# Switched Observer Design for a Class of Non-Linear Systems
## 3 Switched Observer Design
### \(H_{\infty}\) Guaranteed Cost  
Next, we address the non-linear \(H_{\infty}\) switched observer design problem when \(d(t)\
eq 0\) by defining an output performance vector \(z(t)=E\tilde{x}(t)\), with \(z\in\mathbb{R}^{n_{z}}\) and \(E\in\mathbb{R}^{n_{z}\times n_{x}}\), which, along with the error dynamics, yields  
\[\dot{\tilde{x}}(t)=\left(A(\alpha(t))+L_{\sigma(t)}C\right)\tilde{x}(t)+B_{d}d(t) \tag{32}\]  
\[z(t)=E\tilde{x}(t).\]Then, the \(\mathcal{L}_{2}\)-gain of the non-linear error dynamics (often referred as the \(H_{\infty}\) cost) can be defined as follows:  
\[J_{\infty}:=\sup_{0\
eq d\in\mathcal{L}_{2}}\frac{\|z\|_{2}^{2}}{\|d\|_{2}^{2}}. \tag{33}\]  
which allows us to formulate the following problem.  
**Problem 2**.: _Determine a switching strategy \(\sigma(\cdot):\mathbb{R}^{n_{y}}\rightarrow\{1,...,n_{p}\}\), with \(\sigma(\cdot)=\phi(y(t))\), which guarantees that the origin of the unforced estimation error dynamics, i.e., the system defined in (10) with \(d(t)=0\), is locally exponentially stable while ensuring that_  
\[\sup_{d\in\mathcal{L}_{2}}\frac{\|z\|_{2}^{2}}{\|d\|_{2}^{2}}<\rho \tag{34}\]  
_for a given \(\rho>0\)._  
A solution for Problem 2 is proposed in the following theorem  
**Theorem 2**.: _Consider the error dynamics in (32). Let \(\mu\) and \(\gamma\) be given positive scalars. Suppose there exists \(P_{i}\in\mathbb{S}^{n_{x}}\), \(Q_{i}\in\mathbb{S}^{n_{y}}\), \(M_{i}\in\mathbb{R}^{n_{x}\times n_{x}}\), \(i=1,\ldots,n_{p}\), and \(\rho\in\mathbb{R}\) such that the following is satisfied:_  
\[P_{i}>0,\quad i=1,\ldots,n_{p} \tag{35}\] \[\begin{bmatrix}\Pi_{i,j}&PB_{d}&E^{T}\\ B_{d}^{T}P&-\rho I&0\\ E&0&-I\end{bmatrix}<0,\quad i,j=1,\ldots,n_{p} \tag{36}\]  
_where_  
\[\Pi_{i,j}=\mathrm{H}_{\mathrm{e}}\{P_{i}A_{j}+M_{i}C\}+\mu\;C^{T}(Q_{j}-Q_{i})C.\]  
_Then, the switching strategy defined in (13) with \(L_{i}=P_{i}^{-1}M_{i}\), \(i=1,\ldots,n_{p}\), guarantees that the origin of the estimation error dynamics defined in (8) with \(d(t)=0\) is locally exponentially stable and \(J_{\infty}\leq\rho\)._  
Proof.: Multiplying (35) by \(\alpha_{j}\), summing up all terms for all \(j\in\{1,...,n_{p}\}\), yields  
\[\begin{bmatrix}\Pi(\alpha)_{i,j}&PB_{d}&E^{T}\\ B_{d}^{T}P&-\rho I&0\\ E&0&-I\end{bmatrix}<0 \tag{37}\]  
with  
\[\Pi(\alpha)_{i,j}=\mathrm{H}_{\mathrm{e}}\{PA(\alpha)+M_{i}C\}+C^{T}\Bigg{(} \sum_{j=1}^{n_{p}}\pi_{ji}(\alpha)Q_{j}\Bigg{)}C.\]  
Performing the Schur complement in (37) and rearranging the terms, we obtain  
\[\begin{bmatrix}\Pi(\alpha)_{i,j}&PB_{d}\\ B_{d}^{T}P&-\rho I\end{bmatrix}+\begin{bmatrix}E^{T}\\ 0\end{bmatrix}[E&0]<0. \tag{38}\]Pre- and post-multiplying (38) by the vector \([x(t)^{T}\,d(t)^{T}]\) and its transpose, respectively, yields  
\[\hat{x}^{T}(t)\,\left((A(\alpha)+L_{i}C)^{T}P_{i}+P_{i}(A(\alpha)+L _{i}C)+E^{T}E\right)\hat{x}(t)+\hat{x}^{T}(t)PB_{d}d(t)\\ +d^{T}(t)B_{d}P\hat{x}(t)-\rho d^{T}(t)d(t)<-\hat{y}^{T}(t)\left( \sum_{j=1}^{n_{p}}\pi_{ji}(\alpha)Q_{j}\right)\hat{y}(t). \tag{39}\]  
According to the definition in \(i\in\mathcal{I}(\hat{y}(t))\), from (39), the following condition is true  
\[D^{+}V(t)+z^{T}(t)z(t)-\rho d^{T}(t)d(t)<0. \tag{40}\]  
Rearranging and integrating each element in the inequality above from \(t=0\) to \(t\to\infty\) yields  
\[\|z(t)\|^{2}-\rho\|d(t)\|^{2}<V(0)-V(\infty) \tag{41}\]  
and under zero initial conditions \(V(0)=0\), one has  
\[\frac{\|z(t)\|_{2}^{2}}{\|d(t)\|_{2}^{2}}<\rho. \tag{42}\]  
Hence, the proof is concluded.  
**Remark 1**.: _Notice that the term \(\mu\)\(C^{T}(Q_{j}-Q_{i})C\) included in the conditions (23) and (35) of the previous theorems may be cast as BMIs since \(Q_{i}\) and \(\mu\) are decision variables. However, if the parameter \(\mu>0\) is set to be known, those conditions become standard LMIs. Further, a proposed algorithm will exploit the optimisation of the parameters \(\mu\) in order to achieve suboptimal values for the guaranteed \(H_{\infty}\,\text{cost}\)._