## 7 Networks
### Synchronization
#### 7.1.1 Observer-Based Synchronization  
In addition to the observers described in Section 5.1, Zhang et al. Zhang et al. (2019) proposed a switching observer-based synchronization scheme for piecewise linear discrete chaotic systems such as skew tent maps and Lozi maps based on piecewise quadratic Lyapunov functions. In this scheme, the observer switches according to the output of the drive system.  
For piecewise linear chaotic systems of the form  
\[\left\{\begin{array}{l}x(t+1)=A_{l}x(t)+\alpha_{l}\quad for\quad x(t)\in R_{ l},\\ y(t)=Cx(t),\end{array}\right. \tag{92}\]  
the observer is defined as follows:  
\[\left\{\begin{array}{l}\hat{x}(t+1)=A_{l}\hat{x}(t)+\alpha_{l}+K_{l}(y(t)- \hat{y}(t))\quad for\quad x(t)\in R_{l},\\ \hat{y}(t)=Cx(t),\end{array}\right. \tag{93}\]  
where \(\hat{x}(t)\in\mathbb{R}^{n}\) is the observer state, \(\left\{R_{l}\right\}_{l\in L}\subseteq\mathbb{R}^{n}\) denotes a partition of the state space of the chaotic system into a number of closed polyhedral subspaces, \(y(t)\in\mathbb{R}^{m}\) and \(\hat{y}(t)\in\mathbb{R}^{m}\) are the outputs of the chaotic system and observer, respectively, and \(K_{l}\in\mathbb{R}^{n\times m}\) is a switching gain matrix that is yet to be determined.  
For synchronization, \(y(t)=Cx(t)\) is the synchronization signal to be sent to the receiver (observer) and the observer switches the dynamics according to the state of chaotic systems. However, it is well known that not all of the state variables are measurable in practice. Thus, Zhang et al. divided the space of the output \(y(t)\) into several polyhedral cells \(\left\{S_{l}\right\}\), \(l\in L\subseteq\mathbb{R}^{m}\) to meet the following two requirements:  
1. The number of cells of the space of the output \(y(t)\) is exactly equal to that of the state of the chaotic drive system.
2. The observer is guaranteed to acquire the information about which cell in the state of chaotic system is from the synchronization signal \(y(t)\); in other words, the output \(y(t)\) is in cell \(S_{l}\) if and only if \(x(t)\) is in cell \(R_{l}\).  
To prove the synchronization property, Zhang et al. defined the synchronization error as \(e(t)=x(t)-\hat{x}(t)\). Combining (92) and (93), they obtained to the following error dynamics:  
\[e(t+1)=(A_{l}-K_{l}C)e(t)=F_{l}e(t)\quad for\quad y\in S_{l}. \tag{94}\]  
Then, they obtained the following switching feedback stabilization result for the synchronization error dynamics (94).  
**Theorem 3** (Zhang et al. (2019)).: _The synchronization error dynamics (94) are asymptotically stable with the switching feedback gain \(K_{l}\), \(l\in L\) if there exists a set of positive matrices \(P_{l}\), \(l\in L\) such that the following linear matrix inequalities are satisfied:_  
\[(A_{l}-K_{l}C)^{T}P_{l}(A_{l}-K_{l}C)-P_{l}<0,\quad l\in L \tag{95}\]\[(A_{j}-K_{j}C)^{T}P_{j}(A_{j}-K_{j}C)-P_{j}<0,\quad l,j\in\Omega\cap L \tag{96}\]  
_where \(\Omega\) is the set that represents all possible transitions from one region to another, that is,_  
\[\big{\{}\Omega=l,j|y(t)\,\in S_{j},y(t-1)\in S_{l},j\
eq l\big{\}}. \tag{97}\]  
They used the values \(a=1.7\) and \(b=0.5\) with initial conditions \(x_{1}=-0.2,x_{2}=0.2\) for the Lozi map to illustrate the synchronization of two maps.