## 5 Applications in Control Theory
### Observers  
Observers are a primary tool used in control theory. Consider the problem of estimating the current state \(x_{k}\) of a discrete-time nonlinear dynamical system described by a system of first-order difference equations:  
\[\left\{\begin{array}{l}x_{k+1}=f(x_{k}),\\ y_{k+1}=h(x_{k}),\end{array}\right. \tag{69}\]  
where, from past observations, \(y_{s}\) and \(s\leq k\), where \(k\in\{0,1,2,\ldots\}\).  
The vector fields \(f:\mathbb{R}^{n}\longrightarrow\mathbb{R}^{n}\) and \(h:\mathbb{R}^{n}\longrightarrow\mathbb{R}^{p}\) are continuous with \(p\leq n\). The variable \(x\in\mathbb{R}^{n}\) stands for the state of the system, while the variable \(y\in\mathbb{R}^{p}\) represents the measurement output.  
An observer is a system driven by observations  
\[\hat{x}_{k+1}=\hat{f}(\hat{x}_{k},y_{k}) \tag{70}\]  
such that the error \(\hat{x}_{k}=x_{k}-\hat{x}_{k}\) goes to zero as \(k\longrightarrow\infty\).  
In [109], Xiao provided a direct method for constructing a change of coordinates for the design of nonlinear discrete-time observers along with an explicit expression of this change of coordinates. The used the Lozi (3) and Henon (2) maps to perform simulations for chaotic systems to illustrate the method.  
Instead, Voortman et al. [110] designed a data rate-constrained observer for a dynamical system which functions in both discrete time and continuous time. The system is connected to a remote location by a communication channel that can transmit limited amounts of data per unit of time. The objective of the observer is to provide estimates of the state at the remote location through messages that are sent via the channel. The observer is designed such that it is robust to losses in the communication channel. It was tested through simulations for the Lozi map and the Lorenz system. As the Lozi map is not differentiable, a smoothed version was used:  
\[\Phi_{\alpha}\left(\begin{array}{c}x\\ y\end{array}\right)=\left(\begin{array}{c}1-af_{\alpha}(x)+by,\\ x,\end{array}\right), \tag{71}\]  
where  
\[f_{\alpha}(x)=\left\{\begin{array}{ll}|x|,&if|x|\geq\alpha,\\ \frac{x^{2}}{2\alpha}+\frac{x}{2\alpha},&if|x|<\alpha,\end{array}\right. \tag{72}\]  
with \(\alpha<<1\).  
By combining results from several other papers, the authors provided an analytical bound on the channel rate that depends on the Lyapunov dimension rather than the upper box dimension. These analytical bounds were computed for the smoothed Lozi map and the Lorenz system. Under the following Assumption 1, they proved Theorem 2 below.  
**Assumption 1**.: \(a,b,\alpha>0,1-a<b<1\) _, \(\alpha<(a+1-b)^{-1}\)_  
**Theorem 2** ([110]).: _Let Assumption 5 hold and let \(S_{0}\) be a compact invariant set of the smoothed Lozi map (71). Then, the associated dynamical system is observable on the set \(S_{0}\) via any communication channel with capacity_  
\[c<\frac{\left[2 log_{2}\left(\frac{\sqrt{\alpha^{2}+4b}-a}{2}\right)-log_{2}b \right]\left[log_{2}\left(\frac{\sqrt{\alpha^{2}+4b}-a}{2}\right)\right]}{log _{2}\left(\frac{\sqrt{\alpha^{2}+4b}-a}{2}\right)}.\]  
In addition, they computed the Lyapunov dimension for the smoothed Lozi map. Simulations of the observer on both systems proved that the theoretical rate is closely related to the actual rate required to implement the observer.