Anticipating Synchronization and State Predictor for Nonlinear Systems

Abstract

This chapter discusses a synchronization problem, anticipating synchronization, and its application in control design for nonlinear systems with delays. The anticipating synchronization phenomena was initially reported by Voss (2000), and Oguchi and Nijmeijer (2005) then generalized it from the framework of control theory. This chapter revisits the anticipating synchronization problem and introduces a state predictor based on synchronization. Furthermore, we discuss recent progress on predictor design for nonlinear systems with delays.

Appendix: Stability of Systems with Uncertainties

We consider the following system with uncertainties and/or perturbation.

$$\begin{aligned} {\left\{ \begin{array}{ll} \dot{x}(t)=f(t,x_t)+g(t,x_t) , &{} \forall t\ge 0 \\ x_{t_0}=\phi (t) &{} \forall t\in [-\tau , 0], \end{array}\right. } \end{aligned}$$

(5.33)

where $x \in {\mathbb R}^n$, $x_t(\theta )=x(t+\theta )$ for $\theta \in [-\tau ,0]$, $\phi \in {\mathscr {C}}[-\tau ,0]$, and $g(t,x_t)$ denotes the uncertainty and/or perturbation.

For a function $\phi \in {\mathscr {C}}([a,b],{\mathbb R}^n)$, define the continuous norm $\Vert \cdot \Vert _c$ by $\Vert \phi \Vert _c:=\sup _{\theta \in [a, b]}\Vert \phi (\theta )\Vert$, where $\Vert \cdot \Vert$ denotes the Euclidean norm of a vector.

Suppose now that the nominal system

$$\begin{aligned} \dot{x}(t)=f(t,x_t) \end{aligned}$$

(5.34)

has a delay-dependent asymptotically stable equilibrium point at the origin. Then the following theorem is well-known.

Theorem 5.3

([9]) Let there exist a continuous functional $V(t,\phi ):{\mathbb R}\times {\mathscr {C}} [-\tau , 0] \rightarrow {\mathbb R}$ such that

$$ w_1(\Vert \phi (0)\Vert ) \le V(t,\phi ) \le w_2(\Vert \phi \Vert _c),$$

$w_1(r)\rightarrow \infty$ as $r\rightarrow \infty$ and $\dot{V}(x_t)\le 0.$

Let Z be the set of those elements from ${\mathscr {C}}[-\tau ,0]$ for which $\dot{V}=0$ and Q is the largest invariant set situated in Z. Then all solutions of (5.34) tend to Q as $t\rightarrow \infty$. In particular, if the set Q has the only zero element, then the trivial solution of (5.34) is asymptotically stable.

Here $\dot{V}$ is the right derivative of V along the solutions of (5.34), i.e.,

$$ \dot{V}(x_t):=\lim _{h\rightarrow 0+}\sup \frac{1}{h}(V(x_{t+h})-V(x_t)).$$

To derive a sufficient condition for robust stability of perturbed systems, we use the following type of Lyapunov–Krasovskii functional throughout this section.

$$\begin{aligned} V(x_t)=V_0(x(t))+V_1(x_t), \end{aligned}$$

(5.35)

where $V_0$ is a positive-definite function of x(t), and $V_1(x_t)$ consists of the sum of the integrals of the functional $x(t+\theta )$, such as

$$\begin{aligned} V_1(x_t):=&\int _{t-\tau }^t\bar{V}_1(x(\theta ))d\theta +\int _{-\tau }^0\int _{t+s}^t\bar{V}_2(x(\theta ))d\theta ds \\&+\cdots . \end{aligned}$$

Note that the derivative of $V_1(x_t)$ with respect to time does not contain $\dot{x}$.

Then we obtain the following result.

Theorem 5.4

Let $x=0$ be a delay-dependent asymptotically stable equilibrium point of the nominal system (5.34). Let $V(t,x_t):{\mathbb R}\times {\mathscr {C}}\rightarrow {\mathbb R}$ be a Lyapunov–Krasovskii functional (5.35) that satisfies the following conditions:

• (5.4a) $\alpha _1\Vert \psi (0)\Vert ^2\le V(t,\psi )\le \alpha _2\Vert \psi \Vert _c^2$

• (5.4b) the time derivative of V along the trajectories of the unperturbed system (5.34) satisfies $\dot{V}_{(5.34)}(t,\psi )\le -\alpha _3\Vert \psi (0)\Vert ^2$

where $\dot{V}_{(5.34)}$ means the derivation of V along the solution of (5.34) and $\psi \in Q_H:=\{\psi \in {\mathscr {C}}[-\tau , 0]:\Vert \psi \Vert _c<H\}$. In addition, suppose that

• (5.4c) $\Bigl \Vert \frac{\partial V_0(x)}{\partial x}\Bigr \Vert \le \alpha _4\Vert x\Vert$

• (5.4d) $\Vert g(t,0)\Vert =0$

• (5.4e) $\Vert g(t,\psi )\Vert <\gamma \Vert \psi (0)\Vert$

• (5.4f) $\gamma <\frac{\alpha _3}{\alpha _4}$

where $\alpha _i$, $i=1,\ldots ,4$ and $\gamma$ are positive constants. Then, the origin is an asymptotically stable equilibrium point of the perturbed system (5.33). Moreover, if all the assumptions hold globally, then the origin is globally asymptotically stable.

Proof

From the condition (5.4d), $x=0$ is also an equilibrium point of the perturbed system (5.33). The existence of V satisfying condition (5.4a) and (5.4b) guarantees that the zero solution of the unperturbed system (5.34) is asymptotically stable from the Lyapunov–Krasovskii theorem [9, 20]. The time derivative of V along the trajectories of (5.33) satisfies

$$\begin{aligned} \dot{V}(x_t)&\le -\alpha _3\Vert x(t)\Vert ^2+L_gV_0 \\&\le -\alpha _3\Vert x(t)\Vert ^2+\alpha _4\gamma \Vert x(t)\Vert ^2 \\&\le -(\alpha _3-\alpha _4\gamma )\Vert x(t)\Vert ^2. \end{aligned}$$

From the Lyapunov–Krasovskii theorem, if $\gamma < \frac{\alpha _3}{\alpha _4}$, the zero solution of the system (5.33) is also asymptotically stable.

Example 5.1

We consider the following Chua’s circuit with time-delay [21].

$$\begin{aligned} \dot{x}=&\left[ \begin{matrix} -\alpha (1+b) &{} \alpha &{} 0 \\ 1 &{} -1 &{} 1 \\ 0 &{} -\beta &{} -\gamma \\ \end{matrix}\right] x+ \left[ \begin{matrix} -\alpha (a-b) \\ 0 \\ 0 \\ \end{matrix}\right] \varphi (x_1)+\left[ \begin{matrix} 0 \\ 0 \\ -\beta \\ \end{matrix}\right] \varepsilon \sin (\eta x_1(t-\tau )) \nonumber \\ :=&A_0x+B\varphi (x_1)+B_1\xi (x_1(t-\tau )) \\ y(t)=&\left[ \begin{matrix} 1 &{} 0 &{} 0 \\ \end{matrix}\right] x :=Cx, \nonumber \end{aligned}$$

(5.36)

where $\varphi (x_1)=\frac{1}{2}(|x_1+1|-|x_1-1|)$ and $\xi (x_1(t))=\sin (\eta x_1(t))$ with $\alpha =10$, $\beta =19.53$, $\gamma =0.1636$, $a=-1.4325$, $b=-0.7831$, $\eta =0.5$, $\varepsilon =0.2$ and $\tau =0.019$.

If we construct the following predictor,

$$ \varSigma _s: {\left\{ \begin{array}{ll} \dot{z}=A_{0}z+B\varphi (z_1)+B_1\xi (y) +K\{Cz(t-\tau )-y\} \\ z(t)=z_0, \ t\in [-2\tau ,0], \end{array}\right. } $$

then the dynamics of the prediction error $e(t)=z(t-\tau )-x(t)$ is given by

$$\begin{aligned} \dot{e}&=A_{0}e+KCe(t-\tau )+B\{\varphi (e_1+x_1)-\varphi (x_1)\} \nonumber \\&\triangleq A_{0}e+KCe(t-\tau )+B\phi (x_1,e_1). \end{aligned}$$

(5.37)

Fig. 5.8

Prediction Error (Theorem 5.4)

By choosing a Lyapunov–Krasovskii functional as

$$\begin{aligned} V(e_t)=&e(t)^TPe(t)+\int _{0}^{\tau }\int _{t-s}^t \phi ^TB^TRB\phi duds+\sum _{j=0}^1\int _{j\tau }^{(j+1)\tau }\int _{t-s}^te^T(u)Q_je(u)duds, \end{aligned}$$

and a coupling gain as $K=\left[ \begin{matrix}-12.1, -2.25, 3.71 \\ \end{matrix}\right] ^T$, anticipating synchronization of the unperturbed systems can be accomplished. Now, we consider the effect of a perturbation on anticipating synchronization. We assume that $A_0$ in both the master and slave systems contains uncertainties as follows.

$${A}_0=\left[ \begin{matrix} -\alpha (1+b)+\varDelta &{} \alpha &{} 0 \\ 1 &{} -1+\varDelta &{} 1 \\ 0 &{} -\beta &{} -\gamma +\varDelta \\ \end{matrix}\right] .$$

The prediction error dynamics is then obtained by

$$\begin{aligned} \dot{e}&:=A_{0}e+KCe(t-\tau )+B\phi (x_1,e_1)+g, \end{aligned}$$

where $g=\varDelta Ie(t)$ In this case, if $\varDelta$ is bounded, the perturbation satisfies the conditions (5.4d) and (5.4e). In addition, we can obtain the corresponding parameters of condition (5.4f) of Theorem 5.4 as follows: $\gamma <0.0594$. Considering $x(0)=[1, -2, 0]^T$, $x(\theta )=0$ for $\theta <0$ as the initial condition of the master system, $z(\theta )=0$ for $\theta <0$ as that of the slave system, and the perturbation $\varDelta =0.05$, the behavior of the prediction error $e(t)=z(t-\tau _1)-x(t)$ is depicted in Fig. 5.8. Since the prediction error converges to the origin, we know that the effect of the perturbation vanishes, and the slave system works effectively as a predictor of the master system.