# Robust Quadratic Optimal Control for Discrete-Time Linear Systems with Non-Stochastic Noises
## 4 Simulation Analysis  
This section illustrates the features of the non-stochastic robust control law proposed in this paper by the following two examples compared with the output feedback robust model predictive control(OFRMPC) in [16]. The proposed robust optimal control algorithm can be run as follows. **Step 1. ** Set \(k=0\), input \(\hat{x}(0)\) and \(P(0)\). **Step 2. ** Update true state feasible set \(\Omega(\hat{x}(k+1),P(k+1))\) according to Equations (7)-(13). **Step 3. ** Calculate control law \(u(k)\) by solving optimal problem (17). **Step 4. ** Apply \(u(k)\) to the system. If \(k=N-1\), stop; Otherwise, set \(k=k+1\) and go back to **Step 2. **  
**Example 1**. : _Consider the following two-dimension uncertain linear system,_  
\[x(k+1) = \left[ \begin{array}{cc}0.60&0.70\\ 0.25&0.59\end{array} \right]x(k)+\left[ \begin{array}{cc}3\\ 0.3\end{array} \right]u(k)+w(k),k=0,1,\cdots,29,\] \[y(k) = \left[ \begin{array}{cc}0&1\end{array} \right]x(k)+v(k),k=1, \cdots,30,\]  
_where \(x(k)\in R^{2}\), \(u(k)\in R\), \(y(k)\in R\), \(w(k)\in R^{2}\), and \(v(k)\in R\) are the state, the control input, the measured output, the process noise and the observation noise, respectively._  
The noises are independent mutually and belong to the ellipsoid sets,  
\[w(k)\in\Omega(0,W(k)),\]  
\[v(k)\in\Omega(0,V(k)),\]where  
\[W(k)=\left[ \begin{array}{cc}(0.1 arctan(k))^{2}&0\\ 0&(0.1 arctan(k))^{2}\\ \end{array} \right],\]  
\[V(k)=(0.1 arctan(k))^{2}.\]  
The original state \(x(0)\) belongs to the following ellipsoid set,  
\[x(0)\in\Omega(\hat{x}(0),P(0)),\]  
where  
\[\hat{x}(0)=\left[ \begin{array}{cc}1\\ 1\\ \end{array} \right],\]  
\[P(0)=\left[ \begin{array}{cc}25&0\\ 0&25\\ \end{array} \right].\]  
The matrices in performance index are with  
\[Q(k)=\left[ \begin{array}{cc}0.8&0\\ 0&0.8\\ \end{array} \right],\]  
\[R(k)=1,\]  
and they are selected as non-negative definite and positive definite symmetric matrices of appropriate dimensions, respectively. By using Sedumi toolbox, LMI toolbox and Optimization Toolbox in MATLAB, the simulation results are as follows. Figures 1 and 2 show the comparison of the system state and the estimated states between the proposed method and the OFRMPC method. Figures 3 and 4 display the estimated errors of system states for the proposed method and OFRMPC method. Here, the estimated states are the centers of the estimated state feasible sets. It can be seen that the estimated states with the proposed method are closer to the true state. Figure 4: The estimated error of system state x2 for the proposed method and OFRMPC method. Figure 3: The estimated error of system state x1 for the proposed method and OFRMPC method. Figure 2: The system state x2 and estimated state for the proposed method and OFRMPC method. Figure 5 displays the evolution of true state feasible sets for the proposed method. The blue line means the bound of estimated state feasible ellipsoidal sets. It shows that the feasible states sets from instant 2 become smaller and smaller. Meanwhile, they contain the true states all the time and this can be seen directly from the partial enlarged detail in the figure. Figure 6 shows the curve of the control law for the proposed method and the OFRMPC method. It is observed that the control law proposed in this paper tends to zero and the control law for OFRMPC fluctuates around zero. The upper bound \(\lambda\) of the cost function at each time is shown in Figure 7 and they become small gradually. And the quadratic cost function of the proposed method is 49.753, while the quadratic cost function for OFRMPC method is 55.324. This implies that the proposed method has better control performance. Figure 5: The estimated states and the evolution of the system states feasible sets for the proposed method. Figure 6: The control law for the proposed method and OFRMPC method. **Example 2**.