TY - JOUR
AU - Lin, Qian
AB - Abstract This paper deals with the global finite-time synchronization of a class of third-order chaotic systems with some intersecting nonlinearities, which cover many famous chaotic systems. First, a simple, continuous and dimension-reducible control by the name of the variable-substitution and feedback control is designed to construct a master–slave finite-time synchronization scheme. Then, a global finite-time synchronization criterion for the synchronization scheme is proven and the synchronization time is analytically estimated. Subsequently, the criterion and optimization technique are applied to the well-known brushless direct current motor (BLDCM) system and the classic Lorenz system, respectively, further obtaining some new optimized synchronization criteria in the form of algebra. Two numerical examples for the BLDCM system and a numerical example for the Lorenz system are simulated and analyzed to verify the effectiveness of the theoretical results obtained in this paper. 1. Introduction In the past three decades, chaos synchronization has become an interesting and important research branch of nonlinear science due to its potential applications in many fields such as secret communication (Chithra & Raja Mohamed, 2017; Vaseghi et al., 2018), bionomics (Blasius et al., 1999), laser (Wedekind & Parlitz, 2008) and so on (Du et al., 2011; Huang et al., 2017). The pioneering work on chaos synchronization can be traced back to the work of Pecora & Carroll (1990), when an extraordinary control technique, called variable-substitution control (VSC) or Pecora-Carroll control (PC) control later, was proposed to realize the asymptotical synchronization of the master–slave chaotic systems by substituting the signal of one or several state variables of the slave system with the counterpart of the corresponding state variables of the master one. Motivated by this work, many researchers have made unremitting efforts to investigate chaos synchronization by introducing various control techniques such as VSC (Chen et al., 2009; Liu et al., 2009; Pecora & Carroll, 1990), linear feedback control (Chen et al., 2010), impulsive control (Chen et al., 2011), adaptive control (Lei et al., 2007), etc. (Aghababa & Akbari, 2012; Ahmad et al., 2016a; Cai et al., 2007; Njah et al., 2010). It is noteworthy that those control techniques as above can only ensure the asymptotical synchronization of two chaotic systems, that is, when the time tends to be infinite, the synchronization of two chaotic systems can be achieved. However, in some real-world applications such as attitude tracking control of spacecraft (Du et al., 2011), synchronization for permanent-magnet synchronous motor (Chen et al., 2015) and robot manipulators (Su & Zheng, 2011), it is more reasonable that the master–slave systems should be synchronized in finite or predetermined time. For this reason, the finite-time synchronization of chaotic systems has emerged as a valuable research topic and has aroused wide attention from the researchers. Fig. 1. Open in new tabDownload slide Open in new tabDownload slide Open in new tabDownload slide The comparison between global finite-time synchronization and globally asymptotical synchronization for Example 1. (a) The master BLDCM system; (b) The uncontrolled slave BLDCM system; (c)-(d) The synchronization evolutions and the control inputs of two types of synchronization via the VSFC (20) with |$\alpha =7/10$| and |$K=\mathit{\operatorname{diag}}\Big\{8.2,27.5\Big\}$| and via the VSC |${x}_d={x}_1$|⁠; (e)-(f) The synchronization evolutions and the control inputs of two types of synchronization via the VSFC (27) with |$\alpha =3/5$| and |$K=\mathit{\operatorname{diag}}\Big\{2,2\Big\}$| and via the VSC |${x}_d={x}_3$|⁠. Fig. 1. Open in new tabDownload slide Open in new tabDownload slide Open in new tabDownload slide The comparison between global finite-time synchronization and globally asymptotical synchronization for Example 1. (a) The master BLDCM system; (b) The uncontrolled slave BLDCM system; (c)-(d) The synchronization evolutions and the control inputs of two types of synchronization via the VSFC (20) with |$\alpha =7/10$| and |$K=\mathit{\operatorname{diag}}\Big\{8.2,27.5\Big\}$| and via the VSC |${x}_d={x}_1$|⁠; (e)-(f) The synchronization evolutions and the control inputs of two types of synchronization via the VSFC (27) with |$\alpha =3/5$| and |$K=\mathit{\operatorname{diag}}\Big\{2,2\Big\}$| and via the VSC |${x}_d={x}_3$|⁠. Fig. 2. Open in new tabDownload slide Open in new tabDownload slide Open in new tabDownload slide The comparison between global finite-time synchronization and globally asymptotical synchronization for Example 2. (a) The master BLDCM system; (b) The uncontrolled slave BLDCM system; (c)-(d) The synchronization evolutions and the control inputs of two types of synchronization via the VSFC (23) with |$\alpha =2/5$| and |$K=\mathit{\operatorname{diag}}\Big\{3,3\Big\}$| and via the VSC |${x}_d={x}_2$|⁠; (e)-(f) The synchronization evolutions and the control inputs of two types of synchronization via the VSFC (27) with |$\alpha =1/2$| and |$K=\mathit{\operatorname{diag}}\Big\{1.2,3.1\Big\}$| and via the VSC |${x}_d={x}_3$| Fig. 2. Open in new tabDownload slide Open in new tabDownload slide Open in new tabDownload slide The comparison between global finite-time synchronization and globally asymptotical synchronization for Example 2. (a) The master BLDCM system; (b) The uncontrolled slave BLDCM system; (c)-(d) The synchronization evolutions and the control inputs of two types of synchronization via the VSFC (23) with |$\alpha =2/5$| and |$K=\mathit{\operatorname{diag}}\Big\{3,3\Big\}$| and via the VSC |${x}_d={x}_2$|⁠; (e)-(f) The synchronization evolutions and the control inputs of two types of synchronization via the VSFC (27) with |$\alpha =1/2$| and |$K=\mathit{\operatorname{diag}}\Big\{1.2,3.1\Big\}$| and via the VSC |${x}_d={x}_3$| Fig. 3. Open in new tabDownload slide Open in new tabDownload slide Open in new tabDownload slide The comparison between global finite-time synchronizations for Example 3. (a) The master Lorenz system; (b) The uncontrolled slave Lorenz system; (c)-(d) The synchronization evolutions and the control inputs of two finite-time synchronizations via the ANC (39) with |$\alpha =1/2$|⁠,|${k}_1=20$|⁠,|${k}_2=20$| and |${k}_3=4$| designed in Li & Tian (2003) and via the VSFC (30) with |$\alpha =1/2$| and |$K=\mathit{\operatorname{diag}}\Big\{20,4\Big\}$|⁠; (e)-(f) The synchronization evolutions and the control inputs of two finite-time synchronizations via the ANC (39) with |$\alpha =1/2$|⁠,|${k}_1=20$|⁠,|${k}_2=20$| and |${k}_3=4$| designed in Li & Tian (2003) and via the VSFC (33) with |$\alpha =1/2$| and |$K=\mathit{\operatorname{diag}}\Big\{20,4\Big\}$|⁠. Fig. 3. Open in new tabDownload slide Open in new tabDownload slide Open in new tabDownload slide The comparison between global finite-time synchronizations for Example 3. (a) The master Lorenz system; (b) The uncontrolled slave Lorenz system; (c)-(d) The synchronization evolutions and the control inputs of two finite-time synchronizations via the ANC (39) with |$\alpha =1/2$|⁠,|${k}_1=20$|⁠,|${k}_2=20$| and |${k}_3=4$| designed in Li & Tian (2003) and via the VSFC (30) with |$\alpha =1/2$| and |$K=\mathit{\operatorname{diag}}\Big\{20,4\Big\}$|⁠; (e)-(f) The synchronization evolutions and the control inputs of two finite-time synchronizations via the ANC (39) with |$\alpha =1/2$|⁠,|${k}_1=20$|⁠,|${k}_2=20$| and |${k}_3=4$| designed in Li & Tian (2003) and via the VSFC (33) with |$\alpha =1/2$| and |$K=\mathit{\operatorname{diag}}\Big\{20,4\Big\}$|⁠. In the past decade, some researchers have made some important contributions to the field of the finite-time synchronization of chaotic systems, obtaining some great results (Aghababa, 2012; Aghababa et al., 2011; Aghababa & Aghababa, 2012, 2013a,b; Aghababa & Feizi, 2012a,b; Ahmad et al., 2016b; Ao et al., 2019; Cai et al., 2011; Dong & Xian, 2016; Khanzadeh & Pourgholi, 2016; Li & Cao, 2015; Li & Tian, 2003; Li & Wu, 2016; Liu, 2012; Luo & Su, 2017; Ni et al., 2014; Sangpet & Kuntanapreeda, 2020; Tran & Kang, 2015a,b; Vincent & Guo, 2011; Wang et al., 2009a,b, 2019, 2020; Wu et al., 2016; Yang & Wu, 2012; Zhang et al., 2017). In Liu (2012); Wang et al. (2009b), the control Lyapunov function (CLF)-based control was employed to accomplish finite-time synchronization of two unified chaotic systems and finite-time synchronization of two four-dimensional Rabinovich hyperchaotic systems, respectively. With the aid of the sliding mode control, the finite-time synchronizations of some fractional-order chaotic systems were studied in Aghababa (2012); Khanzadeh & Pourgholi (2016); Li & Wu (2016). The authors in Aghababa et al. (2011); Aghababa & Feizi (2012a,b); Ni et al. (2014) have designed several sliding mode controls to successfully realize the finite-time synchronizations of two different chaotic systems, respectively. In Tran & Kang (2015b), a novel observer-based control was presented to investigate the projective finite-time synchronization of chaotic (hyperchaotic) systems with uncertainties and external disturbances. In Li & Cao (2015), the authors have devised an observer-based output feedback control to realize the finite-time synchronization between two high-order nonlinear systems. With the help of the active nonlinear control (ANC), the finite-time synchronization of two chaotic systems was achieved in Li & Tian (2003). The authors of Cai et al. (2011) utilized the ANC to acquire the generalized finite-time synchronization of chaotic systems with different order. A simple adaptive control has been put forward by Vincent & Guo (2011) to realize the finite-time synchronization of chaotic (hyperchaotic) systems. In the work of Wu et al. (2016), by taking the uncertain parameters into account, a novel adaptive control was applied to achieve the finite-time synchronization of nonlinear resource management systems. The work of Zhang et al. (2017) addressed the problem of global finite-time synchronization of two different dimensional chaotic systems under the adaptive updated control. In Tran & Kang (2015a), a novel robust adaptive chatter-free finite-time control has been provided for the achievement of finite-time antisynchronization of different uncertain hyperchaotic systems. The problem of finite-time synchronization between two different uncertain chaotic systems with unknown parameters and input nonlinearities has been investigated by employing a robust adaptive control in Aghababa & Aghababa (2012). The work of Aghababa & Aghababa (2013a) has introduced an adaptive control to tackle with the finite-time synchronization between two chaotic nonlinear gyroscopes in the present of model uncertainties, external disturbances and unknown parameters. In Aghababa & Aghababa (2013b), an adaptive robust finite-time control was suggested for the investigation of finite-time synchronization between two different chaotic systems with model uncertainties, external disturbances and unknown parameters. The authors of Ahmad et al. (2016b) have drawn into a nonlinear feedback control to finite-time synchronization of the chaotic systems with different orders. In Dong & Xian (2016), the finite-time quasi-synchronization of two different Lur’e systems with parameter mismatches has been studied by devising the intermittent control. The work of Wang et al. (2009a) realized the finite-time synchronization of the unified chaotic system with uncertain parameters by making use of a stepping linear feedback control. In Yang & Wu (2012), the global finite-time synchronization of a class of second-order nonautonomous chaotic systems has been achieved with the assistance of a generalized linear state-error feedback control. By designing a new twisting control, the authors of Luo & Su (2017) have successfully synchronized the chaotic systems with model uncertainty in finite time. In Sangpet & Kuntanapreeda (2020), a feedback passivation control was proposed for realizing finite-time synchronization of hyperchaotic systems. Ao et al. (2019) investigated the finite-time synchronization by applying the impulsive control method. Wang et al. (2019) introduced a kind of nonlinear control to achieving the finite-time synchronization of memristor chaotic systems. In Wang et al. (2020), a finite-time function project synchronization control method for chaotic wind power systems was presented. From the reviews as above, it is known that the key to the problem of achieving finite-time synchronization of chaotic systems is to design a suitable control. In the existing literatures as discussed above, some controls have been devised to achieve the finite-time synchronization of two chaotic systems, e.g. CLF-based control (Wang et al., 2009b; Liu, 2012), sliding mode control (Aghababa, 2012; Aghababa et al., 2011; Aghababa & Feizi, 2012a,b; Khanzadeh & Pourgholi, 2016; Li & Wu, 2016; Ni et al., 2014), observer-based control (Li & Cao, 2015; Tran & Kang, 2015b), ANC (Cai et al., 2011; Li & Tian, 2003), adaptive control (Aghababa & Aghababa, 2012, 2013a,b; Tran & Kang, 2015a; Vincent & Guo, 2011; Wu et al., 2016; Zhang et al., 2017), nonlinear feedback control (Ahmad et al., 2016b), intermittent control (Dong & Xian, 2016), stepping linear feedback control (Wang et al., 2009a), generalized linear state-error feedback control (Yang & Wu, 2012), twisting control (Luo & Su, 2017), feedback passivation control (Sangpet & Kuntanapreeda, 2020), impulsive control (Ao et al., 2019) and so on (Wang et al., 2019, 2020). Nevertheless, as far as we know, few other controls are designed to achieve finite-time synchronization of chaotic systems. Moreover, it is worth noticing that a control available to achieve finite-time synchronization is not always a good one, which can be judged by two basic standards: For one thing, to be easily implemented and reduce the cost of synchronization, the control should be as simple as possible. For another, the designed control should be continuous so as to effectively avoid the chattering phenomena and even the failure of the synchronization in the synchronization process (Ni et al., 2014). However, unfortunately, those two basic standards of good controls are usually ignored in the control designs for achieving the finite-time synchronization of chaotic systems. For instance, some of the designed controls for finite-time synchronization contain many nonlinear terms, which are utilized to eliminate the nonlinear terms in the error system obtained from the master–slave chaotic systems. On the one hand, this will lead to information loss of the master–slave chaotic systems. On the other hand, this will trigger the problem that the controls designed are too complex to be easily implemented in practical engineering applications. Motivated by the above discussion, this paper seeks to design a simple, continuous and dimension-reducible control by the name of variable-substitution and feedback control (VSFC) to achieve global finite-time synchronization of a class of chaotic systems with some intersecting nonlinearities, which cover many famous chaotic systems. The VSFC designed in this paper possesses some remarkable merits as shown in Remark 2, incorporating two fundamental standards of a good control as above. The rest of the paper is organized as follows. In Section 2, a class of third-order chaotic system with some intersecting nonlinearities, which covers many famous chaotic systems, is described and the master–slave finite-time synchronization scheme for this class of chaotic systems is constructed by designing the VSFC. In Section 3, with the help of the finite-time stability theory for the nonlinear systems, a sufficient criterion for the global finite-time synchronization of this class of third-order chaotic systems is rigorously proven and the synchronization time is analytically estimated simultaneously. In Section 4, the criterion and optimization technique are applied to the well-known brushless direct current motor (BLDCM) system and the classic Lorenz system, respectively, further acquiring some optimized criteria in the form of algebra for the global finite-time synchronization. In order to verify the effectiveness of the optimized criteria, by computer simulations, two numerical examples for the BLDCM systems are provided for comparison between the global finite-time synchronization via the VSFC designed in this paper and the globally asymptotical synchronization via the VSC proposed in Liu et al. (2009), and a numerical example is presented for comparing the global finite-time synchronization via the VSFC with the global finite-time synchronization via the ANC designed in Li & Tian (2003). Finally, some conclusive remarks are presented in Section 5. 2. Chaotic system and synchronization scheme A class of third-order chaotic system considered here is as follows: $$\begin{equation} \dot{x}= Ax+f(x), \end{equation}$$(1) where the state variable |$x={\Big({x}_1,{x}_2,{x}_3\Big)}^T\in{R}^3$|⁠, the constant coefficient matrix |$A={\Big({a}_{ij}\Big)}_{3\times 3}\in{R}^{3\times 3}$| and the nonlinearity |$f(x)={\Big({b}_1{x}_2{x}_3+{m}_1,\kern0.5em {b}_2{x}_1{x}_3+{m}_2,\kern0.5em {b}_3{x}_1{x}_2+{m}_3\Big)}^T\in{R}^3$| with |${m}_i\in R\Big(i=1,2,3\Big)$| and |${b}_i\ne 0\Big(i=1,2,3\Big)$|⁠. Note that system (1) covers many famous chaotic systems such as the generalized Lorenz system (Wu et al., 2007), the generalized Lorenz-like system (Lü et al., 2004), the Newton–Leipnik system (Jia, 2008) and others (Li & Xu, 2001; Liu & Chen, 2003; Liu et al., 2009). A master–slave finite-time synchronization scheme for two identical third-order chaotic systems described by (1) can be constructed under the control |$u\Big(x,z\Big)$|⁠, as follows: $$\begin{equation} \left\{\begin{array}{l} Master:\kern1em \dot{x}= Ax+f(x),\\{} Slave:\kern1em \dot{z}= Az+f(z)+u\left(x,z\right),\end{array}\right. \end{equation}$$(2) where |$x={\Big({x}_1,{x}_2,{x}_3\Big)}^T$| and |$z={\Big({z}_1,{z}_2,{z}_3\Big)}^T$| are the state variables of the master and slave chaotic systems, respectively. The master–slave chaotic systems as above achieve global finite-time synchronization if for any initial conditions |$x(0)={x}_0\in{R}^3$| of the master chaotic system and |$z(0)={z}_0\in{R}^3$| of the slave chaotic system, there exists the constant |$T\Big({x}_0,{z}_0\Big)>0$|⁠, making that |$\underset{t\to T\Big({x}_0,{z}_0\Big)}{\lim}\Big\Vert x(t)-z(t)\Big\Vert =0$| and |$\Big\Vert x(t)-z(t)\Big\Vert \equiv 0$| for |$t\ge T\Big({x}_0,{z}_0\Big)$|⁠, where |$\Big\Vert \cdot \Big\Vert$| denotes the Euclidean norm and |$T\Big({x}_0,{z}_0\Big)$| is the synchronization time. Our objective is to design a suitable control |$u\Big(x,z\Big)$| to make that the master–slave finite-time synchronization scheme (2) achieves global finite-time synchronization. In what follows, a so-called VSFC will be designed to guarantee the objective as above. Firstly, the state variable |$x$| of the master system is rewritten as |$x={\Big({x_d}^T,{x_r}^T\Big)}^T$| with the driving vector |${x}_d\in{R}^{n_d}$| and the responsive vector |${x}_r\in{R}^{n_r}$|⁠, |${n}_d+{n}_r=3$|⁠. Similarly, the state variable |$z$| of the slave system can also be divided into |$z={\Big({z_d}^T,{z_r}^T\Big)}^T$| where |${z}_d$| and |${z}_r$| possess the same dimensions as those of |${x}_d$| and |${x}_r$|⁠, respectively. The system matrix |$A={\Big({A}_{ij}\Big)}_{3\times 3}$| with |${A}_{11}\in{R}^{n_d\times{n}_d}$|⁠, |${A}_{12}\in{R}^{n_d\times{n}_r}$|⁠, |${A}_{21}\in{R}^{n_r\times{n}_d}$| and |${A}_{22}\in{R}^{n_r\times{n}_r}$|⁠. The nonlinearity |$f(x)={\Big({f_d}^T(x),{f_r}^T(x)\Big)}^T$| with |${f}_d(x)={\Big({f}_{d_1}(x),\cdots, {f}_{d_{n_d}}(x)\Big)}^T\in{R}^{n_d}$| and |${f}_r(x)={\Big({f}_{r_1}(x),\cdots, {f}_{r_{n_r}}(x)\Big)}^T\in{R}^{n_r}$|⁠. Now, we design the VSFC |$u={\Big({u}_d^T,{u}_r^T\Big)}^T$| with |${u}_d\in{R}^{n_d}$| and |${u}_r\in{R}^{n_r}$| in the form of $$\begin{equation} {u}_d:{z}_d={x}_d,\kern0.5em {u}_r={Kg}_r\left({x}_r,{z}_r\right),\kern0.5em \forall t\ge 0. \end{equation}$$(3) Control (3) consists of two parts. The first part, |${u}_d:{z}_d={x}_d$|⁠, implies that the continuously extracted signal of the driving variables |${x}_d$| of the master system will be transmitted to the slave system to substitute the counterpart of the corresponding variables |${z}_d$|⁠, forming an open-loop control. The second part, |${u}_r={Kg}_r\Big({x}_r,{z}_r\Big)\in{R}^{n_r}$|⁠, is a feedback control where the feedback gain |$K\in{R}^{n_r\times{n}_r}$| is a constant diagonal matrix to be determined and $$\begin{equation} {g}_r\left({x}_r,{z}_r\right)={\left({\left|{x}_{r_1}-{z}_{r_1}\right|}^{\alpha}\mathit{\operatorname{sign}}\left({x}_{r_1}-{z}_{r_1}\right),\cdots, {\left|{x}_{r_{n_r}}-{z}_{r_{n_r}}\right|}^{\alpha}\mathit{\operatorname{sign}}\left({x}_{r_{n_r}}-{z}_{r_{n_r}}\right)\right)}^T\in{R}^{n_r}, \end{equation}$$(4) where |$\alpha \in \Big(0,1\Big)$| and the sign function $$ \mathit{\operatorname{sign}}\left(\theta \right)=\left\{\begin{array}{cc}1,& \theta >0,\\{}0,& \theta 0,\\{}-1,& \theta <0.\end{array}\right. $$ Obviously, the continuous function |${g}_r\Big({x}_r,{z}_r\Big)$| for |${x}_r$| and |${z}_r$|⁠, is non-Lipschitz at |${x}_r={z}_r$|⁠, and meanwhile, |${g}_r\Big({x}_r,{z}_r\Big)\equiv 0$| for |${x}_r={z}_r$|⁠. Now, we design a suitable driving vector |${x}_d$| and a feedback gain diagonal matrix |$K\in{R}^{n_r\times{n}_r}$|⁠, such that scheme (2) can achieve the global finite-time synchronization. Based on control (3), scheme (2) can thus be rewritten as $$\begin{equation} \left\{\begin{array}{l} Master:\kern0.5em {\dot{x}}_d={A}_{11}{x}_d+{A}_{12}{x}_r+{f}_d(x),\kern1em {\dot{x}}_r={A}_{21}{x}_d+{A}_{22}{x}_r+{f}_r(x),\\{} Slave:\kern0.5em {z}_d={x}_d,\kern1.5em {\dot{z}}_r={A}_{21}{x}_d+{A}_{22}{z}_r+{f}_r(z)+{u}_r.\end{array}\right. \end{equation}$$(5) Defining an error variable |${e}_r={x}_r-{z}_r$|⁠, from (5) we have $$\begin{equation} {\displaystyle \begin{array}{c}{\dot{e}}_r={A}_{22}\left({x}_r-{z}_r\right)+{f}_r(x)-{f}_r(z)-{Kg}_r\left({x}_r,{z}_r\right)\\{}=\left({A}_{22}+L(x)\right){e}_r-{Kg}_r\left({x}_r,{z}_r\right),\end{array}} \end{equation}$$(6) where |${f}_r(x)-{f}_r(z)=L(x)\Big({x}_r-{z}_r\Big)$| in which $$ L(x)=\left\{\begin{array}{l}\left(\begin{array}{cc}0& {b}_{r1}{x}_d\\{}{b}_{r2}{x}_d& 0\end{array}\right), when\ {n}_d=1,\kern0.5em \\{}0,\begin{array}{cccc}\begin{array}{ccc}& & \end{array}& & when\ {n}_d=2,3,&\ \end{array}\end{array}\right. $$ with |${b}_{r_j}\in \Big\{{b}_1,{b}_2,{b}_3\Big\}$|⁠, |$j=1,2$|⁠. Obviously, the global finite-time synchronization of scheme (5) can be converted into the global finite-time stability of the error system (6) at the origin |${e}_r=0$|⁠. Remark 1 Note that the non-Lipschitz property of |${g}_r\Big({x}_r,{z}_r\Big)$| is a necessary prerequisite for finite-time stability of the error system (6) at |${e}_r=0$| (Wang et al., 2009b). Remark 2 It is worth noticing that, control (3) possesses some remarkable merits: first, the control does not contain any other nonlinear term except the non-Lipschitz function (4), which is indispensable for achieving finite-time synchronization, meaning that it is with simple structure; second, both the variable-substitution part (⁠|${u}_d$|⁠) and the feedback part (⁠|${u}_r$|⁠) of the control are globally continuous on |${R}^3$|⁠, which implies that the controlled slave system can be evolved without jump discontinuity. Worth mentioning that in the evolution of the synchronization, jump discontinuity of the slave system at some time instants can induce an unavoidable chattering phenomena, even make the synchronization fail (Ni et al., 2014); third, under the VSFC (3), the synchronization time can be analytically estimated, as seen in Section 3; fourth, it follows from (3), (5) and (6) that the control (3) can not only immediately make the driving variables |${x}_d$| of the master system be synchronized with the corresponding variables |${z}_d$| of the slave system at time |$t=0$|⁠, but also reduce the dimension of the error system originating from the master and slave chaotic systems. Remark 3 It is worthy of note that the VSFC (3) will degenerate into VSC presented in Liu et al. (2009) if |$K=0$| in (3). Hence, it is interesting to compare the global finite-time synchronization via the VSFC designed in this paper with the globally asymptotical synchronization via the VSC presented in Liu et al. (2009). And the relevant comparison information of synchronization will be listed in the tables and the synchronization evolutions of two types of synchronizations are compared, as illustrated in figures in subsection 4.1. Remark 4 Also, in subsection 4.2, the global finite-time synchronization via the VSFC designed in this paper will be compared with that via the ANC designed in Li & Tian (2003), confirming that the VSFC is much simpler than the ANC in structure and the control inputs of the VSFC are still smaller than those of the ANC. 3. The global finite-time synchronization criteria For the purpose of simplifying the master–slave finite-time synchronization scheme, only the case of single-variable substitution with |${n}_d=1$| is concerned here. In this situation, |${n}_r=2$| and the system matrices of the error system (6) can be depicted by $$\begin{equation} {A}_{22}=\left(\begin{array}{cc}{a}_{r_1{r}_1}& {a}_{r_1{r}_2}\\{}{a}_{r_2{r}_1}& {a}_{r_2{r}_2}\end{array}\right)\in{R}^{2\times 2}, \end{equation}$$(7) $$\begin{equation} L(x)=\left(\begin{array}{cc}0& {b}_{r_1}{x}_d\\{}{b}_{r_2}{x}_d& 0\end{array}\right), \end{equation}$$(8) where |${b}_{r_j}\in \Big\{{b}_1,{b}_2,{b}_3\Big\}$|⁠, |$j=1,2,$| and |${x}_d\in R$|⁠. Lemma 1. (Yu et al., 2005) If |${a}_1,\kern0.5em {a}_2,\cdots, {a}_n$| are all positive and |$\lambda \in \Big(0,2\Big)$|⁠, then $$\begin{equation} {\left(\sum \limits_{i=1}^n{a}_i^2\right)}^{\lambda}\le{\left(\sum \limits_{i=1}^n{a}_i^{\lambda}\right)}^2. \end{equation}$$(9) According to the bounded feature of chaotic attractor (Curran & Chua, 1997), it is clear that there exist the real constants |${\rho}_i$| such that the trajectory of the master chaotic system (1) satisfy $$ \left|{x}_i(t)\right|\le{\rho}_i,\kern1em i=1,2,3,\kern1em \forall t\ge 0. $$ Let |${\rho}_d\in \Big\{{\rho}_1,{\rho}_2,{\rho}_3\Big\}$| be the bound of the chosen single-substituted variable |${x}_d\in R$| of the master chaotic system. A global finite-time synchronization criterion for scheme (5) can then be proven and presented below. Theorem 1. If a positive definite feedback gain matrix |$K=\mathit{\operatorname{diag}}\Big\{{k}_{r1},{k}_{r2}\Big\}$| and a positive definite matrix |$P=\mathit{\operatorname{diag}}\Big\{1,\mu \Big\}$| are selected such that $$\begin{equation} {a}_{r_1{r}_1}<0,\kern0.5em {a}_{r_2{r}_2}<0,\kern0.5em 4{a}_{r_1{r}_1}{a}_{r_2{r}_2}>{s}^2\left(\mu \right) \end{equation}$$(10) ,where $$\begin{equation} s\left(\mu \right)=\left|\frac{a_{r_1{r}_2}}{\sqrt{\mu }}+{a}_{r_2{r}_1}\sqrt{\mu}\right|+\left|\frac{b_{r_1}{\rho}_d}{\sqrt{\mu }}+{b}_{r_2}{\rho}_d\sqrt{\mu}\right|, \end{equation}$$(11) and then the global finite-time synchronization of the master–slave synchronization scheme (5) can be achieved at time $$\begin{equation} T\left({x}_0,{z}_0\right)\le{T}_E=\frac{2}{-{\lambda}_{max}\left(1-\alpha \right)}\ln \left(1-\frac{\lambda_{max}}{2c}V{\left({e}_{r0}\right)}^{\frac{1-\alpha }{2}}\right), \end{equation}$$(12) where |$V\Big({e}_{r0}\Big)={e_{r0}}^T{Pe}_{r0}$| with |${e}_{r0}={x}_r(0)-{z}_r(0)$|⁠, |$\alpha \in \Big(0,1\Big)$|⁠, |$c=\min \Big\{{k}_{r1},{k}_{r2}{\mu}^{\Big(1-\alpha \Big)/2}\Big\}$|⁠, and |${\lambda}_{max}$| is the maximal eigenvalue of the matrix $$\begin{equation} S=\left(\begin{array}{cc}2{a}_{r_1{r}_1}& s\left(\mu \right)\\{}s\left(\mu \right)& 2{a}_{r_2{r}_2}\end{array}\right), \end{equation}$$(13) and $$\begin{equation} {\lambda}_{max}=\left({a}_{r_1{r}_1}+{a}_{r_2{r}_2}\right)+\sqrt{{\left({a}_{r_1{r}_1}-{a}_{r_2{r}_2}\right)}^2+s{\left(\mu \right)}^2}. \end{equation}$$(14) Proof. Choose a Lyapunov function: $$ V\left({e}_r\right)={e_r}^T{Pe}_r,\kern1em 0<P=\mathit{\operatorname{diag}}\left\{1,\mu \right\}. $$ Based on equalities (7) and (8), the time derivative of |$V\Big({e}_r\Big)$| along the trajectory of the error system (6) equals $$\begin{equation} { \begin{array}{l}\dot{V}\left({e}_r\right)={{\dot{e}}_r}^T{Pe}_r+{e_r}^TP{\dot{e}}_r\\{}={e_r}^T\left[{\left({A}_{22}+L\right)}^TP+P\left({A}_{22}+L\right)\right]{e}_r-{\left({Kg}_r\left({x}_r,{z}_r\right)\right)}^T{Pe}_r-{e_r}^T{PKg}_r\left({x}_r,{z}_r\right)\\{}=2{a}_{r_1{r}_1}{e}_{r_1}^2+2{a}_{r_2{r}_2}\mu{e}_{r_2}^2+2\left[\left({a}_{r_1{r}_2}+{b}_{r_1}{x}_d\right)/\sqrt{\mu }+\sqrt{\mu}\left({a}_{r_2{r}_1}+{b}_{r_2}{x}_d\right)\right]{e}_{r_1}{e}_{r_2}\sqrt{\mu }-2{e_r}^T{PKg}_r\left({x}_r,{z}_r\right)\\{}\le 2{a}_{r_1{r}_1}{e}_{r_1}^2+2{a}_{r_2{r}_2}\mu{e}_{r_2}^2+2s\left(\mu \right)\left|{e}_{r_1}\right|\left|{e}_{r_2}\sqrt{\mu}\right|-2{e_r}^T{PKg}_r\left({x}_r,{z}_r\right)\\{}=\left(\left|{e}_{r_1}\right|,\left|{e}_{r_2}\sqrt{\mu}\right|\right)S{\left(\left|{e}_{r_1}\right|,\left|{e}_{r_2}\sqrt{\mu}\right|\right)}^T-2{e_r}^T{PKg}_r\left({x}_r,{z}_r\right)\\{}\le{\lambda}_{\mathrm{max}}{e_r}^T{Pe}_r-2{e_r}^T{PKg}_r\left({x}_r,{z}_r\right),\end{array}} \end{equation}$$(15) where |$S$| and |${\lambda}_{max}$| are defined by (13) and (14), respectively. It is clear that |${\lambda}_{max}<0$| provided that the inequalities defined by (10) hold. Again, in terms of Lemma 1, it is easy to obtain that for |$\alpha \in \Big(0,\kern0.5em 1\Big)$| and |${e}_r\in{R}^2$|⁠, $$\begin{equation} {\displaystyle \begin{array}{c}-2{e_r}^T{PKg}_r\left({x}_r,{z}_r\right)=-2\left({k}_{r1}{\left|{e}_{r_1}\right|}^{1+\alpha }+{k}_{r2}\mu{\left|{e}_{r_2}\right|}^{1+\alpha}\right)\\{}=-2\left({k}_{r1}{\left|{e}_{r_1}\right|}^{1+\alpha }+{k}_{r2}{\mu}^{\left(1-\alpha \right)/2}{\mu}^{\left(1+\alpha \right)/2}{\left|{e}_{r_2}\right|}^{1+\alpha}\right)\\{}\le -2c\left({\left|{e}_{r_1}\right|}^{1+\alpha }+{\mu}^{\left(1+\alpha \right)/2}{\left|{e}_{r_2}\right|}^{1+\alpha}\right)\\{}\le -2c{\left({e_{r_1}}^2+\mu{e_{r_2}}^2\right)}^{\left(1+\alpha \right)/2}\\{}=-2c{\left(V\left({e}_r\right)\right)}^{\left(1+\alpha \right)/2}.\end{array}} \end{equation}$$(16) Hence, combining (15) and (16), one has $$\begin{equation} \dot{V}\left({e}_r\right)\le{\lambda}_{\mathrm{max}}V\left({e}_r\right)-2c{\left(V\left({e}_r\right)\right)}^{\left(1+\alpha \right)/2},\kern0.5em \forall{e}_r\in{R}^2. \end{equation}$$(17) Let |$r(V)=-{\lambda}_{max}V+2{cV}^{\Big(1+\alpha \Big)/2}$|⁠. The function |$r(V)$| is then positive for any |$V\in{R}^{+}$| provided that the inequalities defined by (10) hold. For the defined function |$r(V)$|⁠, there exists a constant |$\varepsilon >0$|⁠, such that $$\begin{equation} {\int}_0^{\varepsilon}\frac{ds}{r(s)}=\frac{2}{-{\lambda}_{max}\left(1-\alpha \right)}\ln \left(1-\frac{\lambda_{max}}{2c}{\varepsilon}^{\frac{1-\alpha }{2}}\right)<+\infty . \end{equation}$$(18) As a result, in accordance with the finite-time stability theorem stated in Moulay & Perruquetti (2008), it is easily concluded that global finite-time synchronization of the master–slave synchronization scheme (5) can be achieved at time |$T\Big({x}_0,{z}_0\Big)$|⁠, which satisfies inequality (12). Based on this, the proof is completed. Remark 5 The sufficient criterion (10) for the global finite-time synchronization is related to a positive scale |$\mu$| to be determined. Whether from a theoretical or practical point of view, it is always desired that the obtained criterion (10) should hold for the most possible cases of the considered chaotic systems. Such a criterion is considered to be sharp. With a view to arriving at the goal, we must select the positive scale |$\mu ={\mu}^{\ast }$| to make |$s\Big(\mu \Big)$| defined by (11) reach its minimum value. Following from (11), it can be concluded that such an optimal value |${\mu}^{\ast }$| for the positive scale |$\mu$| always exists, as shown in the examples of next section. 4. Applications and simulations 4.1 Applying to the BLDCM system As the first application of the results obtained in Section 3, we select the BLDCM system, which is a class of motor system and is usually operated in brushless and electronic commutation (Park et al., 2007). At present, the BLDCM system has been widely applied to many civilian and military fields inclusive of pharmacy (Park et al., 2007), robotics (Asada & Youcef-Toumi, 1978) and aerospace (Murugesan, 1981) due to some advantages over the brushed DC motor, e.g. high dynamic response, high efficiency, long operating life, noiseless operation, higher speed range and so on (Park et al., 2007). With the development of the industry, the requirements for the machine performance and product quality have gradually increased in recent years. As a result, in some real-world application situations, the control for the sole BLDCM system cannot usually satisfy the requirements of the high technology development, while it is usually required that multiple BLDCM systems should work synchronously to adapt the new requirements, for instance, in the application of the liquid medicine filling machine (Wang, 2009). In what follows, we will apply the theoretical results obtained in Section 3 to the considered BLDCM system, which is described as follows (Liu et al., 2009): $$\begin{equation} \dot{x}= Ax+f(x), \end{equation}$$(19) where |$x={\Big({x}_1,{x}_2,{x}_3\Big)}^T\in{R}^3$|⁠, $$ A=\left(\begin{array}{ccc}-1& 0& \rho \\{}0& -\delta & 0\\{}\sigma & 0& -\sigma \end{array}\right),\kern0.5em f(x)=\left(\begin{array}{c}-{x}_2{x}_3+{v}_q\\{}{x}_1{x}_3+{v}_d\\{}\eta{x}_1{x}_2-{T}_L\end{array}\right), $$ with the positive parameters |$\rho$|⁠, |$\delta$|⁠, |$\sigma$|⁠, |${v}_q$|⁠, |${v}_d$|⁠, |$\eta$| and |${T}_L$|⁠. Notably, this system can exhibit complex chaotic behaviours under some special parameters (Liu et al., 2009). Consider the master–slave finite-time synchronization scheme (5) for the BLDCM systems, where a single driving variable |${x}_d={x}_1$| is chosen so that the corresponding VSFC is $$\begin{equation} {u}_d:{x}_d={x}_1,\kern0.5em {u}_r={\left({k}_{r1}{\left|{x}_2-{z}_2\right|}^{\alpha}\mathit{\operatorname{sign}}\left({x}_2-{z}_2\right),{k}_{r2}{\left|{x}_3-{z}_3\right|}^{\alpha}\mathit{\operatorname{sign}}\left({x}_3-{z}_3\right)\right)}^T. \end{equation}$$(20) In this case, the state variables of the master BLDCM systems can be written as |$x={\Big({x_d}^T,{x_r}^T\Big)}^T$| with |${x}_r={\Big(x{}_2,{x}_3\Big)}^T$| and $$ {A}_{22}=\left(\begin{array}{cc}-\delta & 0\\{}0& -\sigma \end{array}\right),\kern1em L(x)=\left(\begin{array}{cc}0& {x}_1\\{}\eta{x}_1& 0\end{array}\right). $$ In this situation, the function defined by (11) can be rewritten as |$s\Big(\mu \Big)={\rho}_1/\sqrt{\mu }+{\eta \rho}_1\sqrt{\mu }$|⁠, which takes the minimum value |$2{\rho}_1\sqrt{\eta }$| at the extreme point |${\mu}^{\ast }=1/\eta$|⁠, where |${\rho}_1$| is a bound of state component |${x}_1$| of the master BLDCM system. From Theorem 1 and Remark 5 one can easily derive the following result. Proposition 1. If the system parameters satisfy $$\begin{equation} \delta \sigma >{{\eta \rho}_1}^2, \end{equation}$$(21) and then the global finite-time synchronization of the master–slave BLDCM systems under the VSFC (20) can be achieved at time $$\begin{equation} T\left({x}_0,{z}_0\right)\le{T}_E=\frac{2}{-{\lambda}_{max}\left(1-\alpha \right)}\ln \left[1-\frac{\lambda_{max}}{2{c}_1}{\left({\left({x}_2(0)-{z}_2(0)\right)}^2+\frac{1}{\eta }{\left({x}_3(0)-{z}_3(0)\right)}^2\right)}^{\frac{1-\alpha }{2}}\right], \end{equation}$$(22) where |$\alpha \in \Big(0,1\Big)$|⁠, |${c}_1=\min \Big\{{k}_{r1},{k}_{r2}{\Big(1/\eta \Big)}^{\Big(1-\alpha \Big)/2}\Big\}$| and |${\lambda}_{max}=-\Big(\delta +\sigma \Big)+\sqrt{{\Big(\delta -\sigma \Big)}^2+4{{\eta \rho}_1}^2}$|⁠. If taking a single driving variable |${x}_d={x}_2$|⁠, the VSFC can be described as $$\begin{equation} {u}_d:{x}_d={x}_2,\kern1em {u}_r={\left({k}_{r1}{\left|{x}_1-{z}_1\right|}^{\alpha}\mathit{\operatorname{sign}}\left({x}_1-{z}_1\right),{k}_{r2}{\left|{x}_3-{z}_3\right|}^{\alpha}\mathit{\operatorname{sign}}\left({x}_3-{z}_3\right)\right)}^T. \end{equation}$$(23) In the case, the state variables of the master system can be presented as |$x={\Big({x_d}^T,{x_r}^T\Big)}^T$| with |${x}_r={\Big(x{}_1,{x}_3\Big)}^T$| and $$ {A}_{22}=\left(\begin{array}{cc}-1& \rho \\{}\sigma & -\sigma \end{array}\right),\kern1em L(x)=\left(\begin{array}{cc}0& -{x}_2\\{}\eta{x}_2& 0\end{array}\right). $$ The function defined by (11) can be rewritten as $$\begin{equation} {s}_2\left(\mu \right)=\rho /\sqrt{\mu }+\sigma \sqrt{\mu }+{\rho}_2\sqrt{\mu}\left|\eta -1/\mu \right|, \end{equation}$$(24) where |${\rho}_2$| is a bound of state component |${x}_2$| of the master BLDCM system. Let |${\mu}^{\ast }$| be a minimal point of |${s}_2\Big(\mu \Big)$| for |$\mu >0$|⁠. From Theorem 1 and Remark 5 one has Proposition 2. If the system parameters satisfy $$\begin{equation} 4\sigma >{\left({s}_2\left({\mu}^{\ast}\right)\right)}^2, \end{equation}$$(25) and then the global finite-time synchronization of the master–slave BLDCM systems under the VSFC (23) can be achieved at time $$\begin{equation} T\left({x}_0,{z}_0\right)\le{T}_E=\frac{2}{-{\lambda}_{max}\left(1-\alpha \right)}\ln \left[1-\frac{\lambda_{max}}{2{c}_2}{\left({\left({x}_1(0)-{z}_1(0)\right)}^2+{\mu}^{\ast }{\left({x}_3(0)-{z}_3(0)\right)}^2\right)}^{\frac{1-\alpha }{2}}\right], \end{equation}$$(26) where |$\alpha \in \Big(0,1\Big)$|⁠, |${c}_2=\min \Big\{{k}_{r1},{k}_{r2}{\Big({\mu}^{\ast}\Big)}^{\Big(1-\alpha \Big)/2}\Big\}$| and |${\lambda}_{max}=-\Big(1+\sigma \Big)+\sqrt{{\Big(1-\sigma \Big)}^2+{\Big({s}_2\Big({\mu}^{\ast}\Big)\Big)}^2}$|⁠. If taking a single driving variable |${x}_d={x}_3$|⁠, the VSFC can be given by $$\begin{equation} {u}_d:{x}_d={x}_3,\kern1em {u}_r={\left({k}_{r1}{\left|{x}_1-{z}_1\right|}^{\alpha}\mathit{\operatorname{sign}}\left({x}_1-{z}_1\right),{k}_{r2}{\left|{x}_2-{z}_2\right|}^{\alpha}\mathit{\operatorname{sign}}\left({x}_2-{z}_2\right)\right)}^T. \end{equation}$$(27) In the case, the state variables of the master system can be decomposed as |$x={\Big({x_d}^T,{x_r}^T\Big)}^T$| with |${x}_r={\Big(x{}_1,{x}_2\Big)}^T$| and $$ {A}_{22}=\left(\begin{array}{cc}-1& 0\\{}0& -\delta \end{array}\right),\kern1em L(x)=\left(\begin{array}{cc}0& -{x}_3\\{}{x}_3& 0\end{array}\right). $$ The function defined by (11) can thus be rewritten as |$s\Big(\mu \Big)={\rho}_3\Big|\sqrt{\mu }-1/\sqrt{\mu}\Big|$|⁠, which achieves its minimum 0 at the extreme point |${\mu}^{\ast }=1$|⁠, where |${\rho}_3$| is a bound of state component |${x}_3$| of the master BLDCM system. Therefore, from Theorem 1 and Remark 5 one has Proposition 3. For any system parameters, the global finite-time synchronization of the master–slave BLDCM systems under the VSFC (27) can be achieved at time $$\begin{equation} T\left({x}_0,{z}_0\right)\le{T}_E=\frac{2}{-{\lambda}_{max}\left(1-\alpha \right)}\ln \left[1-\frac{\lambda_{max}}{2{c}_3}{\left({\left({x}_1(0)-{z}_1(0)\right)}^2+{\left({x}_2(0)-{z}_2(0)\right)}^2\right)}^{\frac{1-\alpha }{2}}\right], \end{equation}$$(28) where |$\alpha \in \Big(0,1\Big)$|⁠, |${c}_3=\min \Big\{{k}_{r1},{k}_{r2}\Big\}$| and |${\lambda}_{max}=-\Big(1+\delta \Big)+\Big|1-\delta \Big|$|⁠. For verifying the effectiveness of the above results, two numerical examples for the BLDCM system in Liu et al. (2009) are introduced to compare the globally asymptotical synchronization via the VSC presented in Liu et al. (2009) with the finite-time synchronization via the VSFC presented in this paper. According to Liu et al. (2009), the initial conditions of the master and slave BLDCM systems are selected as |$x(0)={\Big(-7,\kern0.5em 50,\kern0.5em -8\Big)}^T$| and |$z(0)={\Big(6,\kern0.5em -10,\kern0.5em 20\Big)}^T$|⁠, respectively. For Example 1 given in Table 1, the master and slave BLDCM systems are chaotic and are illustrated in Fig. 1(a–b), from which we know the fact that |${\rho}_1=40$|⁠. By simple computing, it is easy to know that this example satisfies Corollary 1 and Corollary 3 in Liu et al. (2009) as well as Proposition 1 and Proposition 3 in this paper simultaneously, which means that the globally asymptotical synchronization of the master–slave BLDCM systems can be achieved via the VSC |${x}_d={x}_1$| (depicted by the red dashed in Fig. 1(c)) or via the VSC |${x}_d={x}_3$| (depicted by the red dashed in Fig. 1(e)), and also means that the global finite-time synchronization of the master–slave BLDCM systems can be achieved via the VSFC (20) with |$\alpha =1/3$| and |$K=\mathit{\operatorname{diag}}\Big\{8.2,27.5\Big\}$| (depicted by the blue real line in Fig. 1(c)) or via the VSFC (27) with |$\alpha =3/5$| and |$K=\mathit{\operatorname{diag}}\Big\{2,2\Big\}$| (depicted by the blue real line in Fig. 1(e)). And, by means of (22) and (28), the corresponding synchronization time for finite-time synchronization can be estimated as |${T}_E=1.8254$| and |${T}_E=3.1992$|⁠, respectively. It can be seen from Fig. 1(c) and Fig. 1(e) that the finite-time synchronizations via the VSFCs can be achieve in a shorter time than the corresponding asymptotical synchronizations via the VSCs. Also, the control inputs of the VSCs and VSFCs are compared in Fig. 1(d) and Fig. 1(f). The detailed comparison information for globally asymptotical synchronization and global finite-time synchronization has been listed in Table 1. With respect to Example 2, the parameters of the BLDCM systems are listed in Table 2 and the state trajectories of the master and slave BLDCM systems are shown in Fig. 2(a–b), from which we know that |${\rho}_2=50$|⁠. The detailed synchronization comparison information for globally asymptotical synchronization and global finite-time synchronization, including the satisfied criterion, the used control and the synchronization time estimation, has been summarized in Table 2 and the synchronization evolutions of two types of synchronizations and the control inputs of two controls are also compared, as shown in Figure 2. Table 1 Example 1 and the corresponding synchronization comparison information Example 1 . System parameters |$\rho =60$|⁠, |${v}_q=0.168$|⁠, |${v}_d=20.66$|⁠, |$\delta =2$|⁠, |$\eta =0.001$|⁠, |${T}_L=0.53$|⁠, |$\sigma =10$|⁠. Synchronization comparison The asymptotical synchronization Satisfied criterion Corollary 1 in Liu et al. (2009) Corollary 3 in Liu et al. (2009) Used control the VSC |${x}_d={x}_1$| the VSC |${x}_d={x}_3$| Synchronization time |$T\to \infty$| |$T\to \infty$| The finite-time synchronization Satisfied criterion Proposition 1 in this paper Proposition 3 in this paper Used control the VSFC (20) with |$\alpha =7/10$| and |$K=\mathit{\operatorname{diag}}\Big\{8.2,27.5\Big\}$| the VSFC (27) with |$\alpha =3/5$| and |$K=\mathit{\operatorname{diag}}\Big\{2,2\Big\}$| Synchronization time estimation |${T}_E=1.8254$| |${T}_E=3.1992$| Simulation figures Fig 1. (c)-(d) Fig 1. (e)-(f) Example 1 . System parameters |$\rho =60$|⁠, |${v}_q=0.168$|⁠, |${v}_d=20.66$|⁠, |$\delta =2$|⁠, |$\eta =0.001$|⁠, |${T}_L=0.53$|⁠, |$\sigma =10$|⁠. Synchronization comparison The asymptotical synchronization Satisfied criterion Corollary 1 in Liu et al. (2009) Corollary 3 in Liu et al. (2009) Used control the VSC |${x}_d={x}_1$| the VSC |${x}_d={x}_3$| Synchronization time |$T\to \infty$| |$T\to \infty$| The finite-time synchronization Satisfied criterion Proposition 1 in this paper Proposition 3 in this paper Used control the VSFC (20) with |$\alpha =7/10$| and |$K=\mathit{\operatorname{diag}}\Big\{8.2,27.5\Big\}$| the VSFC (27) with |$\alpha =3/5$| and |$K=\mathit{\operatorname{diag}}\Big\{2,2\Big\}$| Synchronization time estimation |${T}_E=1.8254$| |${T}_E=3.1992$| Simulation figures Fig 1. (c)-(d) Fig 1. (e)-(f) Open in new tab Table 1 Example 1 and the corresponding synchronization comparison information Example 1 . System parameters |$\rho =60$|⁠, |${v}_q=0.168$|⁠, |${v}_d=20.66$|⁠, |$\delta =2$|⁠, |$\eta =0.001$|⁠, |${T}_L=0.53$|⁠, |$\sigma =10$|⁠. Synchronization comparison The asymptotical synchronization Satisfied criterion Corollary 1 in Liu et al. (2009) Corollary 3 in Liu et al. (2009) Used control the VSC |${x}_d={x}_1$| the VSC |${x}_d={x}_3$| Synchronization time |$T\to \infty$| |$T\to \infty$| The finite-time synchronization Satisfied criterion Proposition 1 in this paper Proposition 3 in this paper Used control the VSFC (20) with |$\alpha =7/10$| and |$K=\mathit{\operatorname{diag}}\Big\{8.2,27.5\Big\}$| the VSFC (27) with |$\alpha =3/5$| and |$K=\mathit{\operatorname{diag}}\Big\{2,2\Big\}$| Synchronization time estimation |${T}_E=1.8254$| |${T}_E=3.1992$| Simulation figures Fig 1. (c)-(d) Fig 1. (e)-(f) Example 1 . System parameters |$\rho =60$|⁠, |${v}_q=0.168$|⁠, |${v}_d=20.66$|⁠, |$\delta =2$|⁠, |$\eta =0.001$|⁠, |${T}_L=0.53$|⁠, |$\sigma =10$|⁠. Synchronization comparison The asymptotical synchronization Satisfied criterion Corollary 1 in Liu et al. (2009) Corollary 3 in Liu et al. (2009) Used control the VSC |${x}_d={x}_1$| the VSC |${x}_d={x}_3$| Synchronization time |$T\to \infty$| |$T\to \infty$| The finite-time synchronization Satisfied criterion Proposition 1 in this paper Proposition 3 in this paper Used control the VSFC (20) with |$\alpha =7/10$| and |$K=\mathit{\operatorname{diag}}\Big\{8.2,27.5\Big\}$| the VSFC (27) with |$\alpha =3/5$| and |$K=\mathit{\operatorname{diag}}\Big\{2,2\Big\}$| Synchronization time estimation |${T}_E=1.8254$| |${T}_E=3.1992$| Simulation figures Fig 1. (c)-(d) Fig 1. (e)-(f) Open in new tab Table 2 Example 2 and the corresponding synchronization comparison information Example 2 . System parameters |$\rho =0.1$|⁠,|${v}_q=0.168$|⁠,|${v}_d=20.66$|⁠, |$\delta =2$|⁠, |$\eta =1$|⁠, |${T}_L=0.53$|⁠, |$\sigma =2$|⁠. Synchronization comparison The asymptotical synchronization Satisfied criterion Corollary 2 in Liu et al. (2009) Corollary 3 in Liu et al. (2009) Used control the VSC |${x}_d={x}_2$| the VSC |${x}_d={x}_3$| Synchronization time |$T\to \infty$| |$T\to \infty$| The finite-time synchronization Satisfied criterion Proposition 2 in this paper Proposition 3 in this paper Used control the VSFC (23) with |$\alpha =2/5$| and |$K=\mathit{\operatorname{diag}}\Big\{3,3\Big\}$| the VSFC (27) with |$\alpha =1/2$| and |$K=\mathit{\operatorname{diag}}\Big\{1.2,3.1\Big\}$| Synchronization time estimation |${T}_E=3.1206$| |${T}_E=4.0376$| Simulation figures Fig 2. (c)-(d) Fig 2. (e)-(f) Example 2 . System parameters |$\rho =0.1$|⁠,|${v}_q=0.168$|⁠,|${v}_d=20.66$|⁠, |$\delta =2$|⁠, |$\eta =1$|⁠, |${T}_L=0.53$|⁠, |$\sigma =2$|⁠. Synchronization comparison The asymptotical synchronization Satisfied criterion Corollary 2 in Liu et al. (2009) Corollary 3 in Liu et al. (2009) Used control the VSC |${x}_d={x}_2$| the VSC |${x}_d={x}_3$| Synchronization time |$T\to \infty$| |$T\to \infty$| The finite-time synchronization Satisfied criterion Proposition 2 in this paper Proposition 3 in this paper Used control the VSFC (23) with |$\alpha =2/5$| and |$K=\mathit{\operatorname{diag}}\Big\{3,3\Big\}$| the VSFC (27) with |$\alpha =1/2$| and |$K=\mathit{\operatorname{diag}}\Big\{1.2,3.1\Big\}$| Synchronization time estimation |${T}_E=3.1206$| |${T}_E=4.0376$| Simulation figures Fig 2. (c)-(d) Fig 2. (e)-(f) Open in new tab Table 2 Example 2 and the corresponding synchronization comparison information Example 2 . System parameters |$\rho =0.1$|⁠,|${v}_q=0.168$|⁠,|${v}_d=20.66$|⁠, |$\delta =2$|⁠, |$\eta =1$|⁠, |${T}_L=0.53$|⁠, |$\sigma =2$|⁠. Synchronization comparison The asymptotical synchronization Satisfied criterion Corollary 2 in Liu et al. (2009) Corollary 3 in Liu et al. (2009) Used control the VSC |${x}_d={x}_2$| the VSC |${x}_d={x}_3$| Synchronization time |$T\to \infty$| |$T\to \infty$| The finite-time synchronization Satisfied criterion Proposition 2 in this paper Proposition 3 in this paper Used control the VSFC (23) with |$\alpha =2/5$| and |$K=\mathit{\operatorname{diag}}\Big\{3,3\Big\}$| the VSFC (27) with |$\alpha =1/2$| and |$K=\mathit{\operatorname{diag}}\Big\{1.2,3.1\Big\}$| Synchronization time estimation |${T}_E=3.1206$| |${T}_E=4.0376$| Simulation figures Fig 2. (c)-(d) Fig 2. (e)-(f) Example 2 . System parameters |$\rho =0.1$|⁠,|${v}_q=0.168$|⁠,|${v}_d=20.66$|⁠, |$\delta =2$|⁠, |$\eta =1$|⁠, |${T}_L=0.53$|⁠, |$\sigma =2$|⁠. Synchronization comparison The asymptotical synchronization Satisfied criterion Corollary 2 in Liu et al. (2009) Corollary 3 in Liu et al. (2009) Used control the VSC |${x}_d={x}_2$| the VSC |${x}_d={x}_3$| Synchronization time |$T\to \infty$| |$T\to \infty$| The finite-time synchronization Satisfied criterion Proposition 2 in this paper Proposition 3 in this paper Used control the VSFC (23) with |$\alpha =2/5$| and |$K=\mathit{\operatorname{diag}}\Big\{3,3\Big\}$| the VSFC (27) with |$\alpha =1/2$| and |$K=\mathit{\operatorname{diag}}\Big\{1.2,3.1\Big\}$| Synchronization time estimation |${T}_E=3.1206$| |${T}_E=4.0376$| Simulation figures Fig 2. (c)-(d) Fig 2. (e)-(f) Open in new tab Table 3 Example 3 and the corresponding synchronization comparison information Example 3 . System parameters |$d=10$|⁠,|$g=8/3$|⁠,|$r=28$|⁠. Synchronization comparison The finite-time synchronization in Li & Tian (2003) Satisfied criterion Any parameters Used control the ANC (39) with |$\alpha =1/2$|⁠,|${k}_1=20$|⁠,|${k}_2=20$| and |${k}_3=4$| designed in Li & Tian (2003) Synchronization time estimation |$-$| The finite-time synchronization Satisfied criterion Proposition 4 in this paper Proposition 5 with |$\zeta =10$| in this paper Used control the VSFC (30) with |$\alpha =1/2$| and |$K=\mathit{\operatorname{diag}}\Big\{20,4\Big\}$| the VSFC (33) with |$\alpha =1/2$| and |$K=\mathit{\operatorname{diag}}\Big\{20,4\Big\}$| Synchronization time estimation |${T}_E=2.2016$| |${T}_E=1.3259$| Simulation figures Fig 3. (c)-(d) Fig 3. (e)-(f) Example 3 . System parameters |$d=10$|⁠,|$g=8/3$|⁠,|$r=28$|⁠. Synchronization comparison The finite-time synchronization in Li & Tian (2003) Satisfied criterion Any parameters Used control the ANC (39) with |$\alpha =1/2$|⁠,|${k}_1=20$|⁠,|${k}_2=20$| and |${k}_3=4$| designed in Li & Tian (2003) Synchronization time estimation |$-$| The finite-time synchronization Satisfied criterion Proposition 4 in this paper Proposition 5 with |$\zeta =10$| in this paper Used control the VSFC (30) with |$\alpha =1/2$| and |$K=\mathit{\operatorname{diag}}\Big\{20,4\Big\}$| the VSFC (33) with |$\alpha =1/2$| and |$K=\mathit{\operatorname{diag}}\Big\{20,4\Big\}$| Synchronization time estimation |${T}_E=2.2016$| |${T}_E=1.3259$| Simulation figures Fig 3. (c)-(d) Fig 3. (e)-(f) Open in new tab Table 3 Example 3 and the corresponding synchronization comparison information Example 3 . System parameters |$d=10$|⁠,|$g=8/3$|⁠,|$r=28$|⁠. Synchronization comparison The finite-time synchronization in Li & Tian (2003) Satisfied criterion Any parameters Used control the ANC (39) with |$\alpha =1/2$|⁠,|${k}_1=20$|⁠,|${k}_2=20$| and |${k}_3=4$| designed in Li & Tian (2003) Synchronization time estimation |$-$| The finite-time synchronization Satisfied criterion Proposition 4 in this paper Proposition 5 with |$\zeta =10$| in this paper Used control the VSFC (30) with |$\alpha =1/2$| and |$K=\mathit{\operatorname{diag}}\Big\{20,4\Big\}$| the VSFC (33) with |$\alpha =1/2$| and |$K=\mathit{\operatorname{diag}}\Big\{20,4\Big\}$| Synchronization time estimation |${T}_E=2.2016$| |${T}_E=1.3259$| Simulation figures Fig 3. (c)-(d) Fig 3. (e)-(f) Example 3 . System parameters |$d=10$|⁠,|$g=8/3$|⁠,|$r=28$|⁠. Synchronization comparison The finite-time synchronization in Li & Tian (2003) Satisfied criterion Any parameters Used control the ANC (39) with |$\alpha =1/2$|⁠,|${k}_1=20$|⁠,|${k}_2=20$| and |${k}_3=4$| designed in Li & Tian (2003) Synchronization time estimation |$-$| The finite-time synchronization Satisfied criterion Proposition 4 in this paper Proposition 5 with |$\zeta =10$| in this paper Used control the VSFC (30) with |$\alpha =1/2$| and |$K=\mathit{\operatorname{diag}}\Big\{20,4\Big\}$| the VSFC (33) with |$\alpha =1/2$| and |$K=\mathit{\operatorname{diag}}\Big\{20,4\Big\}$| Synchronization time estimation |${T}_E=2.2016$| |${T}_E=1.3259$| Simulation figures Fig 3. (c)-(d) Fig 3. (e)-(f) Open in new tab As analyzed above, it is shown that the theoretical results accord closely with the simulation results, verifying that the theoretical results obtained are effective. 4.2 Applying to the Lorenz system As the second application, we will apply the theoretical results obtained in Section 3 to the classic Lorenz system, which is expressed in the matrix form of Wu et al. (2007): $$\begin{equation} \dot{x}= Ax+f(x), \end{equation}$$(29) with |$x={\Big({x}_1,{x}_2,{x}_3\Big)}^T\in{R}^3$|⁠, $$ A=\left(\begin{array}{ccc}-d& d& 0\\{}r& -1& 0\\{}0& 0& g\end{array}\right),\kern1em f(x)=\left(\begin{array}{c}0\\{}-{x}_1{x}_3\\{}{x}_1{x}_2\end{array}\right), $$ where |$d$|⁠, |$r$| and |$g$| are the positive parameters. Given some special parameter values, this system exhibits the complex chaos behaviour (Wu et al., 2007). Using the similar methods as Subsection 4.1, it follows from Theorem 1 and Remark 5 that Proposition 4. For any positive parameters |$d$|⁠, |$r$| and |$g$|⁠, the global finite-time synchronization of the master–slave Lorenz systems under the VSFC $$\begin{equation} {u}_d:{x}_d={x}_1,\kern1em {u}_r={\left({k}_{r1}{\left|{x}_2-{z}_2\right|}^{\alpha}\mathit{\operatorname{sign}}\left({x}_2-{z}_2\right),{k}_{r2}{\left|{x}_3-{z}_3\right|}^{\alpha}\mathit{\operatorname{sign}}\left({x}_3-{z}_3\right)\right)}^T, \end{equation}$$(30) can be achieved at time $$\begin{equation} T\left({x}_0,{z}_0\right)\le{T}_E=\frac{2}{-{\lambda}_{max}\left(1-\alpha \right)}\ln \left[1-\frac{\lambda_{max}}{2{c}_4}{\left({\left({x}_2(0)-{z}_2(0)\right)}^2+{\left({x}_3(0)-{z}_3(0)\right)}^2\right)}^{\frac{1-\alpha }{2}}\right], \end{equation}$$(31) where |$\alpha \in \Big(0,1\Big)$|⁠, |${c}_4=\min \Big\{{k}_{r1},{k}_{r2}\Big\}$| and |${\lambda}_{max}=\Big|g-1\Big|-g-1$|⁠. Proposition 5. If there exists an arbitrarily small positive real |$\zeta$| such that $$\begin{equation} 4 gd>{\zeta}^2, \end{equation}$$(32) and then the global finite-time synchronization of the master–slave Lorenz systems under the VSFC $$\begin{equation} {u}_d:{x}_d={x}_2,\kern1em {u}_r={\left({k}_{r1}{\left|{x}_1-{z}_1\right|}^{\alpha}\mathit{\operatorname{sign}}\left({x}_1-{z}_1\right),{k}_{r2}{\left|{x}_3-{z}_3\right|}^{\alpha}\mathit{\operatorname{sign}}\left({x}_3-{z}_3\right)\right)}^T, \end{equation}$$(33) can be achieved at time $$\begin{equation} T\left({x}_0,{z}_0\right)\le{T}_E=\frac{2}{-{\lambda}_{max}\left(1-\alpha \right)}\ln \left[1-\frac{\lambda_{max}}{2{c}_5}{\left({\left({x}_1(0)-{z}_1(0)\right)}^2+{\left(\frac{\zeta }{\rho_2}\right)}^2{\left({x}_3(0)-{z}_3(0)\right)}^2\right)}^{\frac{1-\alpha }{2}}\right], \end{equation}$$(34) where |$\alpha \in \Big(0,1\Big)$|⁠, |${c}_5=\min \Big\{{k}_{r1},{k}_{r2}{\Big(\frac{\zeta }{\rho_2}\Big)}^{\Big(1-\alpha \Big)}\Big\}$| and |${\lambda}_{max}=\sqrt{{\Big(g-d\Big)}^2+{\zeta}^2}-g-d$|⁠. Proposition 6. If the system’s parameters satisfy $$\begin{equation} 1-r-{\rho}_3>0, \end{equation}$$(35) and then the global finite-time synchronization of the master–slave Lorenz systems under the VSFC $$\begin{equation} {u}_d:{x}_d={x}_3,\kern1em {u}_r={\left({k}_{r1}{\left|{x}_1-{z}_1\right|}^{\alpha}\mathit{\operatorname{sign}}\left({x}_1-{z}_1\right),{k}_{r2}{\left|{x}_2-{z}_2\right|}^{\alpha}\mathit{\operatorname{sign}}\left({x}_2-{z}_2\right)\right)}^T, \end{equation}$$(36) can be achieved at time $$\begin{equation} T\left({x}_0,{z}_0\right)\le{T}_E=\frac{2}{-{\lambda}_{max}\left(1-\alpha \right)}\ln \left[1-\frac{\lambda_{max}}{2{c}_6}{\left({\left({x}_1(0)-{z}_1(0)\right)}^2+\frac{d}{r+{\rho}_3}{\left({x}_2(0)-{z}_2(0)\right)}^2\right)}^{\frac{1-\alpha }{2}}\right], \end{equation}$$(37) where |$\alpha \in \Big(0,1\Big)$|⁠, |${c}_6=\min \Big\{{k}_{r1},{k}_{r2}{\Big(\frac{d}{r+{\rho}_3}\Big)}^{\Big(1-\alpha \Big)/2}\Big\}$|⁠, |${\lambda}_{max}=\sqrt{{\Big(1-d\Big)}^2+4d\Big(r+{\rho}_3\Big)}-d-1$|⁠, |${\rho}_3$| is a bound of state component |${x}_3$| of the master Lorenz system. In Li & Tian (2003), the ANC is also proposed to achieve the finite-time synchronization of two Lorenz systems. Under the notations of this paper, the master–slave synchronization scheme proposed in Li & Tian (2003) can be rewritten as follows: $$\begin{equation} \left\{\begin{array}{l} Master:\kern1em \dot{x}= Ax+f(x),\\{} Slave:\kern1em \dot{z}= Az+f(z)+u\left(x,z\right),\end{array}\right. \end{equation}$$(38) where |$x={\Big({x}_1,{x}_2,{x}_3\Big)}^T$|⁠, |$z={\Big({z}_1,{z}_2,{z}_3\Big)}^T$|⁠, the ANC is $$\begin{equation} u\left(x,z\right)=\left(\begin{array}{c}d\left(-{x}_1+{x}_2+{z}_1-{z}_2\right)+{k}_1{\left|{x}_1-{z}_1\right|}^{\alpha}\mathit{\operatorname{sign}}\left({x}_1-{z}_1\right)\\{}r\left({x}_1-{z}_1\right)-{x}_2+{z}_2-{x}_1{x}_3+{z}_1{z}_3+{k}_2{\left|{x}_2-{z}_2\right|}^{\alpha}\mathit{\operatorname{sign}}\left({x}_2-{z}_2\right)\\{}g\left(-{x}_3+{z}_3\right)+{x}_1{x}_2-{z}_1{z}_2+{k}_3{\left|{x}_3-{z}_3\right|}^{\alpha}\mathit{\operatorname{sign}}\left({x}_3-{z}_3\right)\end{array}\right), \end{equation}$$(39) with the constant |${k}_i\Big(i=1,2,3\Big)$| and |$\alpha \in \Big(0,1\Big)$|⁠. It is concluded by Li & Tian (2003) that for arbitrary parameters of the master and slave Lorenz systems, the global finite-time synchronization of the master–slave Lorenz systems under the ANC (39) can be achieved. Remark 6 It is worth noticing that, besides the non-Lipschitz functions, the ANC (39) contain many other nonlinear terms, which are used to eliminate the nonlinear terms in the error system obtained from the master–slave Lorenz systems. Clearly, the ANC (39) is much more complex than the VSFC designed in this paper. This will lead to the plight that the control designed is too complex to be easily implemented in practical engineering applications and also lead to the information loss of the master–slave Lorenz systems in the synchronization process. In the following, by computer simulations, a numerical example coming from Li & Tian (2003), where the master and slave Lorenz systems’ parameters are chosen as |$d=10$|⁠, |$g=8/3$| and |$r=28$|⁠, is introduced to compare the global finite-time synchronization via the VSFC designed in this paper with the global finite-time synchronization via the ANC designed in Li & Tian (2003). In the simulations, the initial conditions of the master–slave Lorenz systems are arbitrarily selected as |$x(0)={\Big(1.2,\kern0.5em 3.7,\kern0.5em 15\Big)}^T$| and |$z(0)={\Big(-15.7,\kern0.5em 66.3,\kern0.5em 30.2\Big)}^T$|⁠, respectively. For Example 3 shown in Table 3, the master–slave Lorenz systems are chaotic and it is true that |${\rho}_2=30$|⁠, as illustrated in Fig. 3(a–b). After a simple calculation, this example agrees with Proposition 4 and Proposition 5 in this paper simultaneously, which means that the global finite-time synchronization of the master–slave Lorenz systems can be achieved via the VSFC (30) with |$\alpha =1/2$| and |$K=\mathit{\operatorname{diag}}\Big\{20,4\Big\}$| or via the VSFC (33) with and . In the same time, under the given parameters, the finite-time synchronization can also be achieved via the ANC (39) with , , and designed in Li & Tian (2003). The finite-time synchronization via the VSFC (30) and that via the control (39) are compared, as shown in Fig. 3(c). And the control inputs of the VSFC (30) and the ANC (39) are also compared with each other in Fig. 3(d). Then, the similar comparisons between the VSFC (33) and the control (39) are also fulfilled in Fig. 3(e–f). The corresponding synchronization time are estimated, as summarized in Table 3. The detailed synchronization comparison information has been listed in Table 3. The simulation results show that the finite-time synchronization via the VSFCs can achieve synchronization in a shorter time than that via the ANCs, whereas the control inputs of the VSFC are still smaller than those of the ANC. 5. Conclusions A simple, continuous and dimension-reducible VSFC has been designed to investigate the global finite-time synchronization of a class of third-order chaotic systems with some intersecting nonlinearities. The master–slave finite-time synchronization scheme under the VSFC for this class of chaotic systems is established, which is followed by a proven criterion for the global finite-time synchronization as well as the mathematically estimated synchronization time expressed in an explicit formula. Subsequently, the obtained criterion and optimization technique are applied to the BLDCM system and the Lorenz system respectively, and some optimized synchronization criteria in the form of algebra are further obtained. Finally, three numerical examples are simulated to verify that the theoretical results are consistent with the simulation results. Funding National Natural Science Foundation of China (grant no. 11202239); Science and Technology Foundation (grant no. 2101133); Natural Science Foundation of Naval University of Engineering (grant no. 425317Q063). Conflict of interests No potential conflict of interest was declared by the authors. Acknowledgement The authors really appreciate the valuable comments from the editors and reviewers. References Aghababa , M. P. ( 2012 ) Finite-time chaos control and synchronization of fractional-order chaotic (hyperchaotic) systems via fractional nonsingular terminal sliding mode technique . Nonlinear Dynam. , 9 , 247 – 261 . Google Scholar OpenURL Placeholder Text WorldCat Aghababa , M. P. & Aghababa , H. P. ( 2012 ) A general nonlinear adaptive control scheme for finite-time synchronization of chaotic systems with uncertain parameters and nonlinear inputs . 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Google Scholar Crossref Search ADS WorldCat © The Author(s) 2021. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved. This article is published and distributed under the terms of the Oxford University Press, Standard Journals Publication Model (https://academic.oup.com/journals/pages/open_access/funder_policies/chorus/standard_publication_model)
TI - The global finite-time synchronization of a class of chaotic systems via the variable-substitution and feedback control
JF - IMA Journal of Mathematical Control and Information
DO - 10.1093/imamci/dnaa041
DA - 2021-02-13
UR - https://www.deepdyve.com/lp/oxford-university-press/the-global-finite-time-synchronization-of-a-class-of-chaotic-systems-QKDbhNsYBh
SP - 1
EP - 1
VL - Advance Article
IS - 
DP - DeepDyve
ER -