1 Introduction

A hyperjerk system [1] is a dynamical system described by an nth ordinary differential equation where n is 4 or up to, i.e.

$$\begin{aligned} \frac{{\mathrm{d}}^nx}{{\mathrm{d}}t^n}=J\left( \frac{{\mathrm{d}}^{n-1}x}{{\mathrm{d}}t^{n-1}},\ldots ,\frac{{\mathrm{d}}x}{{\mathrm{d}}t},x\right) \quad (n\geqslant 4). \end{aligned}$$
(1)

where x denotes displacement, \(\frac{{\mathrm{d}}x}{{\mathrm{d}}t}\) denotes velocity, \(\frac{{\mathrm{d}}^2x}{{\mathrm{d}}t^2}\) is acceleration, \(\frac{{\mathrm{d}}^3x}{{\mathrm{d}}t^3}\) is jerk, and \(\frac{{\mathrm{d}}^4x}{{\mathrm{d}}t^4},\ldots , \frac{{\mathrm{d}}^nx}{{\mathrm{d}}t^n}\) are hyperjerks. When \(n=3\), system (1) degenerates into a jerk system. The main concerns of the researchers are the chaotic or hyperchaotic performance, control and synchronization of jerk system or hyperjerk system with such simple structures [2,3,4,5,6,7,8,9,10,11].

As a synchronization strategy, the concept of projective synchronization of chaotic (or hyperchaotic) system was first proposed by Mainieri and Rehacek [12], where the introduction of a scaling factor extended the range of synchronization. When projective synchronization is realized, the state outputs of the drive system and the response system are not only phase locked, but also the amplitude of the corresponding state evolves according to a fixed proportional relationship. Because projective synchronization can be applied to chaotic secure communication, binary number can be extended to base m number to achieve faster transmission, so many scholars have done in-depth research on projective synchronization [13,7, 11, 17], backstep** control [7, 17, 18], interval observer method [24] proposed the exponential reaching law method to design a sliding mode controller, which can not only accelerate the approach speed but also provide a measure for the reduction of chattering to some extent [25, 26]. For the purpose of reaching the stable state of the control system in limited time, terminal sliding mode control is proposed by Zak [27]. Many researchers have applied the terminal sliding control to various control systems. Fei et al. [21, 23] designed the fractional-order sliding mode controller based on neural network.

In [28], Kawamura et al. proposed disturbance observer to compensate the uncertainty and disturbance of the servo motor in the controlling system. Some literature is posted on this method [29,30,31]. Considering the observer error and some other disturbance which could not compensate with observer, such as white noise, fuzzy system is a great idea [32] to smooth the discontinuity of the sliding mode control. Usually the control precision in a fuzzy system can be enhanced by increasing the amount of fuzzy rules. This can lead to a problem of a great deal of computations and even an ”rules explosion.” Variable universe adaptive fuzzy control method (VUAFC) introduced by Li [33, 34] can avoid the problem of ”rules explosion” and improve the control precision by on-line contraction-expansion of variable universes.

Inspired by the above control methods, in this paper, a variable universe adaptive fuzzy sliding mode projective Synchronization (VUAFSMPS) based on disturbance observer is proposed for hyperjerk system. A sliding mode surface with nonlinear term is proposed. We quantitatively analyze the effect of the disturbance observer proposed in [28] and apply it to compensate the low time-varying disturbance of hyperjerk system. Variable universe adaptive fuzzy control method is used to smooth the switching term.

The organization of this work is as follows. In Sect. 2, problem description is given. In Sect. 3, variable universe adaptive fuzzy sliding mode projective synchronization is introduced in detail. In particular, we use linear system theory to analyze the observer error of disturbance observer. Simulation results given in Sect. 4. Finally, the conclusion is drawn.

2 Problem description

Hyperjerk system has a very simple structure. It is quite interesting because it presents a complex dynamic. In the literature, there are some hyperjerk systems presenting chaotic (hyperchaotic) phenomena. Here we consider the projective synchronization.

The drive system is depicted as

$$\begin{aligned} \left\{ \begin{aligned}&{\dot{x}}_1=y_1,\\&{\dot{y}}_1=z_1,\\&{\dot{z}}_1=w_1,\\&{\dot{w}}_1=f_d(x_1,y_1,z_1,w_1)+d_1(t)+n_1(t) \end{aligned}\right. \end{aligned}$$
(2)

where \(x_1, y_1, z_1, w_1\) are state variables, \(f_d\) is continuous function, \(d_1(t)\) is an unknown low time-varying external disturbance, \(\mid d_1(t)\mid \leqslant D_1\), \(D_1\) is a positive constant, \(n_1(t)\) is white noise, \(n_1(t)\leqslant N_1\).

The response system is described as

$$\begin{aligned} \qquad \left\{ \begin{aligned}&{\dot{x}}_2=y_2,\\&{\dot{y}}_2=z_2,\\&{\dot{z}}_2=w_2,\\&{\dot{w}}_2=f_r(x_2,y_2,z_2,w_2)+d_2(t)+n_2(t)+u \end{aligned}\right. \end{aligned}$$
(3)

where \(x_2, y_2, z_2, w_2\) are state variables, \(f_r\) is continuous function, \(d_2(t)\) is an unknown low time-varying external disturbance, \(\mid d_2(t)\mid \leqslant D_2\), \(D_2\) is a positive constant, \(n_2(t)\leqslant N_2\), u is the controller to be designed.

Suppose the scaling factor is \(\alpha\). The errors are given by

$$\begin{aligned} \left\{ \begin{aligned}&e_x=x_2-\alpha x_1,\\&e_y=y_2-\alpha y_1,\\&e_z=z_2-\alpha z_1,\\&e_w=w_2-\alpha w_1. \end{aligned}\qquad \qquad \right. \end{aligned}$$
(4)

Differentiating (4), the error dynamic system is

$$\begin{aligned} \qquad \qquad \left\{ \begin{aligned}&{\dot{e}}_x=e_y,\\&{\dot{e}}_y=e_z,\\&{\dot{e}}_z=e_w,\\&{\dot{e}}_w=f_r-\alpha f_d+(d_2(t)+n_2(t))-\alpha (d_1(t)+n_1(t))+u \end{aligned}\right. \end{aligned}$$
(5)

Our goal is designing suitable input u such that system (3) synchronizes (2) up to the scaling factor \(\alpha\), i.e., system (5) converges to zero, especially, \(\varvec{e}=(e_x,e_y,e_z,e_w)\rightarrow 0\) as \(t\rightarrow \infty\).

3 Design of fuzzy sliding mode projective synchronization based on disturbance observer

The sliding mode surface with nonlinear term is proposed in Sect. 3.1. The observer of slow time-varying disturbance is quantitatively analyzed in Sect. 3.2. Variable universe adaptive fuzzy controller is restated in Sect. 3.3. The composite controller is introduced in Sect. 3.4.

3.1 Sliding mode control

Under the view of traditional sliding mode control, the control system (5) can be stabilized with the sliding mode surface as

$$\begin{aligned} s=e_w+c_3e_z+c_2e_y+c_1e_x \end{aligned}$$
(6)

where the coefficients \(c_1,c_2,c_3\) satisfy the Hurwitz polynomial \(p^3+c_3p^2+c_2p+c_1\), p is Laplace operator. By changing the first term of Eq. (6), we can get a sliding mode surface with nonlinear term, i.e.

$$\begin{aligned} s=e_w^{p/q}+c_3e_z+c_2e_y+c_1e_x \end{aligned}$$
(7)

where \(q<p<2q\), p, q are prime numbers, the coefficients \(c_1, c_2, c_3\) is to be designed such that the sliding surface (7) is stable when \(s=0\).

Differentiating Eq. (7), we get

$$\begin{aligned} \begin{aligned} {\dot{s}}&=\frac{p}{q}{e}_w^{p/q-1}{\dot{e}}_w+c_3e_w+c_2e_z+c_1e_y\\&=\frac{p}{q}e_w^{p/q-1}\left( {\dot{e}}_w+c_3\frac{q}{p}e_w^{2-p/q}+c_2\frac{q}{p}e_w^{1-p/q}e_z+c_1\frac{q}{p}e_w^{1-p/q}e_y\right) \end{aligned} \end{aligned}$$
(8)

Considering the error dynamic system (5), ignoring the external disturbances, the equivalent control term is solved by setting \({\dot{s}}=0\)

$$\begin{aligned} u_{eq}=-f_r+\alpha f_d-\frac{c_3q}{p}e_w^{2-p/q}-\frac{c_2q}{p}e_w^{1-p/q}e_z-\frac{c_1q}{p}e_w^{1-p/q}e_y \end{aligned}$$
(9)

The switching control term is

$$\begin{aligned} u_{erl}=-Msign(s)-\gamma s \end{aligned}$$
(10)

Then the comprehensive controller is designed as

$$\begin{aligned} \begin{aligned} u&=u_{eq}+u_{erl}\\&=-f_r+\alpha f_d-\frac{c_3q}{p}e_w^{2-p/q}-\frac{c_2q}{p}e_w^{1-p/q}e_z-\frac{c_1q}{p}e_w^{1-p/q}e_y\\&\quad -Msign(s)-\gamma s \end{aligned}\end{aligned}$$
(11)

As is well known, the main cause of chattering is the existence of switching gain M. Switching term is used to overcome the disturbances and uncertainty. It is a good idea by using disturbance observer to compensate the disturbance.

3.2 Observer of slow time-varying disturbance

In this section, we give the disturbance observer based on Kawamura’s work [28] and analyze the observation error quantitatively.

For \(d_1(t)\) of system (2), we design the disturbance observer as

$$\begin{aligned} \left\{ \begin{aligned}&\dot{{\hat{d}}}_1=-k_1({\hat{w}}_1-w_1),\\&\dot{{\hat{w}}}_1=f_d+{\hat{d}}_1-k_2({\hat{w}}_1-w_1),\\ \end{aligned}\right. \end{aligned}$$
(12)

where \({\hat{d}}_1\) is the estimation of \(d_1\), \({\hat{w}}_1\) is the estimation of \(w_1\), \(k_1>0, k_2>0\).

For \(d_2(t)\) of system (3), we design the disturbance observer as follows

$$\begin{aligned} \qquad \left\{ \begin{aligned}&\dot{{\hat{d}}}_2=-k_3({\hat{w}}_2-w_2),\\&\dot{{\hat{w}}}_2=f_r+{\hat{d}}_2-k_4({\hat{w}}_2-w_2)+u,\\ \end{aligned}\right. \end{aligned}$$
(13)

where \({\hat{d}}_2\) is the estimation of \(d_2\), \({\hat{w}}_2\) is the estimation of \(w_2\), \(k_3>0, k_4>0\).

We define observation errors \({\tilde{d}}_i={\hat{d}}_i-d_i, {\tilde{w}}_i={\hat{w}}_i-w_i, (i=1,2)\), now we calculate them quantificationally. From system (2) and system (12), we have

$$\begin{aligned} \left\{ \begin{aligned}&\dot{{\tilde{d}}}_1=-k_1{\tilde{w}}_1-{\dot{d}}_1\\&\dot{{\tilde{w}}}_1={\tilde{d}}_1-k_2{\tilde{w}}_1 \end{aligned}\right. \qquad \qquad \end{aligned}$$
(14)

From system (3) and system (13), we have

$$\begin{aligned} \left\{ \begin{aligned}&\dot{{\tilde{d}}}_2=-k_3{\tilde{w}}_2-{\dot{d}}_2\\&\dot{{\tilde{w}}}_2={\tilde{d}}_2-k_4{\tilde{w}}_2 \end{aligned}\right. \qquad \qquad \end{aligned}$$
(15)

By using vector notation, we define

$$\begin{aligned} E_1=\begin{pmatrix} {\tilde{d}}_1\\ {\tilde{w}}_1 \end{pmatrix}, A=\begin{pmatrix} 0 &{} -k_1\\ 1 &{} -k_2 \end{pmatrix}, B=\begin{pmatrix} -1\\ 0 \end{pmatrix} \end{aligned}$$

and rewrite the system (14) as one 2-dimensional first-order vector differential equation

$$\begin{aligned} {\dot{E}}_1=AE_1+B{\dot{d}}_1 \end{aligned}$$
(16)

Now we diagonalize the system matrix A. The eigenvalues of A are \(\lambda _1=\frac{-k_2+\sqrt{k_2^2-4k_1}}{2}\) and \(\lambda _2=\frac{-k_2-\sqrt{k_2^2-4k_1}}{2}\).

If we define

$$\begin{aligned} \begin{aligned} D&=\begin{pmatrix} \frac{-k_2+\sqrt{k_2^2-4k_1}}{2}&{}\quad 0 \\ 0&{}\quad \frac{-k_2-\sqrt{k_2^2-4k_1}}{2} \end{pmatrix},\\ P&=\begin{pmatrix} \frac{k_2+\sqrt{k_2^2-4k_1}}{2} &{}\quad \frac{k_2-\sqrt{k_2^2-4k_1}}{2}\\ 1 &{}\quad 1 \end{pmatrix},\\ P^{-1}&=\frac{1}{\sqrt{k_2^2-4k_1}}\begin{pmatrix} 1&{}\quad \frac{-k_2+\sqrt{k_2^2-4k_1}}{2} \\ -1&{}\quad \frac{k_2+\sqrt{k_2^2-4k_1}}{2} \end{pmatrix},\end{aligned} \end{aligned}$$
(17)

then

$$\begin{aligned}\begin{aligned} P^{-1}AP=D,\,\, A=PDP^{-1} \end{aligned} \end{aligned}$$

Defining \(E^*_1=P^{-1}E_1\), then system (16) change to

$$\begin{aligned} {\dot{E}}^*_1=DE^*_1+P^{-1}B{\dot{d}}_1. \end{aligned}$$
(18)

For D is diagonalizable, from linear system theory, we know the solution of system (18) is

$$\begin{aligned} E^*_1=e^{Dt}E^*_1(t_0)+\int ^{t}_{0}e^{D(t-\tau )}P^{-1}B{\dot{d}}_1(\tau ){\mathrm{d}}\tau \end{aligned}$$
(19)

Because \(\lambda _1=\frac{-k_2+\sqrt{k_2^2-4k_1}}{2}, \lambda _2=\frac{-k_2-\sqrt{k_2^2-4k_1}}{2}\), then

$$\begin{aligned} E^*_1= & {} \begin{pmatrix} e^{\lambda _1t}&{}0\\ 0&{}e^{\lambda _2t} \end{pmatrix}E^*_1(t_0)\\&+\frac{1}{\sqrt{k^2_2-4k_1}} \begin{pmatrix} e^{\lambda _1t}&{}0\\ 0&{}e^{\lambda _2t} \end{pmatrix} \int _{0}^{t} \begin{pmatrix} -e^{-\lambda _1\tau }\\ e^{-\lambda _2\tau } \end{pmatrix}{\dot{d}}_1(\tau ){\mathrm{d}}\tau \end{aligned}$$

If \({\dot{d}}_1(t)=0\), \(E_1^*\rightarrow 0\) as \(t\rightarrow \infty\).

Suppose \(0<\mid {\dot{d}}_1(t)\mid \leqslant N_1\), we have

$$\begin{aligned} \begin{aligned} \mid e^{\lambda _1t}\int _{0}^{t}-e^{-\lambda _1\tau }{\dot{d}}_1(\tau ){\mathrm{d}}\tau \mid \leqslant N_1e^{\lambda _1t}\int _{0}^{t}e^{-\lambda _1\tau }{\mathrm{d}}\tau =\frac{N_1}{\lambda _1}(e^{\lambda _1t}-1)\\ \mid e^{\lambda _2t}\int _{0}^{t}e^{-\lambda _2\tau }{\dot{d}}_1(\tau ){\mathrm{d}}\tau \mid \leqslant N_1e^{\lambda _2t}\int _{0}^{t}e^{-\lambda _2\tau }{\mathrm{d}}\tau =\frac{N_1}{\lambda _2}(e^{\lambda _2t}-1) \end{aligned}\end{aligned}$$
(20)

From linear system theory, we know \(E_1^*\leqslant \frac{-N_1}{\sqrt{k^2_2-4k_1}}\begin{pmatrix} \tfrac{1}{\lambda _1}\\ \tfrac{1}{\lambda _2} \end{pmatrix}\) as \(t\rightarrow \infty\), i.e.

$$\begin{aligned} E_1\leqslant \frac{N_1}{2k_1\sqrt{k_2^2-4k_1}}\begin{pmatrix} k_2^2-2k_1\\ 2k_2 \end{pmatrix} \end{aligned}$$
(21)

What we’re interested in is \({\tilde{d}}_1(t)\). Now we know

$$\begin{aligned}&\mid {\tilde{d}}_1(t)\mid \leqslant \frac{N_1(k_2^2-2k_1)}{2k_1\sqrt{k^2_2-4k_1}}=\frac{N_1((\frac{k_2}{k_1})^2-\frac{2}{k_1})}{2\sqrt{(\frac{k_2}{k_1})^2-\frac{4}{k_1}}}\sim \frac{k_2N_1}{2k_1}\nonumber \\&\qquad (k_1\gg k_2>1) \end{aligned}$$
(22)

From (22), we know that if \(k_1\gg k_2\), observer error is very small, that is to say, the disturbance observer works very well.

Example 1

If \({d}_1=\mathrm{sin}t\), then \({\dot{d}}_1(t)=\mathrm{cos}t\), we have

$$\begin{aligned}&\begin{aligned} E^*_{11}&=e^{\lambda _1t}E^*_{11}(t_0)+\tfrac{1}{(1+\lambda _1^2)\sqrt{k^2_2-4k_1}}(-\mathrm{sin}t+\lambda _1\mathrm{cos}t-\lambda _1e^{\lambda _1t})\\&=\left( E^*_{11}(t_0)-\tfrac{\lambda _1}{(1+\lambda _1^2)\sqrt{k_2^2-4k_1}}\right) e^{\lambda _1t}\\&\quad +\tfrac{1}{\sqrt{1+\lambda _1^2}\sqrt{k_2^2-4k_1}}\left( \tfrac{-1}{\sqrt{1+\lambda _1^2}}\mathrm{sin}t+\tfrac{\lambda _1}{\sqrt{1+\lambda _1^2}}\mathrm{cos}t\right) \\&=\left( E^*_{11}(t_0)-\tfrac{\lambda _1}{(1+\lambda _1^2)\sqrt{k_2^2-4k_1}}\right) e^{\lambda _1t}\\&\quad +\tfrac{1}{\sqrt{1+\lambda _1^2}\sqrt{k_2^2-4k_1}}\mathrm{sin}(t+\beta _1) \end{aligned} \\&\begin{aligned} E^*_{12}&=e^{\lambda _2t}E^*_{12}(t_0)+\tfrac{1}{(1+\lambda _2^2)\sqrt{k^2_2-4k_1}}(\mathrm{sin}t-\lambda _2\mathrm{cos}t-\lambda _2e^{\lambda _2t})\\&=\left( E^*_{12}(t_0)-\tfrac{\lambda _2}{(1+\lambda _2^2)\sqrt{k_2^2-4k_1}}\right) e^{\lambda _2t}\\&\quad +\tfrac{1}{\sqrt{1+\lambda _2^2}\sqrt{k_2^2-4k_1}}\left( \tfrac{1}{\sqrt{1+\lambda _2^2}}\mathrm{sin}t-\tfrac{\lambda _2}{\sqrt{1+\lambda _2^2}}\mathrm{cos}t\right) \\&=\left( E^*_{12}(t_0)-\tfrac{\lambda _2}{(1+\lambda _2^2)\sqrt{k_2^2-4k_1}}\right) e^{\lambda _2t}\\&\quad +\tfrac{1}{\sqrt{1+\lambda _2^2}\sqrt{k_2^2-4k_1}}\mathrm{sin}(t+\beta _2) \end{aligned} \end{aligned}$$

Based on linear system theory, we know \(E^*_1\rightarrow \begin{pmatrix} \tfrac{1}{\sqrt{1+\lambda _1^2}\sqrt{k^2_2-4k_1}}\mathrm{sin}(t+\beta _1)\\ \tfrac{1}{\sqrt{1+\lambda _2^2}\sqrt{k^2_2-4k_1}}\mathrm{sin}(t+\beta _2) \end{pmatrix}\).

From \(E_1=PE_1^*\), we have

$$\begin{aligned} E_1\rightarrow \begin{pmatrix} \frac{-\sqrt{(1+k_2^2-k_1)^2+k_1^2k_2^2}}{1+k_1^2+k_2^2-2k_1}\mathrm{sin}(t+\beta _1)\\ \frac{1}{\sqrt{(1-k_1)^2+k_2^2}}\mathrm{sin}(t+\beta _2) \end{pmatrix} \end{aligned}$$
(23)

Note 1: From (23), we know that the period of the observer error is the same as the period of the disturbance, the observer error boundary of \({\tilde{d}}_1\) is \(\frac{\sqrt{(1+k_2^2-k_1)^2+k_1^2k_2^2}}{1+k_1^2+k_2^2-2k_1}\) and the observer error boundary of \({\tilde{\omega }}_1\) is \(\frac{1}{\sqrt{(1-k_1)^2+k_2^2}}\).

The changes of observer error boundary with parameters evolving are shown as Table 1.

Table 1 Observer errors with parameters \(k1,\ k2\) changing
Full size table

Table 1 further illustrates that as long as appropriate parameters \(k_1\) and \(k_2\) are selected, the observer can achieve better compensation. It is also demonstrated that the observation error is persistent as long as the derivative of the disturbance is not zero. Once again, it is proved that the observation error is related to the parameter proportion.

We also get the similar conclusions about the observer error system (15).

3.3 Variable universe adaptive fuzzy controller

From the discussion of the above subsection, we know the switching gain in sliding mode control can be significantly reduced by using the disturbance observer method. At the same time, as long as the derivative is not equal to zero, the observation error is persisting steadily existing. And if the disturbances don’t have derivatives (inevitable, white noise, for instance), the observer cannot be employed to compensate them. To further reduce chattering is still a problem. Because the fuzzy system can estimate the switching gain effectively according to the reaching condition, this action can reduce chattering. Usually the control precision in a fuzzy system can be improved by increasing the amount of fuzzy rules. Meanwhile, this can lead to a large number of computations and even an ”rules explosion.” VUAFC method improves the control precision by on-line contraction-expansion of variable universes without increasing the fuzzy rules. Variable universe adaptive fuzzy control method can establish high control precision with a few fuzzy rules.

Variable universe adaptive fuzzy controller was first proposed by Li [33, 34]. Because this is a high precision controller, many scholars have studied it [35,36,37,38]. Suppose \(X_i=[-E_i,E_i]\,\,(i=1,2,\ldots ,m)\) be the universe of input variable \(x_i(i=1,2,\dots ,m)\), \(Y=[-U,U]\) be the universe of the output variable y. \(\{A_{ij}\}(1\leqslant j \leqslant q_i)\) stand for a fuzzy partition on \(X_i\) and \({B_l}(1\leqslant l\leqslant h)\) stand for a fuzzy partition on Y. Suppose the fuzzy inference rules is as:

$$\begin{aligned} \mathrm{If}\,\, x_1\,\, is\,\, A_{1j_1},\,\, x_2\,\, is\,\, A_{2j_2}, \ldots x_m\,\, is\,\, A_{mj_m}\,\, \mathrm{then}\,\, y\,\, is\,\, B_l. \end{aligned}$$
(24)

Suppose \(x_{ij_i}\) is the peakpoint of \(A_{ij_i}\), \(y_l\) is the peakpoint of \(B_l\). The output of canonical fuzzy controller is

$$\begin{aligned} y=\sum _{j_1=1}^{q_1}\cdots \sum _{j_m=1}^{q_m}\prod _{i=1}^{m}A_{ij_i}(x_i)y_l. \end{aligned}$$

By introducing scale factors \(\alpha _i\) and \(\beta\), the universe discourse is changed with the changing variables \(x_i\) and y, i.e. \(X_i(x_i)=[-\alpha _i(x_i)E_i,\,\alpha _i(x_i)E_i]\), \(Y(y)=[-\beta (y)U,\beta (y)U]\). The fuzzy output of the variable universe fuzzy controller is as

$$\begin{aligned} y=\beta \sum _{j_1=1}^{q_1}\cdots \sum _{j_m=1}^{q_m}\prod _{i=1}^{m}A_{ij_i}(\frac{x_i}{\alpha (x_i)})y_l. \end{aligned}$$
(25)

The parameter \(\beta\) can be adjusted by optimization.

3.4 The design of the controller

Based on (25), we firstly design the fuzzy system, the input of the fuzzy system is selected as sliding mode surface s(t). Let the output of the fuzzy system is \(\mu\). The universes of discourse of s(t) and \(\mu\) are \(\mid s_0 \mid \times [-1, 1]\) and \([-1,1]\), respectively.

Suppose the width of fuzzy layer is \(2\delta\), the initial membership function of s(t) is:

$$\begin{aligned} \begin{aligned} NB(s)&=\min \left( 1,\max \left( -\frac{s}{\delta },0\right) \right) ,\\ ZO(s)&=\max \left( 0,\min \left( \frac{s+\delta }{\delta },-\frac{s-\delta }{\delta }\right) \right) ,\\ PB(s)&=\min \left( 1,\max \left( \frac{s}{\delta },0\right) \right) \end{aligned} \end{aligned}$$
(26)

The plot of initial membership function is depicted by Fig. 1.

Fig. 1
figure 1

The initial membership function of input s(t)

Full size image

The initial membership function of \(\mu\) is selected as:

$$\begin{aligned}\begin{aligned} NB(\mu )&=\min \left( 1,\max \left( -\frac{\mu }{0.5},0\right) \right) ,\\ ZO(\mu )&=\max \left( 0,\min \left( \frac{\mu }{0.1},-\frac{\mu -0.3}{0.2}\right) \right) ,\\ PB(\mu )&=\min \left( 1,\max \left( \frac{\mu }{0.5},0\right) \right) \end{aligned} \end{aligned}$$

The plot of initial membership function is depicted by Fig. 2.

Fig. 2
figure 2

The initial membership function of output \(\mu\)

Full size image

The natural initial fuzzy rules are:

$$\begin{aligned}\begin{aligned} R^{(1)}: \text {If}\,\, s(t)\,\, \text {is}\,\, NB\,\, \text {then}\,\, \mu \,\, \text {is}\,\, PB\\ R^{(2)}: \text {If}\,\, s(t)\,\, \text {is}\,\, ZO\,\, \text {then}\,\, \mu \,\, \text {is}\,\, ZO\\ R^{(3)}: \text {If}\,\, s(t)\,\, \text {is}\,\, PB\,\, \text {then}\,\, \mu \,\, \text {is}\,\, PB\end{aligned} \end{aligned}$$

The contraction-expansion factor is as follows

$$\begin{aligned} \alpha (s)=1-\lambda _1 exp(-\lambda _2s^2),\,\,\,\lambda _1\in (0,1),\,\,\lambda _2>0 \end{aligned}$$
(27)

From (25), we know the structure of fuzzy system is

$$\begin{aligned}&\mu =\left( p_1\int _{0}^{t}s{\mathrm{d}}\tau +\beta (0)\right) \left( 0.5NB\left( \frac{s}{\alpha (s)}\right) \right. \nonumber \\&\qquad \left. +\,0.1 ZO\left( \frac{s}{\alpha (s)}\right) +0.5 PB\left( \frac{s}{\alpha (s)}\right) \right) \end{aligned}$$
(28)

We add a coefficient \(\mu\) to the switch term, the control law is designed as

Theorem 1

With the disturbance observers (12) and (13), projective synchronization of system (2) and system (3) up to scaling factor \(\alpha\) can be realized under the following controller

$$\begin{aligned} \begin{aligned} u=&-f_r+\alpha f_d-{\hat{d}}_2(t)+\alpha {\hat{d}}_1(t)-\frac{c_3q}{p}e_w^{2-p/q}\\&-\frac{c_2q}{p}e_w^{1-p/q}e_z-\frac{c_1q}{p}e_w^{1-p/q}e_y\\&-\mu M_2sign(s)-\gamma _2 s \end{aligned}\end{aligned}$$
(29)

where \(M_2\geqslant \mid D_1\mid +\mid \alpha D_2\mid +\mid N_1\mid +\mid \alpha N_2\mid\), \(\mu\) is the output of variable universe adaptive fuzzy controller. The equivalent control part \(u_{eq}=-f_r+\alpha f_d-{\hat{d}}_2(t)+\alpha {\hat{d}}_1(t)-\frac{c_{3}q}{p}e_w^{2-p/q}-\frac{c_2q}{p}e_w^{1-p/q}e_z-\frac{c_1q}{p}e_w^{1-p/q}e_y\) and the reaching control part \(u_{erl}=-\mu M_2sign(s)-\gamma _2 s\).

Note 2: Obviously when \(\mu =1\), \(u=u_{eq}+u_{erl}\). When \(\mu \ne 1\), the control law is fuzzified. We select \(\mid s_0 \mid \times [-1, 1]\) as universe of discourse of s(t) avoiding overflow of input.

4 Numerical simulation

We select the drive and response system given by [39]. The drive system is

$$\begin{aligned} \left\{ \begin{aligned} {\dot{x}}_1&=y_1,\\ {\dot{y}}_1&=z_1,\\ {\dot{z}}_1&=w_1,\\ {\dot{w}}_1&=-2x_1^3-y_1^3+(3x_1-4z_1)z_1^2-0.1w_1+d_1(t)+n_1(t). \end{aligned}\right. \end{aligned}$$
(30)

where \(d_1(t)=3 \sin (3 t)\) is the low time-varying disturbance and \(n_1(t)=0.1rand\) is uniform white noise.

The response system is

$$\begin{aligned} \left\{ \begin{aligned} {\dot{x}}_2&=y_2,\\ {\dot{y}}_2&=z_2,\\ {\dot{z}}_2&=w_2,\\ {\dot{w}}_2&=-2x_2^3-y_2^3+(3x_2-4z_2)z_2^2-0.1w_2+d_2(t)+n_2(t). \end{aligned}\right. \end{aligned}$$
(31)

where \(d_2(t)=\sin (0.2t)+\cos (t)\) is the low time-varying disturbance and \(n_2(t)=0.2rand(t)\) is uniform white noise.

We select the scaling factor \(\alpha =1/2\), and the initial values of drive system and response system are \((3,3,3,-3)^T\) and \((-1,-1,-1,-1)^T\), respectively.

The sliding mode surface is designed as

$$\begin{aligned} s=e_w^{\frac{5}{3}}+3e_z+3e_y+e_x \end{aligned}$$
(32)

Let the parameters \(k_1=1000, k_2=200\) in Eq. (12). Let the parameters \(k_3=1000, k_4=200\) in Eq. (13). Let the parameters \(M_2=10, \gamma _2=50\).

4.1 Sliding mode control with disturbance observer

Regardless of fuzzy system, that is to say, let \(\mu =1\) in Eq. (29).

The synchronization error is depicted as Fig. 3. The evolution of sliding mode surface is depicted as Fig. 4.

Fig. 3
figure 3

The synchronization errors

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Fig. 4
figure 4

The evolution of sliding mode surface s

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The control u is depicted as Fig. 5. For the sake of clarity, the control u is enlarged as Fig. 6. The chattering phenomenon is very obvious.

Fig. 5
figure 5

The evolution of control u

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Fig. 6
figure 6

The enlargement of control u

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The phase diagrams of projective synchronization are depicted as Figs. 7 and 8.

Fig. 7
figure 7

The phase diagram on the \(x_1-y_1-z_1\)(\(x_2-y_2-z_2\)) space

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Fig. 8
figure 8

The phase diagram on the \(y_1-z_1-w_1\)(\(y_2-z_2-w_2\)) space

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Figure3 illustrates the projective synchronization be achieved with \(\mu =1\) of controller (28). Figures 5 and 6 illustrate the presence of chattering.

4.2 Sliding mode control with disturbance observer and variable universe adaptive fuzzy controller

Suppose the parameter \(\delta =1\) of Eq. (26), \(\lambda _1=0.9, \lambda _2=10\) in Eq. (27), with the variable universe adaptive fuzzy controller (28), the simulation results of composite controller (29) are as follows.

The synchronization error is depicted as Fig. 9. The evolution of sliding mode surface s is depicted as Fig. 10.

Fig. 9
figure 9

The evolution of the synchronization errors

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Fig. 10
figure 10

The evolution of sliding mode surface s

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The control u is depicted as Fig. 11. For the sake of clarity, the control u is enlarged as Fig. 12.

Fig. 11
figure 11

The control u

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Fig. 12
figure 12

The enlargement of the control u

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The phase diagrams of projective synchronization are depicted as Figs. 13 and 14.

Fig. 13
figure 13

The phase diagram on the \(x_1-y_1-z1\)(\(x_2-y_2-z_2\)) space

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Fig. 14
figure 14

The phase diagram on the \(y_1-z_1-w_1\)(\(y_2-z_2-w_2\)) space

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From Fig. 9, we know that the projective synchronization is achieved with controller (29). By comparing Figs. 5, 6, 11 and 12, we find, by using VUAFC, chattering phenomenon is eliminated. Meanwhile, from Figs. (4) and (10), we find that s cannot stay at zero.

5 Conclusion

In this paper, we have discussed fuzzy sliding mode projective synchronization of hyperjerk system with disturbances. Disturbance observer has been developed for the low time-varying disturbance. By quantitative analysis, given appropriate observer parameters, the disturbance can be compensated by the observer. For the chattering caused by the white noise and the persistent observer error, we have used VUAFC method to smooth it. VUAFSMPS method which has been designed is built on sliding mode control, disturbance observer and variable universe adaptive fuzzy control. Simulation results have shown that this control method can reduce chattering significantly without increasing the amount of control.