1 Introduction

A chaotic system has disordered and unpredictable behavior; these behaviors occur in mechanical oscillators such as pendulums [1] and some chemical reactions [2]. One of the main characteristics of chaotic systems is that it does not repeat their previous behavior. Chaotic dynamics describe many natural and physical systems. In recent years, scientists have paid attention to chaotic systems and synchronization in different sciences for secure communication and other goals. In synchronization of the chaotic systems have been employed linear feedback [3], nonlinear feedback [4], adaptive control [5], PID [6], sliding mode control [7, 8], backstepping [9], model predictive controller [10], and optimal control [11, 12] methods. The fast synchronization of symmetric Hénon maps using adaptive symmetry control is one of the newest researches in the field of synchronization of chaotic systems [13]. In this study, the feedback controllers are used for the conventional Hénon map and adaptive Hénon map. Adaptive symmetry control is also used to provide a secure communication system [14].

General synchronization is also a developing topic [15, 16]. In [15], an adaptive generalized synchronization method is presented between the circuit and computer implementations of the Rössler system. The parameters re-identification method is used to develop an accurate synchronization method between digital and analog chaotic systems [17]. Chemical reaction kinetic is a science investigating the rate of chemical reactions and effective agents, e.g., catalysts, pressure, temperature, reactants concentration, etc., on them and analyzing the experimental data; this information obtained is registered as quantitative kinetic information. Chemical reaction kinetics is one of the important aspects of research and development in research institutes, industrial labs, and academic centers [18, 19].

Globally fixed-time stability means that all the system state variables converge to zero in a finite time and remain zero after the time exactly. In [20], the authors have investigated robust finite-time global synchronization of chaotic systems with different orders. In [21], a diverse structure based on a second-order sliding mode control algorithm has been employed to synchronize two uncertain chemical chaotic systems. In [22], the authors have expressed stable indirect adaptive interval type-2 Fuzzy Sliding-based Control and synchronization of two different chaotic systems.

Adaptive synchronization of two different chaotic systems containing mutually Lipschitz nonlinearities subject to state time delays is presented in [23]. Also, [24] has investigated optimal and adaptive control for 3D chaotic and 4D hyper-chaotic systems. In [25], generalized synchronization of different dimensional integer-order and fractional-order chaotic systems has been presented. You et al. have designed a Lag synchronization system for two different delayed chaotic systems [26]. In 2016 Song et al. presented an adaptive synchronization system for two time-delayed fractional-order chaotic systems with different orders and structures [27]. In [28], multi-switching combination synchronization methods control three different chaotic systems. Authors have studied modified projective phase synchronization of chaotic complex nonlinear systems [29].

Chattering is an adverse phenomenon that causes reduced performance and increases heat loss in power circuits. Recently, designers have focused on removing or reducing the undesirable effects of this phenomenon. In [30, 31], the control input is designed with this aim. Complete removal of this phenomenon is one of the aims of this paper.

In the field of chaotic chemical systems, Changin and Yusen have presented ways to chaos model and control [32]. In 1996 Geysermans and Baras expressed to modeling and controlling chaotic chemical systems in the macroscopic scale as well as probability actuator structure is designed, and pedicure simulation on these systems was investigated [33]. Jia et al. have studied adaptive output feedback control of nonlinear time-delay systems with application to chemical reactor systems [34].

This paper presents a Nonsingular Fixed-time Sliding Mode Control (NFSMC) for synchronizing the Willamowski–Rossler reaction chaotic system. The proposed control method solves some challenging problems in the control systems to provide a fast, robust, accurate, and chattering-free control signal for the proposed chaotic master–slave system. The proposed method will be compared with a Nonsingular Terminal Sliding Mode Control (NTSMC) to show the power of the proposed control system. The novelties of the proposed method can be listed as follows:

  • synchronizing the chaotic chemical systems proved by the fixed-time Lyapunov-based stabilization method,

  • providing a fast, smooth, and accurate control signal,

  • generating the implementable control signals,

  • providing a chattering-free method,

  • robustness against the model’s uncertainties and external disturbances.

2 Mathematic definitions and lemmas

Definition 1

Function \(si{g}^{a}\left(x\right)\) with the relation between absolute function \(|x|\) and symbol function for \(si{g}^{a}\left(x\right)={\left|x\right|}^{a}sgn(x)\) define. The Signum function is defined as follows [35, 36]:

$$sgn\left(x\right)=\left\{\begin{array}{c}1 ; x>0\\ 0 ; x=0\\ -1 ; x<0\end{array}\right.$$
(1)

Definition 2

The relation between absolute and symbol functions is for \(\left|x\right|=xsgn(x)\) [37].

Lemma 1

For a nonlinear system \(\dot{x}=f\left(x\right), f\left(0\right)=0, x\in D\subseteq {\mathfrak{R}}^{n}, x\left(0\right)={x}_{0}\) and assuming the constants \({\rho }_{1}\) to \({\rho }_{4}\), \({\rho }_{1}>0, {\rho }_{2}>0, {\rho }_{3}>1, {\rho }_{4}=1-\frac{1}{2{\rho }_{3}}, {\rho }_{5}=1+\frac{1}{2{\rho }_{3}}\) and Lyapunov function \(V\left( x \right): \Re^{n} \to \Re^{ + } \cup\) [4] for scalar continuous radially unbounded function has been, therefore, is \(\dot{V}(x)\le -{\rho }_{1}{V}^{{\rho }_{4}}\left(x\right)-{\rho }_{2}{V}^{{\rho }_{5}}(x)\). So equilibrium \(x=0\) this system is globally fixed-time stable, and state variables of this system converge from each initial condition to zero and the upper bound of its settling time is for \(T\le \pi {\rho }_{3}{\left(\sqrt{{\rho }_{1}{\rho }_{2}}\right)}^{-1}\) and after the time, the system's state variables are exactly zero [38].

Lemma 2

Considering constants \(\left({a}_{1}, {a}_{2}, \dots , {a}_{n}\right)\in \mathfrak{R}\) and choosing \(0<q<2\), then have: \({\left|{a}_{1}\right|}^{q}+{\left|{a}_{2}\right|}^{q}+\dots + {\left|{a}_{n}\right|}^{q}\ge {\left({a}_{1}^{2}+{a}_{2}^{2}+\dots +{a}_{n}^{2}\right)}^{\frac{q}{2}}\) [39, 40].

3 Expression model of the chemical reactions

This section presents the dynamic model of the chaotic chemical system of the Willamowski–Rossler. This model will be used to test the proposed control method in the next sections.

Willamowski and Rossler's (WR) dynamic model for chaotic chemical systems that includes a range of chaotic dynamical behavior is presented. As mentioned in the introduction, this model is used when we have oscillating chemical reactions. All the reactions have included "Elementary" Steps. In Eq. (2) Chemical model is to be expressed.

$$\begin{array}{c}{A}_{1}+X \begin{array}{c}{k}_{1}\\ \rightleftharpoons \\ {k}_{-1}\end{array}2X\\ X+Y \begin{array}{c}{k}_{2}\\ \to \end{array} 2Y\\ \begin{array}{c}{A}_{5}+Y\begin{array}{c}{k}_{3}\\ \to \end{array} {A}_{2}\\ X+Z\begin{array}{c}{k}_{4}\\ \to \end{array}{A}_{3}\\ {A}_{4}+Z\begin{array}{c}{k}_{5}\\ \rightleftharpoons \\ {k}_{-5}\end{array}2Z\end{array}\end{array}$$
(2)

This model involves two constituents, \(X\) and\(Z\), for two autocatalytic steps and three steps X, Z and Y, which is the third constituent. Species\({A}_{1}\),\({A}_{4}\), and \({A}_{5}\) are reactants and\({A}_{2}\), and \({A}_{3}\) are final products whose concentration is fixed to maintain the non-equilibrium thermodynamics of the system. Also, \({k}_{\pm i} ,(i=1, \dots ,5)\) are rate constants that, in modeling for chaotic dynamics \({k}_{-2}, {k}_{-3}\) and \({k}_{-4}\) have been zero [32, 33]. In [32, 33], an ideal mix and a well-stirred reactor is assumed, and thus, equations of macroscopic rate are:

$$\left\{\begin{array}{l}\dot{x}={a}_{1}x-{k}_{-1}{x}^{2}-xy-xz\\ \dot{y}=xy-{a}_{5}y \\ \dot{z}={a}_{4}z-xz-{k}_{-5}{z}^{2}\end{array}\right. $$
(3)

where mole fractions of \(X, Y\) and \(Z\) are displayed \(x, y\) and \(z,\) respectively and parameters \({a}_{1},{a}_{2}\) and \({a}_{3}\) refer to \({k}_{1}, {k}_{3}\) and \({k}_{5}\). \({k}_{2}\) and \({k}_{4}\) are selected equal to one. In Fig. 1, this model is shown in conditions of \({a}_{1}=30, {a}_{4}=16.5, {a}_{5}=10, {k}_{-1}=0.415, {k}_{-5}=0.5\) that is a chaotic system. In Figs. 2, 3 and 4, the system states are plotted in the time domain. Also, the following parameters can be calculated for this system.

Fig. 1
figure 1

The 3D illustration of the WR model

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Fig. 2
figure 2

\(x\) state of the WR model

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Fig. 3
figure 3

\(y\) state of the WR model

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Fig. 4
figure 4

\(z\) state of the WR model

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Equilibrium points:

$$\left(x, y,z\right)=\left\{\begin{array}{l}\begin{array}{l}\begin{array}{l}\left(0, 0, 0\right)\\ \left({a}_{5}, {a}_{1}-{a}_{5}{k}_{-1}, 0\right)\end{array}\\ \left({a}_{5}, -\frac{{a}_{4}-{a}_{5}-{a}_{1}{k}_{-5}+{a}_{5}{k}_{-1}{k}_{-5}}{{k}_{-5}},\frac{{a}_{4}-{a}_{5}}{{k}_{-5}}\right) \\ \left(-\frac{{a}_{4}-{a}_{1}{k}_{-5}}{{k}_{-1}{k}_{-5}-1}, 0, -\frac{{a}_{1} - {a}_{4}{k}_{-1}}{{k}_{-1}{k}_{-5}-1}\right) \end{array}\\ \left(\frac{{a}_{1}}{{k}_{-1}}, 0, 0\right) \\ (0, 0, \frac{{a}_{4}}{{k}_{-5}})\end{array}\right. $$
(4)

Lyapunov exponent: \({\lambda }_{1}=3.0550\times {10}^{-4}, {\lambda }_{2}=3.0316\times {10}^{-4}\), and \({\lambda }_{3}=2.9968\times {10}^{-4}\).

Kaplan-Yorke coefficient: \(D=2+ \frac{{\lambda }_{1}+ {\lambda }_{2}}{\left| {\lambda }_{3}\right|}=2+\frac{6.0866\times {10}^{-4}}{2.9968\times {10}^{-4}}=4.03103\).

Consider two similar WR systems as follows:

$$\left\{\begin{array}{l}{\dot{x}}_{1}={a}_{1}{x}_{1}-{k}_{-1}{x}_{1}^{2}-{x}_{1}{y}_{1}-{x}_{1}{z}_{1}\\ {\dot{y}}_{1}={x}_{1}{y}_{1}-{a}_{5}{y}_{1} \\ {\dot{z}}_{1}={a}_{4}{z}_{1}-{x}_{1}{z}_{1}-{k}_{-5}{z}_{1}^{2}\end{array}\right. $$
(5)

and

$$\left\{\begin{array}{l}{\dot{x}}_{2}={a}_{1}{x}_{2}-{k}_{-1}{x}_{2}^{2}-{x}_{2}{y}_{2}-{x}_{2}{z}_{2}+{d}_{1}(t,x)+{u}_{1}\\ {\dot{y}}_{2}={x}_{2}{y}_{2}-{a}_{5}{y}_{2}+{d}_{2}(t,x)+{u}_{2} \\ {\dot{z}}_{2}={a}_{4}{z}_{2}-{x}_{2}{z}_{2}-{k}_{-5}{z}_{2}^{2}+{d}_{3}(t,x)+{u}_{3}\end{array}\right. $$
(6)

where the system in Eq. (5) is the master and the system in Eq. (6) is the slave system, and the aim is that the state variables of the slave system track the state variables of the master, therefore three control inputs \({u}_{1}, {u}_{2}\) and \({u}_{3}\) are designed in some ways by NFSMC that this aim achieves zero.

At first, considering \({e}_{1}={x}_{2}-{x}_{1}, {e}_{2}={y}_{2}-{y}_{1}\) and \({e}_{3}={z}_{2}-{z}_{1}\) is clear, if \({e}_{\mathrm{1,2},3}\to 0\) in the fixed time, the goal of control has been met, and as a result, in the fixed time \({x}_{2}\to {x}_{1},{y}_{2}\to {y}_{1}\) and \({z}_{2}\to {z}_{1}\).

$$\left\{\begin{array}{l}{\dot{e}}_{1}={f}_{11}-{f}_{21}+{d}_{1}\left(t,x\right)+{u}_{1} \\ {\dot{e}}_{2}={f}_{12}-{f}_{22}+{d}_{2}(t,x)+{u}_{2} \\ \begin{array}{l}{\dot{e}}_{3}={f}_{13}-{f}_{23}+{d}_{3}\left(t,x\right)+{u}_{3} \\ \begin{array}{l}{f}_{11}=\left({a}_{1}{x}_{2}-{k}_{-1}{x}_{2}^{2}-{x}_{2}{y}_{2}-{x}_{2}{z}_{2}\right) \\ {f}_{12}=\left({x}_{2}{y}_{2}-{a}_{5}{y}_{2}\right) \\ {f}_{13}=\left({a}_{4}{z}_{2}-{x}_{2}{z}_{2}-{k}_{-5}{z}_{2}^{2}\right) \end{array}\\ \begin{array}{l}{f}_{21}=\left({a}_{1}{x}_{1}-{k}_{-1}{x}_{1}^{2}-{x}_{1}{y}_{1}-{x}_{1}{z}_{1}\right) \\ {f}_{22}=\left({x}_{1}{y}_{1}-{a}_{5}{y}_{1}\right) \\ {f}_{23}=({a}_{4}{z}_{1}-{x}_{1}{z}_{1}-{k}_{-5}{z}_{1}^{2} )\end{array}\end{array}\end{array}\right.$$
(7)

As in most systems, parameters cannot be determined exactly, creating uncertainties and external disturbances in systems. Therefore, assuming this is also the same event could happen. \(d_{1,2,3} \left( {t,x} \right)\) are the model of the system's uncertainties and external disturbances models that have been added to equations. In many cases, despite uncertainties and external disturbances, the exact model is not available, but the upper bounds of the model can be predicted. Equations (8) and (9) uncertainties and external disturbances model and their derivations with upper bounds are illustrated.

$$ \left\{ {\begin{array}{*{20}c} {\left| {d_{1} \left( {t,x} \right)} \right| \le \eta_{11} } \\ {\left| {d_{2} \left( {t,x} \right)} \right| \le \eta_{21} } \\ {\left| {d_{3} \left( {t,x} \right)} \right| \le \eta_{31} } \\ \end{array} } \right. $$
(8)

and

$$ \left\{ {\begin{array}{*{20}c} {\left| {\dot{d}_{1} \left( {t,x} \right)} \right| \le \eta_{12} } \\ {\left| {\dot{d}_{2} \left( {t,x} \right)} \right| \le \eta_{22} } \\ {\left| {\dot{d}_{3} \left( {t,x} \right)} \right| \le \eta_{32} } \\ \end{array} } \right. $$
(9)

The upper bounds of the uncertainties and external disturbances will be used in the next section to design the proposed control method. From the mathematical point of view, Eqs. (8) and (9) mean that the uncertainties and external disturbances are bounded, and their upper bounds are available.

4 Designing of the proposed control method

In this section, proportion input control for the presented system in Eq. (7) will be designed and proven using the NFSMC method. Proving input control by the NFSMC method includes two parts. In the first part (reaching phase), the stability of the designed sliding surface can be proven, and then in the second part (settling phase), the reaching of the sliding surface should be proven; since this paper is intended to prove the fixed time, in both parts, stability for a fixed time should be proof.

Theorem 1

The nonlinear system (7), the conditions described in (8 and 9) and the sliding surfaces defined in Eq. (10), and the designed control input in Eq. (11) is stable for a fixed time, and the upper bound of stability time (settling time) achieving of the sliding surface is for \(T \le \mathop \sum \nolimits_{i = 1}^{3} \pi r_{3} \left( {\sqrt {\left( {k_{i1} k_{i2} } \right)} } \right)^{ - 1}\).

$$ \left\{ {\begin{array}{*{20}c} {s_{1} = \dot{e}_{1} + c_{11} sig^{{n_{11} }} \left( {e_{1} } \right) + c_{12} sig^{{n_{12} }} \left( {e_{1} } \right)} \\ {s_{2} = \dot{e}_{2} + c_{21} sig^{{n_{21} }} \left( {e_{2} } \right) + c_{22} sig^{{n_{22} }} \left( {e_{2} } \right)} \\ {s_{3} = \dot{e}_{3} + c_{31} sig^{{n_{31} }} \left( {e_{3} } \right) + c_{32} sig^{{n_{32} }} \left( {e_{3} } \right)} \\ \end{array} } \right. $$
(10)

where \(c_{i1} ,c_{i2}\) are positive and constant numbers and \(n_{i1} , n_{i2} , \left( {i = 1, 2, 3} \right)\) are positive numbers and lower than one, which follow in the equation \( \left\{ {\begin{array}{*{20}c} {n_{i1} = N } \\ {n_{i2} = \frac{N}{2 - N}} \\ \end{array} } \right.;\;\left( {N = \left( {0,1} \right)} \right)\).

$$ \left\{ {\begin{array}{*{20}c} {u_{i} = u_{{eq_{i} }} + u_{{r_{i} }} } \\ {u_{{eq_{i} }} = - f_{i} - c_{i1} sig^{{n_{i1} }} \left( {e_{i} } \right) - c_{i2} sig^{{n_{i2} }} \left( {e_{i} } \right) } \\ {\dot{u}_{{r_{i} }} = - k_{i1} sig^{{\alpha_{i1} }} \left( {s_{i} } \right) - k_{i2} sig^{{\alpha_{i2} }} \left( {s_{i} } \right) - \eta_{i2} sgn\left( {s_{i} } \right)} \\ \end{array} } \right. $$
(11)

where \(i = 1, 2, 3\) and \(k_{i1} , k_{i2}\) are positive functions, \(\alpha_{i1} , \alpha_{i2}\) are positive numbers and lower than one and \(f_{i}\) presented in Eq. (12).

$$ \left\{ {\begin{array}{*{20}c} {f_{1} = \left( {f_{11} } \right) - \left( {f_{21} } \right) } \\ {f_{2} = \left( {f_{12} } \right) - \left( {f_{22} } \right) } \\ {f_{3} = \left( {f_{13} } \right) - \left( {f_{23} } \right) } \\ \end{array} } \right. $$
(12)

Proof

To prove the first part, fixed-time stability sliding surfaces must be proof. If we assume that the system reached the sliding surface, which means \(s_{i} = 0\). Then we have \(\dot{e}_{i} + c_{i1} sig^{{n_{i1} }} \left( {e_{i} } \right) + c_{i2} sig^{{n_{i2} }} \left( {e_{i} } \right) = 0\). In this case, it is clear that we can write \(\dot{e}_{i} = - c_{i1} sig^{{n_{i1} }} \left( {e_{i} } \right) - c_{i2} sig^{{n_{i2} }} \left( {e_{i} } \right)\). Now a simple Lyapunov function as \(V_{i} \left( x \right) = \frac{1}{2}e_{i}^{2}\) can help prove the settling phase. The derivation of the Lyapunov function will be as \(\dot{V}\left( x \right) = e_{i} \dot{e}_{i} = - c_{i1} \left| {e_{i} } \right|^{{n_{i1} + 1}} - c_{i2} \left| {e_{i} } \right|^{{n_{i2} + 1}}\). It can be written as \(\dot{V}\left( x \right) = - c_{i1} V^{{\frac{{n_{i1} + 1}}{2}}} - c_{i2} V^{{\frac{{n_{i2} + 1}}{2}}}\). Considering Lemma 1, the fixed-time stability of the reaching phase is done.

To prove the second part, consider the Lyapunov function \(V\left( x \right) = \frac{1}{2}\mathop \sum \nolimits_{i = 1 }^{3} s_{i}^{2}\) which this Lyapunov function is eligibly presented Lyapunov function in lemma 1 that its derivative is \(\dot{V}\left( x \right) = \mathop \sum \nolimits_{i = 1 }^{3} s_{i} \dot{s}_{i}\). To continue with applying designed input to Eq. (7) and replacing \({\dot{\text{e}}}_{{\text{i}}}\) s in sliding surface equations conclude \(s_{i} = u_{{r_{i} }} + d_{i}\) and then the derivative of this new sliding surface has \(\dot{s}_{i} = \dot{u}_{{r_{i} }} + \dot{d}_{i}\). To continue, this equation should replace in \(\dot{V}\left( x \right)\) equation which will be the result:

$$ \dot{V}\left( x \right) = \mathop \sum \limits_{i = 1 }^{3} s_{i} ( - k_{i1} sig^{{\alpha_{i1} }} \left( {s_{i} } \right) - k_{i2} sig^{{\alpha_{i2} }} \left( {s_{i} } \right) - \eta_{i2} sgn\left( {s_{i} } \right) + \dot{d}_{i} ) $$
(13)

after production \({\text{s}}_{{\text{i}}}\) in parentheses and according to \(s_{i} \dot{d}_{i} \le \left| {s_{i} } \right|\left| {\dot{d}_{i} } \right|\) and \(\left| {\left| {\dot{d}_{1} \left( {t,x} \right)} \right|} \right| \le \eta_{12}\) also, according to Definitions 1 and 2 conclude:

$$ \dot{V}\left( x \right) \le \mathop \sum \limits_{i = 1 }^{3} - k_{i1} \left| {s_{i} } \right|^{{\alpha_{i1} + 1}} - k_{i2} \left| {s_{i} } \right|^{{\alpha_{i2} + 1}} $$
(14)

according to \(\alpha_{i1} , \alpha_{i2}\), that are numbers between zero and one, as a result \(\alpha_{i2} + 1, \alpha_{i2} + 1\) are numbers between zero and two, and according to lemma2 can write \(\dot{V}\left( x \right) \le \mathop \sum \nolimits_{i = 1 }^{3} - \left( {\sqrt 2 } \right)^{{\alpha_{i1} + 1}} k_{i1} s_{i}^{{\frac{{\alpha_{i1} + 1}}{2}}} - \left( {\sqrt 2 } \right)^{{\alpha_{i2} + 1}} k_{i2} s_{i}^{{\frac{{\alpha_{i2} + 1}}{2}}}\). In the end, control parameters choose as follows:

$$ r_{1} = \left( {\sqrt 2 } \right)^{{\alpha_{i1} + 1}} k_{i1} > 0,r_{2} = \left( {\sqrt 2 } \right)^{{\alpha_{i2} + 1}} k_{i2} > 0, r_{4} = \alpha_{i1} = 1 - \frac{1}{{r_{3} }}, r_{5} = \alpha_{i2} = 1 + \frac{1}{{{\varvec{r}}_{3} }} $$
(15)

where \({\text{r}}_{3} > 1\), with these options can conclude:

$$ \dot{V}\left( x \right) \le - r_{1} V^{{r_{4} }} - r_{2} V^{{r_{5} }} $$
(16)

according to Lemma 1, system (6) for fixed time is stable, and its stability time (or its settling time) is \(T \le \pi r_{3} \left( {\sqrt {\left( {r_{1} r_{2} } \right)} } \right)^{ - 1}\). The proof is done

Remark 1.

If the aim of controller design is in order to anti-synchronization, dynamic errors define as \(e_{1} = x_{2} + x_{1} , e_{2} = y_{2} + y_{1}\), and \(e_{3} = z_{2} + z_{1}\), and the process of designing in a way that was to be repeated. For this goal, the control input is as follows:

$$ \left\{ {\begin{array}{*{20}c} {u_{i} = u_{{eq_{i} }} + u_{{r_{i} }} } \\ {u_{{eq_{i} }} = f_{i} - c_{i1} sig^{{n_{i1} }} \left( {e_{i} } \right) - c_{i2} sig^{{n_{i2} }} \left( {e_{i} } \right) } \\ {\dot{u}_{{r_{i} }} = - k_{i1} sig^{{\alpha_{i1} }} \left( {s_{i} } \right) - k_{i2} sig^{{\alpha_{i2} }} \left( {s_{i} } \right) - \eta_{i2} sgn\left( {s_{i} } \right)} \\ \end{array} } \right. $$
(17)

where:

$$ \left\{ {\begin{array}{*{20}c} {f_{1} = - \left( {f_{11} } \right) - \left( {f_{21} } \right) } \\ {f_{2} = - \left( {f_{12} } \right) - \left( {f_{22} } \right) } \\ {f_{3} = - \left( {f_{13} } \right) - \left( {f_{23} } \right) } \\ \end{array} } \right. $$
(18)

for obtaining more information about anti-synchronization, see [41, 42].

Proposition 1

The NTSMC control method presented in [43] can be modified for this paper’s goal. The control inputs of this method can be written as follows:

$$ \left\{ {\begin{array}{*{20}c} {u_{i} = u_{{eq_{i} }} + u_{{r_{i} }} } \\ {u_{{eq_{i} }} = f_{i} - c_{i1} sig^{{n_{i1} }} \left( {e_{i} } \right) } \\ {\dot{u}_{{r_{i} }} = - k_{i1} sig^{{\alpha_{i1} }} \left( {s_{i} } \right) - \eta_{i2} sgn\left( {s_{i} } \right)} \\ \end{array} } \right. $$
(19)

where \(i = 1, 2, 3\) and \(k_{i1}\) are positive functions, \(\alpha_{i1}\) are positive numbers and lower than one and \(f_{i}\) presented in Eq. (12). Also, the sliding surfaces are as follows:

$$ \left\{ {\begin{array}{*{20}c} {s_{1} = \dot{e}_{1} + c_{11} sig^{{n_{11} }} \left( {e_{1} } \right)} \\ {s_{2} = \dot{e}_{2} + c_{21} sig^{{n_{21} }} \left( {e_{2} } \right)} \\ {s_{3} = \dot{e}_{3} + c_{31} sig^{{n_{31} }} \left( {e_{3} } \right)} \\ \end{array} } \right. $$
(20)

where \(c_{i1}\) are positive constants, and \(n_{i1} , \left( {i = 1, 2, 3} \right)\) are positive numbers lower than one.

5 Numerical simulation

In this section, the proposed control method is tested and compared with an NTSMC presented in [43] and explained in Proposition 1. The solver method for the simulation is ode4 (Runge–Kutta) with 0.001 sample time. The control parameters are selected as the same numbers for both methods as follows:

$$ c_{i1} = 7,c_{i2} = 10,k_{i1} = k_{i2} = 0.5,N = 0.7, r_{3} = 1.2 $$
(21)

For a logical comparison, two performance criteria of IAE (integral of the absolute value of error) and ITSV (integral of the time square value) are defined as follows:

$$ \left\{ {\begin{array}{*{20}c} {IAE_{e} = \smallint \left| {e\left( t \right)} \right|dt } \\ {ITSV_{u} = \smallint tu^{2} \left( t \right)dt} \\ \end{array} } \right. $$
(22)

The defined performance criteria are calculated for the synchronization errors and the control signals. The results of the calculation are presented in Table 1.

Table 1 The comparison results
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The results are shown in the same figure for every variable to show the difference between control methods. Figure 5 shows the synchronization errors of both control methods. Figures 6, 7, and 8 show the system’s states. The control inputs are shown in Fig. 9. The 3D illustration of two chaotic systems is also shown in Fig. 10.

Fig. 5
figure 5

The synchronization errors of two control methods

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Fig. 6
figure 6

The variable \({\varvec{x}}\) of two control methods

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Fig. 7
figure 7

The variable \({\varvec{y}}\) of two control methods

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Fig. 8
figure 8

The variable \({\varvec{z}}\) of two control methods

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Fig. 9
figure 9

3D illustration of two chaotic systems in two methods

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Fig. 10
figure 10

The designed control inputs

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The simulation figures and the comparison results show the power of the proposed control method compared with the Finite-time (Terminal) SMC method. All the comparison results in Table 1 have better values, which means the proposed control method is faster, more accurate, and cheaper (as the ITSV values are less than).

6 Conclusion

This paper presented a Nonsingular Fixed-time Sliding Mode Control (NFSMC) method to synchronize two Willamowski–Rossler chaotic systems. The controller is accurate, fast, chatter-free, practicable, and robust against uncertainties and external disturbances. The proposed control method has been tested in Simulink, and a comparison has been made with the Nonsingular Terminal Sliding Mode Control (NTSMC) method. The comparison results have shown that the proposed control method works faster, more accurately, and cheaper. Two performance criteria, IAE and ITSV, have been used to compare the methods. One of the possible fields for the next studies is designing the disturbance observer to estimate the uncertainties and external disturbances and use the estimated values in the controller.