# Robust Quadratic Optimal Control for Discrete-Time Linear Systems with Non-Stochastic Noises
## 4 Simulation Analysis
: _For the position control problem of the industrial robot with six degrees of freedom, because most of the motion in the robot processing is at low speed except the singularity, the coupling of the motion of the robot is weak. In addition, a reasonable path can be planned for the singularity to keep the robot away from its own singularity and avoid strong coupling between movements in different directions. In reference [20], the position control model of industrial robot is simplified as an independent but identical second-order system with disturbance in each direction,_  
\[\left[ \begin{array}{c}\dot{X}\\ \dot{Y}\\ \dot{Z}\end{array} \right]=\left[ \begin{array}{ccc}A_{X}&0&0\\ 0&A_{Y}&0\\ 0&0&A_{Z}\end{array} \right]\left[ \begin{array}{c}X\\ Y\\ Z\end{array} \right]+\left[ \begin{array}{ccc}B_{X}&0&0\\ 0&B_{Y}&0\\ 0&0&B_{Z}\end{array} \right]\left[ \begin{array}{c}u_{X}\\ u_{Y}\\ u_{Z}\end{array} \right]+\left[ \begin{array}{c}d_{X}\\ d_{Z}\end{array} \right],\]  
_where \(X=(X1,X2)^{T}\), \(Y=(Y1,Y2)^{T}\), \(Z=(Z1,Z2)^{T}\) are the displacement and velocity for the center point of the end of the manipulator along \(X\), \(Y\), \(Z\) direction respectively. The system matrices and input matrices in each direction are as follows,_  
\[A_{X}=\left[ \begin{array}{ccc}0&1\\ -1172.5\cdot t^{-2}&-46.2\cdot t^{-1}\end{array} \right],B_{X}=\left[ \begin{array} []{ccc}0\\ 1172.5\cdot t^{-2}\end{array} \right],\]  
\[A_{Y}=\left[ \begin{array}{ccc}0&1\\ -536.8\cdot t^{-2}&-38.9\cdot t^{-1}\end{array} \right],B_{Y}=\left[ \begin{array} []{ccc}0\\ 536.8\cdot t^{-2}\end{array} \right],\]  
\[A_{Z}=\left[ \begin{array}{ccc}0&1\\ -2285.8\cdot t^{-2}&-132.2\cdot t^{-1}\end{array} \right],B_{Z}=\left[ \begin{array} []{ccc}0\\ 2258.8\cdot t^{-2}\end{array} \right],\]  
_where \(t\) is time. The disturbance matrices in each direction are as follows,_  
\[d_{X}=\left[ \begin{array}{c}d_{X1}\\ d_{X2}\end{array} \right],d_{Y}=\left[ \begin{array}{c}d_{Y1}\\ d_{Y2}\end{array} \right],d_{Z}=\left[ \begin{array}{c}d_{Z1}\\ d_{Z2}\end{array} \right],\]  
_which contain the system noises, the un-modeled dynamics and the coupling effect of the motion of the robot in \(X\), \(Y\), \(Z\) directions and are bounded._  
Because the dynamics system in each direction has the same structure, here we take the motion control in \(X\) direction as an example to show the effectiveness of the proposed method. The laser tracker is used to measure the position for the center point of the end  
Figure 7: The curve for the upper bound of the cost function. of the manipulator, the measurement accuracy is \(15\ \upmu\mathrm{m}+5\ \upmu\mathrm{m}/\mathrm{m}\). By discretizing the dynamics system in \(X\) direction with the discrete sampling time \(T=0.01s\), it yields  
\[X(k+1) = \left[ \begin{array}{cc}1&0.01\\ -11.725\cdot k^{-2}&1-0.462\cdot k^{-1}\end{array} \right]X(k)+\left[ \begin{array} []{cc}0\\ 11.725\cdot k^{-2}\end{array} \right]u_{X}(k)+d_{X}(k),\] \[y(k) = \left[ \begin{array}{cc}1&0\end{array} \right]X(k)+v(k).\]  
The noises are independent mutually and belong to the ellipsoid sets,  
\[d_{X}(k)\in\Omega(0,W(k)),\]  
\[v(k)\in\Omega(0,V(k)),\]  
where  
\[W(k)=\left[ \begin{array}{cc}9&0\\ 0&25\end{array} \right],\]  
\[V(k)=4\times 10^{-4}.\]  
The original state \(X(0)\) belongs to the following ellipsoid set,  
\[X(0)\in\Omega(\hat{X}(0),P(0)),\]  
where  
\[\hat{X}(0)=\left[ \begin{array}{c}2\\ 10\end{array} \right],\]  
\[P(0)=\left[ \begin{array}{cc}100&0\\ 0&100\end{array} \right],\]  
and the matrices in performance index are with  
\[Q(k)=\left[ \begin{array}{cc}0.001&0\\ 0&0.001\end{array} \right],\]  
\[R(k)=1.\]  
The simulation results are as follows. Figures 8 and 9 display the estimation errors of displacement and velocity for the center point of the end of the manipulator in \(X\) direction for the proposed method and OFRMPC method.