# Nonlinear Integral Type Observer Design for State Estimation and Unknown Input Reconstruction
###### Abstract  
This paper is concerned with model-based robust observer designs for state observation and its application for unknown input reconstruction. Firstly, a sliding mode observer (SMO), which provides exponential convergence of estimation error, is designed for a class of multivariable perturbed systems. Observer gain matrices  
==WARNING: Truncated because of repetitions==
(\mathbf{\bar{e}}_{2i})\) with \(i=1\cdots p\)._  
_The control gains applied in (37) satisfy_  
\[ \begin{array}{l}\overline{\mathcal{P}}\mathcal{A}^{T}+\mathcal{A}\overline{ \mathcal{P}}-\mathcal{L}^{T}\mathcal{B}^{T}-\mathcal{B}\mathcal{L}+2\beta \overline{\mathcal{P}}<0\\ \mathbf{\bar{\xi}}>\zeta(t)\end{array} \] (38)  
_where \(\beta>0\), \(\overline{\mathcal{P}}=\mathcal{P}^{-1}\), \(\mathcal{P}=\mathcal{P}^{T}>0\), \(\mathcal{L}=\mathcal{K}\overline{\mathcal{P}}\), and the matrices are with the following specific structures_  
\[ \begin{array}{l}\mathcal{A}=\left[\begin{array}{ccc}\overline{\mathbf{A}}_ {11}&0&0\\ 0&0&\mathbf{I}_{p}\\ \overline{\mathbf{A}}_{21}\overline{\mathbf{A}}_{11}&0&0\end{array}\right] \in\mathfrak{R}^{(n+p)\times(n+p)}\\ \mathcal{B}=\left[\begin{array}{c}0\\ 0\\ \mathbf{I}_{p}\end{array}\right]\in\mathfrak{R}^{(n+p)\times p},\mathcal{K}= \left[\begin{array}{cc}0&\mathbf{K}_{\mathbf{I}}&\mathbf{K}_{\mathbf{P}} \end{array}\right]\end{array} \] (39)  
_and_  
\[\begin{array}{l}\varphi=\!\left[\! \begin{array}{c}\varphi_{1}&0\\ 0&\varphi_{2}\end{array} \!\right]\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! +++  
\[\dot{\mathbf{x}}=\mathcal{A}_{\mathcal{C}}\mathbf{x}+\mathcal{B}\Big{(}-\xi\text{sgn}( \overline{\mathbf{\epsilon}}_{2})+\mathbf{D}_{22}\dot{\mathbf{f}}(\mathbf{x},t) \Big{)} \tag{43}\]  
in which  
\[\mathcal{A}_{\mathcal{C}}=\left[ \begin{array}{ccc}\overline{\mathbf{\Lambda}}_{11 }&0&0\\ 0&0&\mathbf{I}_{p}\\ \overline{\mathbf{\Lambda}}_{21}\overline{\mathbf{\Lambda}}_{11}&-\mathbf{K}_{ \mathbf{I}}&-\mathbf{K}_{\mathbf{P}}\end{array} \right]\in\mathfrak{R}^{(n+p) \times(n+p)} \tag{44}\]  
Note that the extended system (43) is only used for stability analysis and control gain determination purposes. No extra state information needs to be involved for the observer implementation.