# Switched Observer Design for a Class of Non-Linear Systems
## 1 Introduction
The convergence and performance analysis is studied in the framework of a semi-definite programming (SDP) problem in terms of LMI conditions which contain more degrees of freedom than the classical approach of invariant structure observer. It is also shown that the proposed method encompasses the switching strategy design and a set of observer gains, which ensure the local exponential stability of the estimation error dynamics while estimating a set of admissible initial conditions. Finally, we extend the observer design methodology to cope with energy bounded exogenous disturbances into an \(H_{\infty}\) setting. Analytical and simulation comparisons are also provided to demonstrate the superiority of the new LMI design methodology compared to previous results. In this context, the contribution of this work can be summarized as follows:  
* A non-linear Luenberger-like switched observer with a guaranteed decay rate and \(H_{\infty}\) cost for the estimation of error dynamics is designed in terms of LMI constraints, considering a class of non-linear systems locally satisfying a Lipschitz condition. The remainder of this paper is organized as follows. Section 2 describes the system dynamics along with some preliminary assumptions introduced in order to ensure the well-posedness of the LPV formulation of the state estimation error dynamics. Section 2.1 proposes LMI design conditions considering the representative LPV system, which guarantees both the local exponential stability of the closed loop system with prescribed decay rate and an \(H_{\infty}\) guaranteed cost. Section 4 applies the proposed methodology to a representative system, and Section 5 draws concluding remarks and points out some possible research lines. The following notation and definitions will be used throughout the paper:  
* \(\mathbb{N}_{+},\mathbb{R}\), \(\mathbb{R}^{n}\), and \(\mathbb{R}^{n_{x}\times n_{y}}\) represent, respectively, the set of non-negative integer numbers, real numbers, \(n\)-dimensional real vectors, and of \(n\times m\) real matrices;
* \(\mathbb{S}^{n}\subset\mathbb{R}^{n\times n}\) represents the subspace of square symmetric matrices;
* For \(A\in\mathbb{R}^{r\times m}\), \(A^{T}\) denotes the transpose of \(A\);
* For \(P\in\mathbb{S}^{n}\), \(P>0\) (\(P<0\)) means that \(P\) is positive definite (negative definite), and \(\mathrm{H}_{\mathrm{e}}\{P\}=P+P^{T}\) represents the Hermitian operator applied to \(P\);
* The set \(\mathrm{Co}(x,\hat{x})=\{\lambda x+(1-\lambda)\hat{x}:\ \lambda\in[0,1]\}\) is the convex hull of \(\{x,\ \hat{x}\}\);
* \(e_{s}(i)=\underbrace{\{0,...,0},\overbrace{\text{$s$ components}}^{\text{iith}},0,...,0}_{\text{$s$ components}}\in\mathbb{R}^{s}\) is a vector of the canonical basis of \(\mathbb{R}^{s}\);* The set of all vectors \(\lambda=[\lambda_{1},...,\lambda_{N}]^{T}\), such that \(\lambda_{i}\geq 0\), \(i\in\{1,...,N\}\) and \(\lambda_{1}+...+\lambda_{N}=1\) is designated by \(\Lambda_{N}\);
* The convex combination of a set of matrices \(Q_{1},...,Q_{N}\in\mathbb{R}^{n\times m}\) is denoted by \[Q_{\lambda}=\sum_{i=1}^{N}\lambda_{i}Q_{i},\] where \(\lambda\in\Lambda_{N}\);
* The subset of Metzler matrices denoted \(\mathcal{M}\) consists of all matrices \(\Pi\in\mathbb{R}^{n\times n}\) with elements \(\pi_{ji}\) such that \(\pi_{ji}\geq 0\;\;\forall\;\;i\
eq j\) and \[\sum_{j=1}^{n_{p}}\pi_{ji}=0,\;i,j\in\{1,...,n_{p}\};\]
* \(\|\cdot\|\) is the Euclidean norm in \(\mathbb{R}^{n}\), for every \(n\in\mathbb{N}_{+}\);
* The set of trajectories \(\zeta(t)\in\mathbb{R}^{n}\) with finite energy, i.e., \(\int_{0}^{\infty}\zeta^{T}(t)\zeta(t)dt<\infty\), is denoted by \(\mathcal{L}_{2}^{n}\). _Instrumental Tools._ The following statements are instrumental for deriving the main theoretical results of this paper. **Lemma 1** (Differential mean value theorem [28]). : _Let \(f:\mathbb{R}^{n_{x}}\rightarrow\mathbb{R}^{n_{f}}\). Let \(x\), \(\hat{x}\in\mathbb{R}^{n_{x}}\). We assume that \(f\) is a differentiable function on \(\mathcal{C}o(x,\hat{x})\).