# A Master-Slave Separate Parallel Intelligent Mobile Robot Used for Autonomous Pallet Transportation
## 3 Synchronization Nonlinear Control
### High-Gain Observer  
High-gain observer is a good tool for the nonlinear control process in which some state variables cannot be measured accurately due to technical or economic reasons. In some practical cases, the tool can be modified to produce an output feedback controller [14]. Suppose that a nonlinear system can be transformed into the form:  
\[\left\{ \begin{array}{rl}\dot{x}=Ax+g(y,u)\\ \dot{y}=Cx\end{array} \right. \tag{7}\]  
Figure 5: Control strategy of slave robot. where \((A,C)\) is observable. This form is special because the nonlinear function \(g\) depends only on the output \(y\) and the control input \(u\). Taking the observer as:  
\[\dot{\hat{x}}=A\hat{x}+g(y,u)+H(y-C\hat{x}) \tag{8}\]  
it can be easily seen that the estimation error \(\widetilde{x}=x-\hat{x}\) satisfies the linear equation:  
\[\dot{\widetilde{x}}=(A-HC)\widetilde{x} \tag{9}\]  
Hence, designing \(H\) such that \(A-HC\) is Hurwitz guarantees asymptotic error convergence, that is \(\lim\limits_{x\rightarrow\infty}\widetilde{x}(t)=0\). Aside from the fact that the observer works only for a special class of nonlinear systems, its main drawback is the assumption that the nonlinear function \(g\) is perfectly known. Any error in modeling \(g\) will be reflected in the estimation error equation. The upper nonlinear observer is suitable for most system. But for the sub-style parallel intelligent mobile robot, slave robot should follow master robot synchronously and timely. There are also many measurement deviations. We should give a special design of the observer gain that makes the observer robust to uncertainties in modeling the nonlinear functions. High-gain observer can guarantee that the output feedback controller recovers the performance of the state feedback controller when the observer gain is sufficiently high. We use the observer in output feedback stabilization. The main results in Section 4 are separation principles that allow us to separate the design into two tasks. First, we design a state feedback controller that stabilizes the system and meets other specifications. Then, an output feedback controller is obtained by replacing the state \(x\) by its estimate \(\hat{x}\) provided by high-gain observer. A key property that makes this separation possible is the design of the state feedback controller to be globally bounded in \(x\). High-gain observer can be used in a wide range of control problems. Consider the second-order nonlinear system:  
\[\left\{ \begin{array}{c}\dot{x}_{1}=x_{2}\\ \dot{x}_{2}=\phi(x,u)\\ y=x_{1}\end{array} \right. \tag{10}\]  
where \(x=[x_{1},x_{2}]^{T}\). Suppose \(u=y(x)\) is a locally Lipschitz state feedback control law that stabilizes the origin \(x=0\) of the closed-loop system. To implement this feedback control using only measurements of the output \(y\), we use the observer:  
\[\left\{ \begin{array}{c}\dot{\hat{x}}_{1}=\hat{x}_{2}+h_{1}(y-\hat{x}_{1})\\ \dot{\hat{x}}_{2}=\phi_{0}(\hat{x},u)+h_{2}(y-\hat{x}_{1})\end{array} \right. \tag{11}\]  
We set \(\widetilde{x}_{1}(0)=0\) and \(\widetilde{x}_{2}(0)=0\). Where \(\phi_{0}(x,u)\) is a nominal model of the nonlinear function \(\phi(x,u)\). The estimation error:  
\[\widetilde{x}=\left[ \begin{array}{c}\widetilde{x}_{1}\\ \widetilde{x}_{2}\end{array} \right]=\left[ \begin{array}{c}x_{1}-\hat{x}_{1 }\\ x_{2}-\hat{x}_{2}\end{array} \right] \tag{12}\]  
Satisfies the equation:  
\[\left\{ \begin{array}{c}\dot{\widetilde{x}}_{1}=-h_{1}\widetilde{x}_{1}+ \widetilde{x}_{2}\\ \dot{\widetilde{x}}_{2}=-h_{2}\widetilde{x}_{1}+\delta(x,\widetilde{x})\end{array} \right. \tag{13}\]where \(\delta(x,\widetilde{x}=\phi(x,\gamma(\hat{x}))-\phi_{0}(\hat{x},\gamma(\hat{x})))\). We want to design the observer gain \(H=\left[h_{1},h_{2}\right]^{\mathsf{T}}\) such that \(\lim_{t\rightarrow\infty}\widetilde{x}(t)=0\). In the absence of the disturbance term \(\delta\), asymptotic error convergence is achieved by designing \(H\) such that:  
\[A_{o}=\left[ \begin{array}{cc}-h_{1}&1\\ -h_{2}&0\end{array} \right] \tag{14}\]  
is Hurwitz.