# Switched Observer Design for a Class of Non-Linear Systems
## 3 Switched Observer Design
### Guaranteed Decay Rate  
The state estimation convergence is evaluated within a Lyapunov framework, considering a minimum-type piecewise quadratic Lyapunov function [29]. The analysis of the corresponding dissipation mechanism leads to a set of LMI conditions for the purpose of the switched observer design. To this end, let \(V:\mathbb{R}^{n_{x}}\rightarrow\mathbb{R}^{+}\) be the (positive definite) weight functional candidate as defined below:  
\[ \begin{split} V(\bar{x}(t))&=\min_{i\in\{1,...,n_{p }\}}\bar{x}^{T}(t)P_{i}\bar{x}(t)\\ &=\min_{\lambda\in\Lambda}\left(\sum_{i=1}^{n_{p}}\lambda_{i} \bar{x}(t)^{T}P_{i}\bar{x}(t)\right)\end{split}  \tag{20}\]  
where \(P_{i}>0\in\mathbb{S}^{n_{x}}\ \forall i\in\{1,...,n_{p}\}\). In order to ensure the convergence of the estimated states, consider the following dissipation inequality:  
\[D^{+}V(t)+2\gamma V(t)\leq 0, \tag{21}\]  
where \(\gamma\) is a positive scalar and \(D^{+}V(t)\) denotes the Dini time derivative [34] of the piecewise Lyapunov function in (20). If (21) is satisfied along the trajectories of (10), the estimation error dynamics are exponentially stable with a guaranteed decay rate \(\gamma\), since it can be readily shown that the following holds  
\[\|\bar{x}(t)\|\leq M_{x}\|\bar{x}(0)\|\mathrm{e}^{-\gamma t} \tag{22}\]  
for every \(\bar{x}(0)\in\bar{\mathcal{X}}\subset\mathbb{R}^{n_{x}}\). **Problem 1**. : _Determine a switching strategy \(\sigma(\cdot):\mathbb{R}^{n_{y}}\rightarrow\{1,...,n_{s}\}\) such that: \(\sigma(\cdot)=\phi(y(t))\), \(\forall,\ t\geq 0\), which makes the origin \(\bar{x}=0\) of the error system (10), supposing that \(d(t)=0\) for \(t\geq 0\), an exponentially stable equilibrium point._  
A solution for Problem 1 is presented in the following theorem. Figure 1: Switched observer implementation scheme. **Theorem 1**. : _Consider the error dynamics defined in (10). Let \(\mu\) and \(\gamma\) be given positive scalars, Suppose there exist \(P_{i}\in\mathbb{S}^{n_{x}}\), \(Q_{i}\in\mathbb{S}^{n_{y}}\), \(M_{i}\in\mathbb{R}^{n_{x}\times n_{y}}\), \(i=1,\ldots,n_{p}\), such that_  
\[P_{i}>0,\quad i=1,\ldots,n_{p} \tag{23}\] \[\mathrm{H}_{\mathrm{e}}\{P_{i}A_{j}+M_{i}C\}+\mu\;C^{T}(Q_{j}-Q_{ i})C+2\gamma P_{i}<0,\quad i,j=1,\ldots,n_{p} \tag{24}\]  
_Then, the estimation error dynamics are locally exponentially stable with a guaranteed decay rate \(\gamma\), with the switching strategy as defined in (13) and the following observer gains:_  
\[L_{i}=P_{i}^{-1}M_{i},\;i=1,...,n_{p}. \tag{25}\]  
Proof. : Multiplying (23) by \(\alpha_{j}\), summing up all terms for all \(j\in\{1,...,n_{p}\}\), and considering (16), yields  
\[(A(\alpha)+L_{i}C)^{T}P_{i}+P_{i}(A(\alpha)+L_{i}C)+2\gamma P_{i}+C^{T}\left( \sum_{j=1}^{n_{p}}\pi_{ji}(\alpha)Q_{j}\right)C<0. \tag{26}\]  
Pre- and post-multiplying (26) by the vector \(x^{T}(t)\) and its transpose, respectively, yields  
\[\tilde{x}^{T}(t)\;\Big{(}\left(A(\alpha)+L_{i}C\right)^{T}P_{i}+P_{i}(A(\alpha )+L_{i}C)+2\gamma P_{i}\Big{)}\;\tilde{x}(t)<-\tilde{y}^{T}(t)\left(\sum_{j=1 }^{n_{p}}\pi_{ji}(\alpha)Q_{j}\right)\tilde{y}(t) \tag{27}\]  
Considering that \(i\in\mathcal{I}(\tilde{y}(t))\), we obtain  
\[\tilde{y}^{T}(t)\left(\sum_{j=1}^{n_{p}}\pi_{ji}(\alpha)Q_{j}\right)\tilde{y}( t)>0 \tag{28}\]  
and then, from (27), the following sufficient conditions are true  
\[\tilde{x}^{T}(t)\;\Big{(}\left(A(\alpha)+L_{i}C\right)^{T}P_{i}+P_{i}(A(\alpha )+L_{i}C)+2\gamma P_{i}\Big{)}\;\tilde{x}(t)<0 \tag{29}\]  
On the other hand, for the quadratic Lyapunov candidate function defined in (20), the RHS of the dissipation inequality (21) satisfies  
\[\|\bar{x}(t)\|\leq M_{x}\|\bar{x}(0)\|\mathrm{e}^{-\gamma t} \tag{22}\]0 \[\|\bar{x}(t)\|\leq M_{x}\|\bar{x}(0)\|\mathrm{e}^{-\gamma t} \tag{22}\]1  
It follows that (29) implies  
\[\|\bar{x}(t)\|\leq M_{x}\|\bar{x}(0)\|\mathrm{e}^{-\gamma t} \tag{22}\]2  
and hence (22) is fulfilled and the proof is completed.