Global asymptotic synchronization of a class of non-linear systems via sampled-data feedback

Abstract

In this paper, we investigate the problem of using sampled-data feedback to synchronize a slave (driven) system with a master (driver) system. Based on the domination approach, both state-feedback and output-feedback control methods using sampled-data are proposed to make the tracking error converge to zero. The problem is of practical importance since in practice the system state is transmitted as sampled signal, and very often only the output is measurable. The effectiveness of the proposed approach is illustrated by the simulation for a chaotic Chua oscillator.

Keywords

Non-linear systems sampled-data control synchronization

Introduction

The history of the analysis of synchronization phenomena can be traced back to the finding of Huygens that two very weakly coupled pendulum clocks (hanging at the same beam) become synchronized in phase (Boccaletti et al., 2002). Other early examples include the synchronized lightning of fireflies and the peculiarities of adjacent organ pipes (Blekhman, 1988). In recent years, increasing attention has been paid to the study of synchronization between systems due to its useful applications in secure communication, biological systems, information processing and chemical reactions. The basic idea of synchronization is to design a suitable controller for the slave (driven) system so that the states of the slave system can track the states of the master (driver) system. The synchronization problem has been well-studied under various conditions and some effective approaches have been proposed, for instance, feedback control in Huang et al. (2009, 2012), adaptive control in Hu and Xu (2008), Elabbasy et al. (2006) and Wei et al. (2014a), impulsive control in Yang and Chua (1997) and Liu (2001), fuzzy logic control in Li and Ge (2011), and intermittent control in Huang and Li (2010).

Most of the results mentioned above focus on the controllers using continuous signal. In practice, however, more and more controllers are being implemented using digital computers, which necessitates the investigation of sampled-data systems, as mentioned in Åström and Wittenmark (2011) and Gyurkovics and Elaiw (2004). Compared with the tremendous results achieved for non-linear systems under a continuous-time controller, there are relatively few works on designing sampled-data ones. Recently, the problem of non-linear system stabilization using sample-data controller was studied in Qian and Du (2012) and Ahrens et al. (2009), and some results for discrete-time systems have been reported by Liu et al. (2012) and Yang and Chen (2002). A hybrid system method was employed in Nešic et al. (2009), showing that global stability of the closed-loop system using a sampled-data output-feedback stabilizer could be guaranteed. For linear systems, since a sampled linear continuous-time system is equivalent to a linear discrete-time system, the stabilization problems can be easily solved, but it is not a trivial problem for non-linear systems. In the case when full-states are not available, the global output-feedback stabilization problem of non-linear systems is a challenging one even in the case of using continuous-time feedback. There are fewer results available for the output-feedback stabilization problem of non-linear systems using discrete-time feedback (Ahrens et al., 2009; Chai and Qian, 2014; Khalil, 2004). Another useful method based on adaptive fuzzy design has been used for systems with backlash, dead-zones, and unknown dynamics (Liu and Tong, 2014, 2015a,b).

An interesting problem is to design a sampled-data controller to achieve synchronization for a more general class of non-linear systems. The problem of sampled-data synchronization has been discussed in previous work based on linear matrix inequalities. For instance, a set of sampled-data synchronization controllers were designed for a class of dynamical networks in Shen et al. (2012) and a sampled-data control approach was proposed for the synchronization of chaotic Lur’e systems in Wu et al. (2013).

In this paper, the domination approach will be applied to solve the problem of global asymptotic sampled-data synchronization. We consider a class of master (driver) systems in the lower-triangular form, which has been widely discussed in the literature (Isidori, 1995)

\begin{matrix} {\overset{\cdot}{x}}_{1} = x_{2} + ϕ_{1} (x_{1}) \\ ⋮ \\ {\overset{\cdot}{x}}_{n - 1} = x_{n} + ϕ_{n - 1} (x_{1}, \dots, x_{n - 1}) \\ {\overset{\cdot}{x}}_{n} = ϕ_{n} (x_{1}, \dots, x_{n}) \\ y = x_{1} \end{matrix}

(1)

where $x = (x_{1}, \dots, x_{n})^{T} \in R$ and y are the system states and system output respectively, $ϕ_{i} (x_{1}, \dots, x_{i})$ , $i = 1, 2, \dots, n$ are continuous functions. We assume that the states of system are bounded. The following two control problems will be solved.

Problem 1. To design a sampled-data state-feedback controller $u (t) = u (t_{k}), \forall t \in [t_{k}, t_{k + 1}), t_{k} = kT, k \in N^{+}$ and a sampling period T for the following slave (response) system

\begin{matrix} {\overset{\cdot}{z}}_{1} = z_{2} + ϕ_{1} (z_{1}) \\ ⋮ \\ {\overset{\cdot}{z}}_{n - 1} = z_{n} + ϕ_{n - 1} (z_{1}, \dots, z_{n - 1}) \\ {\overset{\cdot}{z}}_{n} = ϕ_{n} (z_{1}, \dots, z_{n}) + u \end{matrix}

(2)

where $z = (z_{1}, \dots, z_{n})^{T} \in R$ are the system states, such that the states of system (2) will track the corresponding states in system (1) asymptotically. Problem 1 is illustrated in Figure 1.

Figure 1.

Asymptotic synchronization with sampled-data state-feedback control.

Problem 2. To design a series of sampled-data inputs $s = [s_{1}, s_{2}, \dots, s_{n}]^{T}$ (using system output $x_{1} (t_{k})$ and $z_{1} (t_{k})$ only) and a sampling period T for the following slave (response) system

\begin{matrix} {\overset{\cdot}{z}}_{1} = z_{2} + ϕ_{1} (z_{1}) + s_{1} (x_{1} (t_{k}), z_{1} (t_{k})) \\ ⋮ \\ {\overset{\cdot}{z}}_{n - 1} = z_{n} + ϕ_{n - 1} (z_{1}, \dots, z_{n - 1}) + s_{n - 1} (x_{1} (t_{k}), z_{1} (t_{k})) \\ {\overset{\cdot}{z}}_{n} = ϕ_{n} (z_{1}, \dots, z_{n}) + s_{n} (x_{1} (t_{k}), z_{1} (t_{k})) \end{matrix}

(3)

where $z_{1}$ is the system output, such that the states of system (3) will track the corresponding states in system (1) asymptotically. The slave system (3) can be an artificially built system only using the sampled-data output of the master system. Problem 2 is illustrated in Figure 2.

Figure 2.

Asymptotic synchronization with sampled-data output-feedback control.

The main contributions of this paper are as follows.

In practice, continuous-time controllers are not available, and even the intermittent controllers are not always applicable. On this note, sampled-data controllers are designed to solve the synchronization problems.

We prove that a globally stabilizing output-feedback controller exists and can be designed for the non-linear systems under a general condition.

The domination approach is successfully applied to construct controllers which are linear and easy to implement.

The rest of this paper is organized as follows. First, we construct a linear sampled-data state-feedback controller to ensure the asymptotic convergence of synchronization error. After that, we design a series of sampled-data inputs to achieve asymptotic synchronization. Our conclusion is given in the last section.

Linear sampled-data state-feedback for synchronization

In this section, we will design a state-feedback controller for the slave system (2) to track the states of system (1). In Huang and Li (2010), the intermittent feedback method is applied to the synchronization of systems. In order to achieve synchronization, a series of sampled-data state inputs from the master system is used, and three parameters need to be determined. Compared with Huang and Li (2010), we use sampled-data state-feedback control instead of an intermittent feedback method, so only two parameters need to be chosen during the design process. Moreover, the state-feedback controller we design will be implemented on the last dynamic $x_{n}$ only. Based on the domination approach introduced in Qian (2005), we develop a sampled-data state-feedback controller $u (x_{1} (t_{k}), \dots, x_{n} (t_{k}), z_{1} (t_{k}), \dots, z_{n} (t_{k}))$ to synchronize systems (1) and (2).

Linear continuous-time state-feedback control for synchronization

First, we design a continuous-time controller for the slave system (2), which will be a discrete-time case by using sampled-data control. To achieve our control objective, we assume that the non-linearity satisfies the following condition.

Assumption 1. The function $ϕ_{i}$ ( $i = 1, 2, \dots, n$ ) is a Lipschitz continuous function, i.e. there is a positive constant $C_{0}$ such that

| ϕ_{i} (x_{1}, \dots, x_{i}) - ϕ_{i} (y_{1}, \dots, y_{i}) | \leq C_{0} \sum_{k = 1}^{i} | x_{k} - y_{k} |

(4)

The above Lipschitz condition is a common assumption for output-feedback stabilization which has been used in existing work, e.g. Khalil and Saberi (1987) and Gauthier et al. (1992).

Theorem 1. Under Assumption 1, the problem of synchronization of systems (1) and (2) can be solved by the following continuous-time state-feedback controller

u = L^{n} [K_{1} (x_{1} - z_{1}) + K_{2} \frac{(x_{2} - z_{2})}{L} + \dots + K_{n} \frac{(x_{n} - z_{n})}{L^{n - 1}}]

(5)

where $K_{i} \in R$ are coefficients of a Hurwitz polynomial $s^{n} + K_{1} s^{n - 1} + \dots + K_{n}$ andLis a large enough gain.

Proof. Substituting equation (5) into equation (2) yields

\begin{matrix} {\overset{\cdot}{z}}_{1} = z_{2} + ϕ_{1} (z_{1}) \\ ⋮ \\ {\overset{\cdot}{z}}_{n - 1} = z_{n} + ϕ_{n - 1} (z_{1}, \dots, z_{n - 1}) \\ {\overset{\cdot}{z}}_{n} = ϕ_{n} (z_{1}, \dots, z_{n}) + L^{n} [K_{1} (x_{1} - z_{1}) + K_{2} \frac{(x_{2} - z_{2})}{L} + \dots + K_{n} \frac{(x_{n} - z_{n})}{L^{n - 1}}] \end{matrix}

(6)

By introducing the change of coordinates $ε_{i} = \frac{x_{i} - z_{i}}{L^{i - 1}}$ for $i = 1, \dots, n$ , we can obtain the following error dynamics

\begin{matrix} {\overset{\cdot}{ε}}_{1} = L ε_{2} + {\tilde{ϕ}}_{1} (\cdot) \\ ⋮ \\ {\overset{\cdot}{ε}}_{n - 1} = L ε_{n} + \frac{{\tilde{ϕ}}_{n - 1} (\cdot)}{L^{n - 2}} \\ {\overset{\cdot}{ε}}_{n} = \frac{{\tilde{ϕ}}_{n} (\cdot)}{L^{n - 1}} - L (K_{1} ε_{1} + \dots + K_{n} ε_{n}) \end{matrix}

(7)

where ${\tilde{ϕ}}_{i} = ϕ_{i} (x_{1}, \dots, x_{i}) - ϕ_{i} (z_{1}, \dots, z_{i}), i = 1, \dots, n$ .

Noting that equation (7) can be rewritten in the following form

\overset{\cdot}{ε} = L (A - BK) ε + Φ

(8)

where

A = (\begin{matrix} 0 & 1 & 0 & \dots & 0 \\ 0 & 0 & 1 & \dots & 0 \\ ⋮ & ⋮ & ⋮ & ⋱ & ⋮ \\ 0 & 0 & 0 & \dots & 1 \\ 0 & 0 & 0 & \dots & 0 \end{matrix}), B = (\begin{matrix} 0 \\ 0 \\ ⋮ \\ 1 \end{matrix}), K = [K_{1}, K_{2}, \dots, K_{n}], Φ = (\begin{matrix} {\tilde{ϕ}}_{1} (\cdot) \\ \frac{{\tilde{ϕ}}_{2} (\cdot)}{L} \\ ⋮ \\ \frac{{\tilde{ϕ}}_{n} (\cdot)}{L^{n - 1}} \end{matrix})

By Assumption 1, one obtains, for $i = 1, \dots, n$

\begin{matrix} | Φ_{i} | = | \frac{ϕ_{i} (x_{1}, \dots, x_{i}) - ϕ_{i} (z_{1}, \dots, z_{i})}{L^{i - 1}} | \\ \leq C_{0} \frac{(| x_{1} - z_{1} | + | x_{2} - z_{2} | + \dots + | x_{i} - z_{i} |)}{L^{i - 1}} \end{matrix}

(9)

Then we can achieve

\begin{matrix} ∥ Φ ∥ \leq C_{0} \sqrt{| ε_{1} |^{2} + {(\sum_{i = 1}^{2} | ε_{i} |)}^{2} + \dots + {(\sum_{i = 1}^{n} | ε_{i} |)}^{2}} \\ \leq C_{0} \sqrt{n {(\sum_{i = 1}^{n} | ε_{i} |)}^{2}} \leq n C_{0} ∥ ε ∥ \end{matrix}

(10)

It is not hard to see that there exists a constant ${\hat{C}}_{0}$ such that

∥ Φ ∥ \leq {\hat{C}}_{0} ∥ ε ∥

(11)

where ${\hat{C}}_{0} \geq n C_{0}$ is a positive constant.

By the selection of K, $\hat{A} = A - BK$ is a Hurwitz matrix (i.e. the real part of eigenvalues of matrix $\hat{A} < 0$ ). Hence, there is a positive definite matrix $P = P^{T} > 0$ such that

{\hat{A}}^{T} P + P \hat{A} = - I

(12)

Now we construct a positive definite Lyapunov function

V = ε^{T} P ε

(13)

With the help of equations (7) and (11), we evaluate and estimate the derivative of $V (ε)$

\begin{matrix} \overset{\cdot}{V} & = L ε^{T} ({\hat{A}}^{T} P + P \hat{A}) ε + 2 Φ^{T} P ε \\ \leq - L ∥ ε ∥^{2} + 2 {\hat{C}}_{0} ε^{T} ∥ P ∥ ∥ ε ∥ \\ = - (L - 2 {\hat{C}}_{0} ∥ P ∥) ∥ ε ∥^{2} \end{matrix}

(14)

Obviously, a large enough gain $L > 2 {\hat{C}}_{0} ∥ P ∥ + 1$ yields

\overset{\cdot}{V} \leq - ∥ ε ∥^{2} \leq 0

(15)

which implies that the two systems (1) and (2) synchronize asymptotically.

Linear sampled-data state-feedback control for synchronization

Now we consider the case of using sampled-data control inputs for the slave system (2).

Theorem 2. Under Assumption 1, the problem of synchronization of systems (1) and (2) can be solved by the following sampled-data state-feedback controller

\begin{matrix} u (t) = L^{n} [K_{1} (x_{1} (t_{k}) - z_{1} (t_{k})) + K_{2} \frac{(x_{2} (t_{k}) - z_{2} (t_{k}))}{L} + \dots \\ + K_{n} \frac{(x_{n} (t_{k}) - z_{n} (t_{k}))}{L^{n - 1}}], t \in [kT, (k + 1) T] \end{matrix}

(16)

Proof. When we design the sampled-data controller, the states $x_{i}$ and $z_{i}$ in controller (5) change to $x_{i} (t_{k})$ and $z_{i} (t_{k})$ , which are the values of $x_{i}$ and $z_{i}$ at time instant $t_{k}$ respectively. Therefore the slave system (6) becomes

\begin{matrix} {\overset{\cdot}{z}}_{1} = z_{2} + ϕ_{1} (z_{1}) \\ ⋮ \\ {\overset{\cdot}{z}}_{n - 1} = z_{n} + ϕ_{n - 1} (z_{1}, \dots, z_{n - 1}) \\ {\overset{\cdot}{z}}_{n} = ϕ_{n} (z_{1}, \dots, z_{n}) + L^{n} [K_{1} (x_{1} (t_{k}) - z_{1} (t_{k})) \\ + K_{2} \frac{(x_{2} (t_{k}) - z_{2} (t_{k}))}{L} + \dots + K_{n} \frac{(x_{n} (t_{k}) - z_{n} (t_{k}))}{L^{n - 1}}] \end{matrix}

(17)

Consequently, the error dynamics equation (7) becomes

\begin{matrix} {\overset{\cdot}{ε}}_{1} = L ε_{2} + {\tilde{ϕ}}_{1} (\cdot) \\ ⋮ \\ {\overset{\cdot}{ε}}_{n - 1} = L ε_{n} + \frac{{\tilde{ϕ}}_{n - 1} (\cdot)}{L^{n - 2}} \\ {\overset{\cdot}{ε}}_{n} = \frac{{\tilde{ϕ}}_{n} (\cdot)}{L^{n - 1}} - L [K_{1} ε_{1} (t_{k}) + \dots + K_{n} ε_{n} (t_{k})] \end{matrix}

(18)

and the derivative of Lyapunov function $V (ε)$ turns to

\overset{\cdot}{V} = - L ε^{T} ({\hat{A}}^{T} P + P \hat{A}) ε + 2 Φ^{T} P ε + 2 L ε^{T} P [BK (ε - ε (t_{k}))]

(19)

By Assumption 1, we can see that

\begin{matrix} ∥ \overset{\cdot}{ε} ∥ = ∥ L \hat{A} ε + Φ + LBK (ε - ε (t_{k})) ∥ \\ \leq (L ∥ \hat{A} ∥ + {\hat{C}}_{0}) ∥ ε ∥ + L ∥ K ∥ ∥ ε - ε (t_{k}) ∥ \\ \leq (L ∥ \hat{A} ∥ + {\hat{C}}_{0} + L ∥ K ∥) ∥ ε - ε (t_{k}) ∥ + (L ∥ \hat{A} ∥ + {\hat{C}}_{0}) ∥ ε (t_{k}) ∥ \end{matrix}

(20)

With the aid of Lemma 1 (Appendix 1), we have

∥ ε - ε (t_{k}) ∥ \leq \frac{δ (t - t_{k})}{1 - δ (t - t_{k})} ∥ ε ∥

(21)

where

δ (t - t_{k}) = \frac{L ∥ \hat{A} ∥ + {\hat{C}}_{0}}{L (∥ \hat{A} ∥ + ∥ K ∥) + {\hat{C}}_{0}} \times (e^{(t - t_{k}) (L (∥ \hat{A} ∥ + ∥ K ∥) + {\hat{C}}_{0})} - 1)

(22)

Substituting equation (21) into equation (19), we get

\begin{matrix} \overset{\cdot}{V} \leq - (L - 2 {\hat{C}}_{0} ∥ P ∥) ∥ ε ∥^{2} + 2 L ∥ P ∥ ∥ K ∥ ∥ ε ∥ ∥ ε - ε (t_{k}) ∥ \\ \leq - (L - 2 {\hat{C}}_{0} ∥ P ∥) ∥ ε ∥^{2} + \frac{2 L δ (t - t_{k})}{1 - δ (t - t_{k})} ∥ P ∥ ∥ K ∥ ∥ ε ∥^{2} \end{matrix}

(23)

Clearly, for a fixed $L \geq 2 {\hat{C}}_{0} ∥ P ∥ + 1$ , a small enough T can yield

\begin{matrix} \overset{\cdot}{V} \leq - (L - 2 {\hat{C}}_{0} ∥ P ∥ - 2 L ∥ P ∥ ∥ K ∥ \frac{δ (t - t_{k})}{1 - δ (t - t_{k})}) ∥ ε ∥^{2} \\ \leq - \frac{1}{2} ∥ ε ∥^{2}, \forall t \in [t_{k}, t_{k + 1}) \end{matrix}

(24)

We can conclude that the states of system (2) will asymptotically track the states of system (1) by using the sample-data state-feedback controller (16).

An example

In this section, we will study numerically the synchronization for the original Chua oscillators by applying the theory presented in the previous section. The chaotic synchronization problem has been well-studied under various conditions and some effective approaches have been proposed in recent years (Huang et al., 2012; Huang and Li, 2010; Wang et al., 2010; Wei et al., 2014a,b,c,d,e,f, 2015). As a class of chaotic system, the Chua oscillators intrinsically defy synchronization, because even two identical systems starting from slightly different initial conditions would evolve in time in an unsynchronized manner (the differences in the system states could grow exponentially) (Boccaletti et al., 2002).

The Chua oscillator is given as

\begin{matrix} {\overset{\cdot}{x}}_{1} = α (x_{2} - x_{1} - g (x_{1})) \\ {\overset{\cdot}{x}}_{2} = x_{1} - x_{2} + x_{3} \\ {\overset{\cdot}{x}}_{3} = - β x_{2} \end{matrix}

(25)

where

g (x_{1}) = b x_{1} + 0.5 (a - b) (| x_{1} + 1 | - | x_{1} - 1 |)

(26)

with $a < b < 0$ being two constants. We choose the system parameters as $α = 9.2156$ , $β = 15.9946$ , $a = 1.2495$ and $b = - 0.75735$ .

Under the new coordinates $χ_{1} = x_{1}$ , $χ_{2} = α x_{2}$ , $χ_{3} = α x_{3}$ , system (25) can be rewritten as

\begin{matrix} {\overset{\cdot}{χ}}_{1} = χ_{2} - α (χ_{1} + g (χ_{1})) \\ {\overset{\cdot}{χ}}_{2} = χ_{3} + α χ_{1} - χ_{2} \\ {\overset{\cdot}{χ}}_{3} = - β χ_{2} \end{matrix}

(27)

It’s not hard to see that the above system is in the same form as system (1), $ϕ_{1} (\cdot) = - α (χ_{1} + g (χ_{1}))$ and $ϕ_{2} (\cdot) = α χ_{1} - χ_{2}$ satisfy the Lipschitz condition. Therefore, system (27) fulfils Assumption 1. According to Theorem 2, a sampled-data state-feedback controller can be designed for the slave system as

\begin{matrix} {\overset{\cdot}{Z}}_{1} = Z_{2} - α (Z_{1} + g (Z_{1})) \\ {\overset{\cdot}{Z}}_{2} = Z_{3} + α Z_{1} - Z_{2} \\ {\overset{\cdot}{Z}}_{3} = - β Z_{2} + L^{3} [K_{1} (χ_{1} (t_{k}) - Z_{1} (t_{k})) + K_{2} \frac{χ_{2} (t_{k}) - Z_{2} (t_{k})}{L} \\ + K_{3} \frac{χ_{3} (t_{k}) - Z_{3} (t_{k})}{L^{2}}] \end{matrix}

(28)

with $Z_{1} = z_{1}$ , $Z_{2} = α z_{2}$ , $Z_{3} = α z_{3}$ , such that systems (1) and (2) can synchronize.

In the following simulation, we choose $K = [3, 1, 1]^{T}$ , $L = 2$ and sampling time $T = 0.01$ s. The initial conditions are chosen as $x (0) = [- 2, 1, - 1]^{T}$ for the master system and $z (0) = [3, - 1, 2]^{T}$ for the slave system.

From Figure 3, it can be seen that the slave system can synchronize with the master system by using the proposed sampled-data state-feedback controller. Figure 4 shows that by applying the same controller, synchronization can be achieved under different sampling periods.

Figure 3.

The two Chua systems synchronize under sampled-data state-feedback.

Figure 4.

The tracking errors under different sampling periods.

Linear sampled-data output-feedback for synchronization

In this section, we will develop a series of sampled-data inputs $s_{i} (i = 1, 2, \dots, n)$ and only use measurable states to achieve the synchronization for systems (1) and (3). Like the previous section, we design the continuous-time control inputs first, then extend them to a discrete case.

Linear continuous-time output-feedback control for synchronization

First, we design the continuous-time control inputs for the slave system (3).

Theorem 3. Under Assumption 1, the second synchronization problem of systems (1) and (3) can be solved by applying the following continuous output-feedback on the slave system

s = [s_{1} (e_{1}), s_{2} (e_{1}), \dots, s_{n} (e_{1})]^{T} = \hat{K} e_{1}

(29)

where $\hat{K} = [L K_{1}, L^{2} K_{2}, \dots, L^{n} K_{n}]^{T}$ and $e_{1} = x_{1} - z_{1}$ which is the difference between the outputs of the master system (1) and the slave system (3).

Proof. To prove Theorem 3, we construct the controlled system with a scaling gain $L > 1$ to be determined later and series of gains $K_{i}$ , for $i = 1, \dots, n$

\begin{matrix} {\overset{\cdot}{z}}_{1} = z_{2} + ϕ_{1} (z_{1}) + L K_{1} (x_{1} - z_{1}) \\ ⋮ \\ {\overset{\cdot}{z}}_{n - 1} = z_{n} + ϕ_{n - 1} (z_{1}, \dots, z_{n - 1}) + L^{n - 1} K_{n - 1} (x_{1} - z_{1}) \\ {\overset{\cdot}{z}}_{n} = ϕ_{n} (z_{1}, \dots, z_{n}) + L^{n} K_{n} (x_{1} - z_{1}) \end{matrix}

(30)

By introducing the change of coordinates $ε_{i} = \frac{x_{i} - z_{i}}{L^{i - 1}}$ for $i = 1, \dots, n$ , the error dynamics can be written as

\begin{matrix} {\overset{\cdot}{ε}}_{1} = L ε_{2} + {\tilde{ϕ}}_{1} (.) - L K_{1} ε_{1} \\ ⋮ \\ {\overset{\cdot}{ε}}_{n - 1} = L ε_{n} + \frac{{\tilde{ϕ}}_{n - 1} (.)}{L^{n - 2}} - L K_{n - 1} ε_{1} \\ {\overset{\cdot}{ε}}_{n} = \frac{{\tilde{ϕ}}_{n} (.)}{L^{n - 1}} - L K_{n} ε_{1} \end{matrix}

(31)

where ${\tilde{ϕ}}_{i} = ϕ_{i} (x_{1}, \dots, x_{i}) - ϕ_{i} (z_{1}, \dots, z_{i}), i = 1, \dots, n$ . Under these new coordinates, the above error dynamics can be rewritten as follows

\overset{\cdot}{ε} = L (A - \bar{K}) ε + Φ

(32)

where A, $Φ$ are defined in equation (8) and $\bar{K} = (\begin{matrix} K_{1} & 0 & \dots & 0 \\ K_{2} & 0 & \dots & 0 \\ ⋮ & ⋮ & ⋱ & ⋮ \\ K_{n} & 0 & \dots & 0 \end{matrix})$ .

By Assumption 1, there exists a constant ${\hat{C}}_{0}$ such that

∥ Φ ∥ \leq {\hat{C}}_{0} ∥ ε ∥

(33)

There exist gains K_i’s such that $\bar{A} = A - \bar{K}$ is a Hurwitz matrix (i.e the real part of eigenvalues of matrix $\bar{A} < 0$ ). Hence, there is a positive definite matrix $P = P^{T} > 0$ such that

{\bar{A}}^{T} P + P \bar{A} = - I

(34)

Now we define a positive definite Lyapunov function

V = ε^{T} P ε

(35)

With the help of equations (31) and (33), we evaluate and estimate the derivative of $V (ε)$

\begin{matrix} \overset{\cdot}{V} & = L ε^{T} ({\bar{A}}^{T} P + P \bar{A}) ε + 2 Φ^{T} P ε \\ \leq - L ∥ ε ∥^{2} + 2 {\hat{C}}_{0} ε^{T} ∥ P ∥ ∥ ε ∥ \\ = - (L - 2 {\hat{C}}_{0} ∥ P ∥) ∥ ε ∥^{2} \end{matrix}

(36)

Obviously, a large enough gain L yields

\overset{\cdot}{V} \leq - ∥ ε ∥^{2} \leq 0

(37)

Therefore, it can be concluded that the two systems synchronize asymptotically.

Linear sampled-data output-feedback control for synchronization

Now we consider the case of using sampled-data control inputs for the slave system (3).

Theorem 4. Under Assumption 1, the problem of synchronization of systems (1) and (3) can be solved by applying the following sampled-data output-feedback on the slave system

\begin{matrix} s (t) = [s_{1} (e_{1} (t_{k})), s_{2} (e_{1} (t_{k})), \dots, s_{n} (e_{1} (t_{k})]^{T} \\ = \hat{K} e_{1} (t_{k}), t \in [kT, (k + 1) T] \end{matrix}

(38)

where $e_{1} (t_{k}) = x_{1} (t_{k}) - z_{1} (t_{k})$ is the difference between the outputs of the master system and the slave system at instant $t_{k}$ .

Proof. When we design the sampled-data control inputs for a proper sampling time T, the output $x_{1}$ in the slave system (30) is replaced by its value at time instant $t_{k}$ : $x_{1} (t_{k})$ , and $z_{1}$ changes to $z_{1} (t_{k})$ . Therefore, the slave system (30) becomes

\begin{matrix} {\overset{\cdot}{z}}_{1} = z_{2} + ϕ_{1} (z_{1}) + L K_{1} e_{1} (t_{k}) \\ ⋮ \\ {\overset{\cdot}{z}}_{n - 1} = z_{n} + ϕ_{n - 1} (z_{1}, \dots, z_{n - 1}) + L^{n - 1} K_{n - 1} e_{1} (t_{k}) \\ {\overset{\cdot}{z}}_{n} = ϕ_{n} (z_{1}, \dots, z_{n}) + L^{n} K_{n} e_{1} (t_{k}) \end{matrix}

(39)

and the error dynamics (31) become

\begin{matrix} {\overset{\cdot}{ε}}_{1} = L ε_{2} + {\tilde{ϕ}}_{1} (\cdot) - L K_{1} ε_{1} (t_{k}) \\ ⋮ \\ {\overset{\cdot}{ε}}_{n - 1} = L ε_{n} + \frac{{\tilde{ϕ}}_{n - 1} (\cdot)}{L^{n - 2}} - L K_{n - 1} ε_{1} (t_{k}) \\ {\overset{\cdot}{ε}}_{n} = \frac{{\tilde{ϕ}}_{n} (\cdot)}{L^{n - 1}} - L K_{n} ε_{1} (t_{k}) \end{matrix}

(40)

which can be written in the following form

\overset{\cdot}{ε} = L (A - \bar{K}) ε + L \bar{K} (ε - ε (t_{k})) + Φ

(41)

The derivative of Lyapunov function $V (ε)$ turns to

\begin{matrix} \overset{\cdot}{V} = & (L ε^{T} {\bar{A}}^{T} + Φ^{T} + L (ε - ε (t_{k}))^{T} {\bar{K}}^{T}) P ε \\ + ε^{T} P (L \bar{A} ε + Φ + L \bar{K} (ε - ε (t_{k}))) \\ = & L ε^{T} {\bar{A}}^{T} P ε + L ε^{T} P \bar{A} ε + 2 ε^{T} P Φ + 2 L ε^{T} P \bar{K} (ε - ε (t_{k})) \\ \leq - L ∥ ε ∥^{2} + 2 {\hat{C}}_{0} ε^{T} ∥ P ∥ ∥ ε ∥ + 2 L ε^{T} ∥ P ∥ ∥ \bar{K} ∥ ∥ ε - ε (t_{k}) ∥ \end{matrix}

(42)

Noting that

\begin{matrix} \overset{\cdot}{ε} = & L \bar{A} ε + Φ + L \bar{K} (ε - ε (t_{k})) \\ \leq (L ∥ \bar{A} ∥ + {\hat{C}}_{0}) ∥ ε ∥ + L ∥ \bar{K} ∥ ∥ ε - ε (t_{k}) ∥ \\ \leq (L ∥ \bar{A} ∥ + {\hat{C}}_{0} + L ∥ \bar{K} ∥) ∥ ε - ε (t_{k}) ∥ \\ + (L ∥ \bar{A} ∥ + {\hat{C}}_{0}) ∥ ε (t_{k}) ∥ \end{matrix}

(43)

by Lemma 1 (Appendix 1), we have

∥ ε - ε (t_{k}) ∥ \leq \frac{δ (t - t_{k})}{1 - δ (t - t_{k})} ∥ ε ∥

(44)

where

δ (t - t_{k}) = \frac{L ∥ \bar{A} ∥ + {\hat{C}}_{0}}{L (∥ \bar{A} ∥ + ∥ \bar{K} ∥) + {\hat{C}}_{0}} (e^{(t - t_{k}) (L (∥ \bar{A} ∥ + ∥ \bar{K} ∥) + {\hat{C}}_{0})} - 1)

(45)

Substituting equation (44) into equation (42), we arrive at

\dot{V} \leq - L ∥ ε ∥^{2} + 2 C_{0} ε^{T} ∥ P ∥ ∥ ε ∥ + 2 L ε^{T} ∥ P ∥ ∥ \bar{K} ∥ \frac{δ (t - t_{k})}{1 - δ (t - t_{k})} ∥ ε ∥

(46)

It is not hard to see that a large enough L and a small enough T can yield

\dot{V} \leq - (L - 2 {\hat{C}}_{0} ∥ P ∥ - 2 L ∥ P ∥ ∥ \bar{K} ∥ \frac{δ (t - t_{k})}{1 - δ (t - t_{k})}) ∥ ε ∥^{2} \leq - \frac{1}{2} ∥ ε ∥^{2}

(47)

Then, we can conclude that system (1) is synchronized with system (3) by the sample-data inputs (38) in discrete-time.

An example

In this section, the Chua oscillator has been used again to verify the proposed theorem. From Theorem 4, a slave system with sampled-data inputs can be designed as

\begin{matrix} {\overset{\cdot}{z}}_{1} = z_{2} - α (z_{1} + g (z_{1})) + L K_{1} (χ_{1} (t_{k}) - z_{1} (t_{k})) \\ {\overset{\cdot}{z}}_{2} = z_{3} + α z_{1} - z_{2} + L^{2} K_{2} (χ_{1} (t_{k}) - z_{1} (t_{k})) \\ {\overset{\cdot}{z}}_{3} = - β z_{2} + L^{3} K_{3} (χ_{1} (t_{k}) - z_{1} (t_{k})) \end{matrix}

(48)

For the purpose of simulation, the initial conditions are set to be $x (0) = [- 2, 1, - 1]^{T}$ for the master system and $z (0) = [3, - 1, 2]^{T}$ for the slave system, and the parameters are chosen as $K = [6, 2, 1]^{T}$ , $L = 2$ and sampling time $T = 0.01$ s. The simulation results are shown in Figure 5, from which we can see that the states of the slave system can converge to the states of the master system. In Figure 6, the performances of synchronization using two proposed methods are presented in three-dimensional plots.

Figure 5.

The two Chua systems synchronize under the sampled-data output-feedback.

Figure 6.

The two Chua systems synchronize under sampled-data feedback (three-dimensional).

The results imply that by applying the proposed sampled-data output-feedback, the slave system can synchronize with the master system.

Conclusion

In this paper, we first show that under a certain condition, a linear sampled-data state-feedback controller can be designed for a class of non-linear systems to guarantee that the master system and slave system synchronize in a period. Then, by carefully choosing the sampling period, a series of sampled-data inputs are designed to achieve synchronization. During the design process, we construct a Lyapunov function for the synchronization error dynamics and prove that the dynamics are asymptotically stable after implementing the designed control laws. Numerical simulations of Chua systems are performed to illustrate that the problem of synchronization can be solved under the proposed control methods.

Footnotes

Appendix 1

Lemma 1.(Chu et al., 2015) Suppose that the following system (49)

(49)

\overset{\cdot}{Z} (t) = F (Z (t), u), u = - KZ (t_{k}), t \in [t_{k}, t_{k + 1})

satisfies $∥ F ∥ \leq d_{1} ∥ Z (t) - Z (t_{k}) ∥ + d_{2} ∥ Z (t_{k}) ∥ + d_{3}$ for positive constants $d_{i}$ , $i = 1, 2, 3$ . Then there is a constant $C = d_{3} / d_{2}$ and time-varying function $δ (t - t_{k}) = \frac{d_{2}}{d_{1}} (e^{d_{1} (t - t_{k})} - 1)$ , such that

∥ Z (t) - Z (t_{k}) ∥ \leq \frac{δ (t - t_{k})}{1 - δ (t - t_{k})} (∥ Z (t) ∥ + C)

Proof. Integrating the system (49), we have

∥ Z (t) - Z (t_{k}) ∥ \leq \int_{t_{k}}^{t} ∥ \overset{\cdot}{z} (s) ∥ ds \leq A (t)

where

A (t) = \int_{t_{k}}^{t} (d_{1} ∥ Z (s) - Z (t_{k}) ∥ + d_{2} ∥ Z (t_{k}) ∥ + d_{3}) ds

It is straightforward to see that

\begin{matrix} \overset{\cdot}{A} (t) = d_{1} ∥ Z (t) - Z (t_{k}) ∥ + d_{2} ∥ Z (t_{k}) ∥ + d_{3} \\ \leq d_{1} A (t) + d_{2} ∥ Z (t_{k}) ∥ + d_{3} \end{matrix}

which implies

\begin{matrix} A (t) \leq δ (t - t_{k}) ∥ Z (t_{k}) + C ∥ \\ ∥ Z (t) - Z (t_{k}) ∥ \leq \frac{δ (t - t_{k})}{1 - δ (t - t_{k})} (∥ Z (t) ∥ + C) \end{matrix}

Declaration of conflicting interests

The authors declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported by a NPRP grant from the Qatar National Research Fund (grant number 4-1162-1-181) and the National Natural Science Foundation of China (grant number 61374038).

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