# Nonlinear Integral Type Observer Design for State Estimation and Unknown Input Reconstruction
## 4 Numerical Example  
In this section, a single link robot arm is considered for the SMO designed. The resulting estimation performance will be later compared with the one by PIO. Some advantages of the PIO are going to be carried out. The dynamics of a disturbed single link robot arm with a revolute elastic joint rotating in a vertical plane is described as follows [28]:  
\[\dot{\mathbf{x}} =\left[ \begin{array}{cccc}0&1&0&0\\ -\frac{k}{l_{l}}&-\frac{F_{l}}{l_{l}}&\frac{k}{l_{l}}&0\\ 0&0&0&1\\ -\frac{k}{l_{m}}&0&\frac{k}{l_{m}}&-\frac{F_{m}}{l_{m}}\end{array} \right] \mathbf{x}+\underbrace{\left[ \begin{array}{c}0\\ 0\\ 0\\ 0\\ \frac{1}{l_{m}}\end{array} \right]}_{\mathbf{B}}\mathbf{u}\] \[+\underbrace{\left[ \begin{array}{c}0\\ 1\\ 0\\ 0\end{array} \right]}_{\mathbf{D}}\underbrace{\left[ \begin{array}{c}0\\ -\frac{\delta lM}{l_{l}}\sin(x_{1})+d(t)\\ 0\\ 0\\ 0\\ 0\\ \end{array} \right]}_{\mathbf{F}(\mathbf{x},t)} \tag{53}\] \[\mathbf{y} =\underbrace{\left[ \begin{array}{cccc}0&1&0&0\\ 0&0&1&0\end{array} \right]}_{\mathbf{C}}\mathbf{x}\]  
where the state vector is given by \(\mathbf{x}=\left[ \begin{array}{cccc}x_{1}&x_{2}&x_{3}&x_{4}\end{array}  \right]^{T}\). The states \(x_{1}\), \(x_{2}\), \(x_{3}\) and \(x_{4}\) are the link displacement, the link velocity, the rotor displacement and the rotor angular velocity. The rest of system parameters are summarized in Table 1.  
Before the controlling, a practical condition is imposed to the control problem. That is, only two sensors are available. One is an encoder used to measure angular position and the other is a tachometer used to measure angular velocity. This is important to equip the sensors at proper places. For example, suppose that the encoder and the tachometer are used to measure angular position and angular velocity of the link, respectively. It leads to an output matrix \(\mathbf{C}=\left[ \begin{array}{ccc}1&0&0&0\\ 0&1&0&0\end{array} \right]\). Applying coordinate transformation gives  
\[\mathbf{A}_{11}=\left[ \begin{array}{ccc}0&1.0000\\ -3.7500&-0.0013\end{array} \right]\text{and }\mathbf{A}_{211}=0_{1\times 2} \tag{54}\]  
It is evident that the pair \((\mathbf{A}_{11},\mathbf{A}_{211})\) is not observable, but the system contains two stable invariant zeros \(\lambda(\mathbf{A}_{11})\) located at \(-0.006\)\(\pm\)\(1.9365j\). Note that since the invariant zeros are located in the left-half-plane, the SMO and PIO designs remain applicable through the coordinate transformation technique. However, since the unmovable zeros are close to the imaginary axis, it leads to a slow observation speed. In contrast, when the encoder is used for motor position measurement, it results in the output matrix as the one shown in (53). Applying coordinate transformation gives the system as described in (4), where the resulting system matrices are  
\[\left[ \begin{array}{c}\mathbf{A}_{11}\\ \mathbf{A}_{211}\\ \mathbf{A}_{212}\end{array} \right]=\left[ \begin{array}{cccc}0&0&0&1.0000\\ -3.7500&-0.0013&-3.7500&0\\ 0&1.0000&0&0\\ -3.7500&0&-3.7500&-0.0015\end{array} \right]\]  
\[\left[ \begin{array}{c}\mathbf{B}_{1}\\ \mathbf{B}_{2}\end{array} \right]=\left[ \begin{array}{c}0\\ 0.25\\ 0\\ 0\end{array} \right]\text{, }\mathbf{D}_{22}=\left[ \begin{array}{c}0\\ -1.0\end{array} \right]\text{, }\mathbf{T}_{p}=\left[ \begin{array}{cc}0&1.0\\ -1.0&0\end{array} \right] \tag{55}\]  
It can be observed that the pair \((\mathbf{A}_{11},\mathbf{A}_{211})\) is now fully observable and therefore the observation performance can be improved. For the SMO, consider \(\alpha=1\) and then the control matrices returned by the Matlab LMI-toolbox are