# Nonlinear Integral Type Observer Design for State Estimation and Unknown Input Reconstruction
## 4 Numerical Example
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameter Definitions** & **Value** & **Unit** \\ \hline Link inertia (\(J_{l}\)) & 4 & N\(\cdot\)m\({}^{2}\) \\ Motor rotor inertia (\(J_{m}\)) & 4 & N\(\cdot\)m\({}^{2}\) \\ Elastic constant (\(k\)) & 15 & N\(\cdot\)m/rad \\ Link mass (\(M\)) & 0.15 & kg \\ Gravity constant (\(g\)) & 9.8 & N\(\times\)kg \\ Length for center of mass (\(l\)) & 0.3 & m \\ Viscous coefficient (\(F_{l}\)) & 0.006 & N\(\cdot\)s/rad \\ Viscous coefficient (\(F_{m}\)) & 0.005 & N\(\cdot\)s/rad \\ \hline \hline \end{tabular}
\end{table} 
Table 1: System parameters. (56)  
Based on the given system parameters listed in Table 1, the nonlinearity satisfies \(\|\mathbf{f}(\mathbf{x},t)\|\leq\delta=0.1103\). For the PIO, the control matrices generated by the Matlab LMI toolbox are given as follows:  
\[\varphi= \begin{bmatrix}\mathbf{q}_{1}&0\\ 0&\mathbf{q}_{2}\end{bmatrix} = \begin{bmatrix}0.9163&0.2602&\vdots&0&0&0&0\\ 0.2602&0.1860&\vdots&0&0&0&0\\ 0&0&0.8261&0&0.3339&0\\ 0&0&0.8261&0&0.3339\\ 0&0&0.3339&0&0.3328&0\\ 0&0&0&0.3339&0&0.3328\\ \end{bmatrix} \]  
\[\mathbf{K_{P}}=\left[ \begin{array}{ccc}21.7396&0.0278\\ 0.0283&21.8043\\ \end{array} \right],\mathbf{K_{I}}=\left[ \begin{array}{ccc}22.2876&0.0272\\ 0.0288&22.3538\\ \end{array} \right] \tag{57}\]  
where \(\beta=1\) is applied to achieve a desired decay speed. For the lumped perturbation term \(\mathbf{f}(\mathbf{x},t)\), there exists a positive constant \(\zeta=1.8\) such that \(\|\mathbf{D}_{22}\dot{\mathbf{f}}(\mathbf{x},t)\|\leq\|\mathbf{D}_{22}\| \Big{(}\Big{[}\frac{g[M}{\hbar}x_{2}\cos(x_{1})\Big{]}+\Big{|}\dot{d}(t)\Big{|} \Big{)}\leq\zeta=1.8\) and thereby the robust gains \(\xi_{i}=2\) are considered. To compare with the method [22], in the following simulations, two types of external unknown inputs are considered to simulate \(d(t)\). One is a continuous sine wave and the other is a discontinuous saw-tooth-like wave. The continuous unknown input is generated by \(d(t)\equiv d_{1}(t)=0.25\sin(2\pi t)\) and the discontinuous one is generated by (using a Matlab script) \(d(t)\equiv d_{2}(t)=0.025*(2|10t-20 floor(0.5t)-6.5|-13)\). It is clear that the time derivative of the discontinuous one does not satisfy \(L_{2}\) condition. However, the SMO can still recovery its profile accurately. In the following simulations, the discontinuous control applied in the SMO is replaced by a sigmoid function [20; 27]. The unknown input is estimated by considering the approximation: \(\mathbf{v}_{eq}\approx\rho\frac{\mathbf{p}_{2}\mathbf{s}}{\|\mathbf{p}_{2} \mathbf{s}\|+\varepsilon}\approx\mathbf{A}_{21}\mathbf{e}_{1}+\mathbf{D}_{22} \mathbf{f}(\mathbf{x},t)\), where \(\rho=1\) and \(\varepsilon=0.005\) are used which lead to in \(\eta\approx 0.89\). Figure 1 shows that the observer states approach to the real system states by using the SMO. The corresponding estimation errors are depicted in Figure 2. Note that the values of \(\widetilde{x}_{2}(t)\) are obviously larger than other state estimation errors. This is because that the unknown input directly affects in the direction of \(x_{2}\) and the sigmoid-function-based control applied in the SMO cannot completely eliminate the unknown input. Figure 1: State observation by sliding mode observer (SMO). Examining the sliding trajectories as illustrated in Figure 3. It is obvious that the approaching phase does not occur immediately, but remains being realized in finite time under the approaching condition (33). Put it clearly, since the closed-loop observation error dynamics is exponentially stable, it follows that the circumstance \(\|\mathbf{\bar{A}}_{21}\mathbf{e}_{1}\|<\eta\) is eventually achieved such that the approaching condition (33) is finally fulfilled. Figure 4 demonstrates the unknown disturbance reconstruction by the way of equivalent control injection. Figure 4: Continuous input recovery by SMO. Figure 3: Sliding dynamics. Figure 2: Estimation error responses by SMO.