## 5 Simulation Experiment  
### the Experiment Process  
To illustrate the effectiveness of the proposed methodology, in this section, a simulation experiment on three tank systems(TTS) is introduced. In a lot of literature, TTS have been considered as a benchmark to solve the problems such as system control, identification and fault diagnosis purposes [30; 31; 32; 33]. It is a multi-input, multi-output (MIMO) plant. Most available control techniques have been applied to this benchmark, allowing the comparison of their performances. It possesses a rich nonlinear dynamic, high complexity and strong coupling. The TTS is shown in Figure 3.  
Figure 3: Three tanks system. Consider three tank systems as described in Reference [19]. It is described by a state-space equation with  
\[\left\{ \begin{array}{l}\frac{dh_{1}}{dt}=\frac{1}{A}(Q_{1}-\delta_{1}\text{A}_{ n}\text{sgn}(h_{1}-h_{3})\sqrt{2g\left|h_{1}-h_{3}\right|})\\ \frac{dh_{2}}{dt}=\frac{1}{A}(Q_{2}+\delta_{3}\text{A}_{n}\text{sgn}(h_{3}-h_{ 2})\sqrt{2g\left|h_{3}-h_{2}\right|}-\delta_{2}\text{A}_{n}\sqrt{2g\left|h_{2} \right|})\\ \frac{dh_{3}}{dt}=\frac{1}{A}(\delta_{1}\text{A}_{n}\text{sgn}(h_{1}-h_{3}) \sqrt{2g\left|h_{1}-h_{3}\right|}-\delta_{3}\text{A}_{n}\text{sgn}(h_{3}-h_{2} )\sqrt{2g\left|h_{3}-h_{2}\right|}),\end{array} \right. \tag{15}\]  
where A is the cross section area of tank (\(cm^{2}\)); An is the cross section area of pipe (\(cm^{2}\));Q1,Q2 are incoming mass flow (\(cm^{3}/s\));\(\delta_{i}\), i = 1, 2, 3, are coefficient of flow for valves; i = 1, 2, 3, are the water level (cm) of each tank and can not be measured but can be estimated. The state vector \(X\), input \(U\) and output vector \(Y\) will be  
\[X=\left[h_{1},h_{2},h_{3}\right]^{T}\in R^{3},Y=\left[h_{1},h_{2}\right]\text{, }U=\left[Q_{1},Q_{2}\right]\in R^{2}\text{, }Q_{1}=Q_{2}\]  
The system parameters are \(A=100\), \(A_{n}=0.6\), \(Q_{1}=Q_{2}=30\). It is necessary to monitor the valves because their opening has a great influence on the dynamic characteristics of the system. Then the parametric model of the three tank system can be defined as follows:  
\[\left\{ \begin{array}{l}\frac{dh_{1}}{dt}=\frac{1}{A}(Q_{1}-p_{1}\delta_{1 max} \text{A}_{n}\text{sgn}(h_{1}-h_{3})\sqrt{2g\left|h_{1}-h_{3}\right|})\\ \frac{dh_{2}}{dt}=\frac{1}{A}(Q_{2}+p_{3}\delta_{3 max}\text{A}_{n}\text{sgn}(h _{3}-h_{2})\sqrt{2g\left|h_{1}-h_{3}\right|}-p_{2}\delta_{2 max}\text{A}_{n} \sqrt{2g\left|h_{2}\right|})\\ \frac{dh_{3}}{dt}=\frac{1}{A}(p_{1}\delta_{1 max}\text{A}_{n}\text{sgn}(h_{1}-h _{3})\sqrt{2g\left|h_{1}-h_{3}\right|}-p_{3}\delta_{3 max}\text{A}_{n}\text{sgn }(h_{3}-h_{2})\sqrt{2g\left|h_{3}-h_{2}\right|}),\end{array} \right. \tag{16}\]  
where \(p\) denotes the opening of valve, \(p_{i}\in[0,1]\), \(\delta_{imax}=0.6(i=1,2,3)\), \(h_{max}=100\). It is assumed that the liquid levels \(h_{1}\) and \(h_{2}\) are measurable and the parameters used to represent the valve opening and \(h_{3}\) are unknown. Our aim here is to estimate the parameters \(p\) and state \(x_{3}\) by using adaptive observer based on chaotic DE. In order to verify the effect of fitness function on the performance of the proposed observer, in this paper, the two fitness functions are used to adaptive observer. The one is negative Euclidean distance based and the other is correlation coefficient based. The first fitness function \(f_{t1}\) based on the residual norm is as follows  
\[f_{t1}=max(-\sum_{i=1}^{2}\left\|y_{i}-g_{i}\right\|_{2}).\]  
The second fitness function \(f_{t2}\) based on correlation coefficient \(\rho_{y_{i}g_{i}}\) is as follows  
\[f_{t2}=max\sum_{i=1}^{2}\rho_{y_{i}g_{i}}.\]  
The constraint conditions of the above fitness functions are as follows  
\[s.t.\left[ \begin{array}{l}0\leq p_{1}\leq 1\\ 0\leq p_{2}\leq 1\\ 0\leq p_{3}\leq 1\\ 0\leq\frac{h_{3}}{h_{max}}\leq 1.\end{array} \right]\]  
The initial states of the real system are \(X(0)=[0,0,60]\). The initial states of the adaptive observer \(X^{\prime}(0)=[0,0,0]\). \(X(0)\
eq X^{\prime}(0)\).