# Chaotic Synchronizing Systems with Zero Time Delay and Free Couple via Iterative Learning Control
## 3 Example Illustration and Demonstrated Results
### The Example of Iterative Learning Algorithmto Decide Learning Law  
To exhibit the synchronization of two non-linear systems and verify the algorithm of iterative learning control law in Figure 1, the drive-response systems with non-identical initial conditions are given by the following:  
\[ \begin{array}{l}\dot{\mathbf{x}}_{\mathbf{m}}(t)=\ f(\mathbf{x}_{\mathbf{m}},t)=\mathbf{A}\big{(} x_{\mathbf{m}}^{1}\big{)}\mathbf{x}_{\mathbf{m}}(t),\\ =\left[\begin{array}{ccc}-10&10&0\\ 30&-1&-x_{m}^{1}\\ 0&x_{m}^{1}&-\frac{8}{3}\end{array}\right]\mathbf{x}_{\mathbf{m}}(t),\\ \mathbf{y}_{\mathbf{m}}(t)\ =\mathbf{x}_{\mathbf{m}}(t),\ \mathbf{x}_{\mathbf{m}0}(t=0)=(0.02,\,0. 01,\,0.03),\end{array}  \tag{14}\]  
and  
\[ \begin{array}{l}\dot{\mathbf{x}}_{\mathbf{s}}^{(k)}(t)\ =f\Big{(}\mathbf{x}_{\mathbf{s}}^{(k)},t \Big{)}+\mathbf{B}\mathbf{u}^{(k)}(t)=\mathbf{A}\big{(}x_{m}^{1}\big{)}\mathbf{x}_{\mathbf{s}}^{(k )}(t)+\mathbf{B}\mathbf{u}^{(k)}(t)\\ =\left[\begin{array}{ccc}-10&10&0\\ 30&-1&-x_{m}^{1}\\ 0&x_{m}^{1}&-\frac{8}{3}\end{array}\right]\mathbf{x}_{\mathbf{s}}(t)+\mathbf{B}\mathbf{u}^{(k )}(t),\\ \mathbf{y}_{\mathbf{s}}^{(k)}(t)\ =\mathbf{x}_{\mathbf{s}}^{(k)}(t),\ \mathbf{x}_{\mathbf{s}0}^{(0)}=( 5,\,10,\,15),\end{array}  \tag{15}\]  
From the drive-response system, the dynamical error system is expressed as  
\[ \begin{array}{l}\dot{\mathbf{\Delta}}^{(k)}=\mathbf{A}\big{(}x_{m}^{i}\big{)}\Big{(} \mathbf{x}_{\mathbf{m}}-\mathbf{x}_{\mathbf{s}}^{(k)}\Big{)}^{\mathsf{T}}=\mathbf{A}\big{(}x_{m} ^{i}\big{)}\mathbf{\Delta}^{(k)}+\mathbf{B}\mathbf{u}^{(k)}(t),\\ =\left[\begin{array}{ccc}-10&10&0\\ 30&-1&-x_{m}^{1}\\ 0&x_{m}^{1}&-\frac{8}{3}\end{array}\right]\mathbf{\Delta}^{(k)}+(-1)\mathbf{B}\mathbf{u}^{ (k)}(t),\end{array}  \tag{16}\]  
In this example, given \(\mathbf{C}_{\mathbf{m}}=\mathbf{C}_{\mathbf{s}}=\mathbf{1}\), the output error is \(\mathbf{y}_{\mathbf{m}}-\mathbf{y}_{\mathbf{s}}^{(k)}=\mathbf{\Delta}^{(k)}=\Big{(}\mathbf{x}_{\mathbf{m}} -\mathbf{x}_{\mathbf{s}}^{(k)}\Big{)}^{\mathsf{T}}\) in \(\mathbf{R}^{3}\). The parameters are explained as \(\mathbf{x}_{\mathbf{m}}=(x_{m1},\ x_{m2},\ x_{m3})^{T}\), \(\mathbf{x}_{\mathbf{s}}^{(k)}=\Big{(}x_{s1}^{(k)},x_{s2}^{(k)},x_{s3}^{(k)}\Big{)}^{T}\), and \(\dot{\mathbf{\Delta}}^{(k)}=\Big{(}\dot{\mathbf{x}}_{\mathbf{m}}(t)-\dot{\mathbf{x}}_{\mathbf{s}}^{( k)}(t)\Big{)}^{T}\). The \(x_{m}^{1}\) in polynomial matrix \(\mathbf{A}\big{(}x_{m}^{1}\big{)}\) is the first component of the state vector in the drive system. The state vector in the estimation system is \(\dot{\mathbf{x}}_{\mathbf{m}}(t)=\ \Big{(}\mathbf{A}\big{(}x_{m}^{1}\big{)}^{T}\mathbf{A}\big{(}x_{m} ^{1}\big{)}\Big{)}^{-1}\mathbf{A}\big{(}x_{m}^{1}\big{)}^{T}\ \dot{\mathbf{x}}_{\mathbf{m}} ^{T}\) and the iterative learning law with the initial condition \(\mathbf{u}^{(0)}(t=0)\ =\ \mathbf{\Delta}^{(0)}(t=0)\ =\ \Big{(}\mathbf{x}_{\mathbf{m}0}-\mathbf{x}_{\mathbf{s}0}^{( 0)}\Big{)}^{T}\). The \(\mathbf{u}^{(k)}(t)\) in Equation (6) and the Lyapunov equation are respectively shown as Equations (6) and (8).The matrices \(\mathbf{B_{1}}\) and \(\mathbf{B_{2}}\) in the ILC rule of Equation (6) are:  
\[ \begin{array}{l}\mathbf{B_{1}}=\left[\begin{array}{ccc}0.0034&0.0345&0\\ 0.0140&0.0085&-0.0022\\ 0.0151&0.0056&-3.75\end{array}\right]\text{ and }\mathbf{B_{2}}=\left[\begin{array}{ccc}-10&10&0\\ 30&-1&0\\ 0&0&-\frac{8}{3}\end{array}\right],\end{array}  \tag{17}\]  
The matrix \(\mathbf{B_{2}}\) is from the decomposition of the coefficient matrix of the system in (Equation (14)) and the matrix \(\mathbf{B_{1}}=(\mathbf{M})^{k}(\mathbf{B_{2}})^{-2}\).The parameters \(0\leq k\leq 1\) and the matrix \(\mathbf{M}\) is in [33]. The matrix \(\mathbf{B}\) in Equation (1b) is to select the identity matrix. The results were simulated by MATLAB (R2013b, Math Works, Natick, MA, USA, 2013) to verify the performance of the ILC rule.