## 5 An Illustrative Example  
[PERSON]'s stochastic chaotic system is described by [10; 11],  
\[\left\{ \begin{array}{l}\dot{x}_{1}(t)=a(x_{2}(t)-x_{1}(t))+w_{1,k}(t)\\ \dot{x}_{2}(t)=(c-a)x_{1}(t)-x_{1}(t)x_{3}(t)+cx_{2}(t)+w_{2,k}(t)\\ \dot{x}_{3}(t)=x_{1}(t)x_{2}(t)-bx_{3}(t)+w_{3,k}(t)\end{array} \right. \tag{39}\]  
where the white noise process, \(w_{i,k}(kT)=[w_{1,k}(kT),\,w_{2,k}(kT),\,w_{3,k}(kT)]^{T}\), has zero mean and covariance; \(\text{cov}(w_{i,k})=diag[0.02,\,0.02,\,0.02]\), where \(w_{i,k}(t)=w_{i,k}(kT)\) for \(kT\leq t<(k+1)T\), \(k\) nonnegative integer; the goal is to determine a sampling period \(T\) for the computation of a piecewise constant signal. This system has the attractor and time series shown in Figure 3 where \(a=35\), \(b=3\), and \(c=28\) without noises. In the simulation, the sampling period \(T_{f}=0.005\) s, final simulation time \(t_{f}=60\) s, and initial conditions used are \(x_{1}(0)=-10,\,\,x_{2}(0)=0,\,\,x_{3}(0)=37\). Case 1: OKID Method  
Based on OKID method, where \(q=1\), \(n_{\text{min}}=qp=3\), information data \(l=600\); apply a singular value decomposition to \(\overline{H}(0)\) to have \(\sum=diag\)[13.7793, 6.3290, 3.0489, \(1.8062\times 10^{-15},\cdots,8.7938\times 10^{-19}]_{24\times 24}\) and \(\sum_{n_{\text{min}}}=diag[13.7793,\,6.3290,\,30489]\). Then, compute system and observer-gain form in Equations (21)-(23) and matrices as  
\[\hat{G}=\left[ \begin{array}{ccc}1.0888&0.0106&0.1086\\ -0.0054&0.9985&0.0121\\ -0.1407&-0.0132&0.8965\end{array} \right],\,\hat{H}=\left[ \begin{array}{ccc} 0.3069&0.0980&-0.0374\\ -0.4161&-0.0802&0.3939\\ 0.0097&-0.0007&-0.0146\end{array} \right],\]  
\[\hat{C}=\left[ \begin{array}{ccc}-0.0796&0.0295&0.8471\\ -0.8069&-0.0394&0.3114\\ -0.0847&0.8928&-0.0208\end{array} \right],\,F=\left[ \begin{array}{ccc}-0.64 31&1.4053&0.0714\\ -0.0850&0.1182&-1.1103\\ -1.0287&-0.0777&0.0454\end{array} \right].\]  
Finally, simulations of the stochastic chaotic trajectories of [PERSON]'s system, and computation of observer Markov parameters are obtained, as shown in Figures 4a, 5a and 6a; and data of output error \(\widetilde{y}_{dt}(t)\), are shown in Figures 4b, 5b and 6b. Figure 3: (**a**) The deterministic chaotic attractor of [PERSON]’s system, plotted in the \(x_{3}-x_{1}-x_{2}\) space; (**b**) the deterministic chaotic time series of [PERSON]’s system. Figure 4: (**a**) Trajectories of \(y_{1}(t)\) and \(\vartheta_{1}(t)\); (**b**) Data of output error \(\widetilde{y}_{1}(t)\). Figure 5: (**a**) Trajectories of \(y_{2}(t)\) and \(\vartheta_{2}(t)\); (**b**) Data of output error \(\widetilde{y}_{2}(t)\). Case 2: OKID Combined with a Digital Redesign Observer Method  
Based on the new digitally redesigned observer form in Equation (38), the system and observer gain matrices are:  
\[A_{c}=\left[ \begin{array}{ccc}18.4700&2.1597&21.8221\\ -0.8592&-0.2809&2.5970\\ -28.3021&-2.6347&-20.1827\end{array} \right],I_{c}=\left[ \begin{array}{ccc} 1.0018&0.0001&-0.0003\\ 0.0001&1.0000&-0.0001\\ -0.0003&-0.0001&0.9980\end{array} \right]\times 10^{-4},C=I_{3},\,Q=10^{8}I_{3},\,R=I_{3},\]  
\[\mathcal{G}_{d}=\left[ \begin{array}{ccc}0.0099&-0.0253&-0.0005\\ -0.0007&-0.0025&0.0213\\ 0.0254&-0.0016&0.0005\end{array} \right],\tilde{H}_{d}=\left[ \begin{array}{ cccc}0.0133&0.0029&-0.0103\\ 0.0019&0.0004&-0.0013\\ 0.0078&0.0024&-0.0016\end{array} \right],I_{d}=\left[ \begin{array}{ccc}0. 4673&-1.2700&-0.0429\\ 0.0432&-0.1184&1.1159\\ 1.2140&-0.0868&-0.0418\end{array} \right].\]  
Finally, simulations of the stochastic chaotic trajectories of [PERSON]'s system and computation of digital redesign observer parameters are obtained, as shown in Figures 7a, 8a and 9a, and data of output error \(\widetilde{y}_{di}(t)\), are shown in Figures 7b, 8b and 9b. The error range of the outputs values \((\widetilde{y}_{1}(t),\widetilde{y}_{2}(t),\widetilde{y}_{3}(t))\) of the OKID and OKID combined with a digital redesign method individually are shown in Table 1.