the following fractional order L-V system described by [34], [35]

$$\{\begin{array}{cc}\hfill {\mathfrak{D}}^{\nu }{x}_1& =x_1-x_1{x}_2+2x_1^2-2.70x_3{x}_1^2+g_1{u}_1(t),\hfill \\ \hfill {\mathfrak{D}}^{\nu }{x}_2& =-x_2+x_1{x}_2+g_2{u}_2(t),\hfill \\ \hfill {\mathfrak{D}}^{\nu }{x}_3& =-3x_3+2.70x_3{x}_1^2-g_3{u}_3(t).\hfill \end{array}$$ (26)

To check the feasibility of the control method, let the control direction changes at $t=5$ s. Let

$$g_1=\{\begin{array}{cc}\hfill 1.2,& t<5,\hfill \\ \hfill -1.2,& t\ge 5,\hfill \end{array}$$ (27)

$$g_2=\{\begin{array}{cc}\hfill 0.7,& t<5,\hfill \\ \hfill -0.7,& t\ge 5,\hfill \end{array}$$ (28)

$$g_3=\{\begin{array}{cc}\hfill 0.3,& t<5,\hfill \\ \hfill -0.3,& t\ge 5,\hfill \end{array}$$ (29)

The initial values are $x(0)={[2.0,-2.4,3.0]}^T$, and the order is $\nu =0.96$. When $u_i(t)\equiv 0,i=1,2,3$, the model (29) shows the chaotic behavior that is depicted in Fig. 1.

Figure 1.

The chaotic phenomenon of the system (29), in (a) x₁ − x₂ − x₃ space; (b) x₁ − x₂ plane; (c) x₁ − x₃ plane; (d) x₂ − x₃ plane.

For the input saturation, let $\mathrm{sat}({\overline{v}}_i)={\rho }_i({\overline{v}}_i)+\mathrm{\Delta }({\overline{v}}_i)$ with $u_{im}=u_{iM}=5,i=1,2,3$, and

$$\{\begin{array}{cc}\hfill {\rho }_1({\overline{v}}_1)& =8\tanh \left(\frac{{\overline{v}}_1}{8}\right),\hfill \\ \hfill {\rho }_2({\overline{v}}_2)& =5\tanh \left(\frac{{\overline{v}}_2}{5}\right),\hfill \\ \hfill {\rho }_3({\overline{v}}_3)& =9\tanh \left(\frac{{\overline{v}}_3}{9}\right).\hfill \end{array}$$ (30)

The following control parameters will be used in the simulation (in this part, $i=1,2,3$). Firstly, let $k_1=k_2=k_3=2.2$ in the controller (17). Secondly, make $h_i=2.55,{\lambda }_i=1.2$ in the sliding surface (5). Thirdly, let the parameters of the Nussbaum function in Lemma 2 be ${\vartheta }_i=1,p_i=1.55,q_i=0.22$, and the initial conditions be 0.8. Fourthly, the input signals are all $x_1,x_2$ and $x_3$ in the NN. For each input, five Gaussian functions uniformly distributed within the interval [-5, 5] are used. Finally, the learning parameters in the adaptation law (18) are given by ${\eta }_i^{⁎}=0.08,{\eta }_i=5.2$, and the initial condition is ${\hat{\phi }}_1(0)={\hat{\phi }}_i(0)=2$. The reference signal is selected to be $x_r={[\sin t,\cos t,\sin t+\cos t]}^T$.

The simulation results are depicted in Figure 1, Figure 2, Figure 3. Figs. 1(a)-(d) depicts the chaotic behavior of the state response of the fractional order L-V system. Figs. 2(a)-(c) shows the tracking performance, from which one can see that the tracking errors tend to zero rapidly in Fig. 2(d). The actual control inputs are given in Fig. 3 (a). The sliding surfaces $s_1$ and $s_2$ are plotted in Fig. 3 (b) and (c), respectively. It is obvious that the control inputs and the sliding surfaces are all smooth, which also verifies the theoretical results. The time responses of parameters ${\hat{\phi }}_1$, ${\hat{\phi }}_2$ and ${\hat{\phi }}_3$ are given in Fig. 3 (d). Generally speaking, the control effect is consistent with the theoretical results.

Figure 2.

Tracking performance, in (a) x₁ and x_r1; (b) x₂ and x_r2; (c) x₃ and x_r3; (d) e₁, e₂ and e₃.

Figure 3.

Simulation results, in (a) actual control inputs; (b) the sliding surface s₁; (c) the sliding surface s₂; (d) parameters ${\hat{\phi }}_1$, ${\hat{\phi }}_2$ and ${\hat{\phi }}_3$.

5. Conclusions

In this paper, Nussbaum gain control is introduced into FOCSs to handle the case that the control direction is unknown. First, a class of stable sliding mode surfaces are constructed to solve the reachability problem in traditional sliding mode control. Then, combined with the integer order Nussbaum gain control method, the chaos control is realized when the control coefficient is unknown and the input is saturated. It can be seen from the simulation that the proposed approach is feasible. Considering the influence of unknown control direction, limited gain and other uncertain factors simultaneously, how to construct the synchronization controller of FOCSs is a subject that we need to further study in the future.

CRediT authorship contribution statement

Suxia Wang: Methodology, Data curation. Yong Chen: Methodology, Investigation, Data curation, Conceptualization.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Data availability

Data will be made available on request.