# Chaotic Synchronizing Systems with Zero Time Delay and Free Couple via Iterative Learning Control
## 2 The Iterative Learning Control (ILC) Problem Formulation  
### System Description  
The drive and response systems are chaotic systems to synchronize with zero-time delay and couple free. The general formulas of systems are described by Equations (1a) and (1b), respectively. \[ \begin{array}{l}\dot{x}_{m}(t)\ =\ f(x_{m},t)=A\big{(}x_{m}^{i}\big{)}x_{m}(t), \\ \ \ \ \ \ \ \ \ y_{m}(t)\ =\ \mathbf{C_{m}}x_{m}(t),\end{array}  \tag{1a}\]The response system adjusts the error between the drive and tracking systems to synchronization by the control input \(\mathbf{B}\mathbf{u}^{(k)}\). \[ \begin{split}\dot{\mathbf{x}}_{\mathbf{s}}^{(k)}(t)\ =f\Big{(}\mathbf{x}_{\mathbf{s}}^{(k)},t \Big{)}+\mathbf{B}\mathbf{u}^{(k)}(t)=\mathbf{A}\big{(}x_{m}^{i}\big{)}\mathbf{x}_{\mathbf{s}}^{(k) }(t)+\mathbf{B}\mathbf{u}^{(k)}(t),\\ \mathbf{y}_{\mathbf{s}}^{(k)}(t)\ =\ \mathbf{C}_{\mathbf{s}}\mathbf{x}_{\mathbf{s}}^{(k)}(t), \end{split}  \tag{1b}\]  
The state vectors \(\mathbf{x}_{\mathbf{m}}(t)\) and \(\mathbf{x}_{\mathbf{s}}^{(k)}(t)\) and the outputs \(\mathbf{y}_{\mathbf{m}}\) and \(\mathbf{y}_{\mathbf{s}}^{(k)}\) are of the drive-response systems in the state space of \(\mathbf{R}^{\mathbf{n}}\), respectively. Originally, \(\mathbf{f}(\mathbf{x}_{\mathbf{m}},t)\) and \(\mathbf{f}\Big{(}\mathbf{x}_{\mathbf{s}}^{(k)},t\Big{)}\) weresimilar systems. The response system includes the iterative control input signal of \(\mathbf{B}\mathbf{u}^{(k)}(t)\) and the output \(\mathbf{y}_{\mathbf{s}}^{(k)}\), after the \(k\)-th learning of iteration. The \(C_{m}\), \(C_{s}\), and \(B=B^{T}\) are non-singular constant matrices with appropriated dimensions and some entries of the nonsingular polynomial matrix \(\mathbf{A}\big{(}x_{m}^{i}\big{)}\) are replaced by the \(i\)-th component of \(\mathbf{x}_{\mathbf{m}}(t)\),which is the factor of system synchronization, in which \(i=1\), \(2\), \(3\). The input signal sequence \(\Big{\{}\mathbf{u}^{(k)}(t)\Big{\}}_{k=\mathbf{1},\ \mathbf{2},\ \mathbf{2},\ \mathbf{\cdots}}\) in \(\mathbf{R}^{m}\) is the control learning law after the \(k\)-th iterative learning for the response system synchronizing the drive system. \(\mathbf{\Delta}^{(k)}\ =\ \Big{(}\mathbf{x}_{\mathbf{m}}-\mathbf{x}_{\mathbf{s}}^{(k)}\Big{)}^{T}\) is the synchronization error and the output error of the drive-response system is given \(\mathbf{y}_{\mathbf{m}}-\mathbf{y}_{\mathbf{s}}^{(k)}=\mathbf{C}_{\mathbf{m}}\mathbf{x}_{\mathbf{m}}-\mathbf{C}_ {\mathbf{s}}\mathbf{x}_{\mathbf{s}}^{(k)}\). The output error is equal to the synchronization error when \(\mathbf{C}_{\mathbf{m}}=\mathbf{C}_{\mathbf{s}}=\mathbf{1}\) is set.The dynamical system synchronization error between the drive-response systems is described in Equations (1a) and (1b). \[\dot{\mathbf{\Delta}}^{(k)}=\dot{\mathbf{x}}_{\mathbf{m}}(t)-\dot{\mathbf{x}}_{\mathbf{s}}^{(k)}( t)=\mathbf{A}\Big{(}x_{m}^{i}\Big{)}\Big{(}\mathbf{x}_{\mathbf{m}}-\mathbf{x}_{\mathbf{s}}^{(k)} \Big{)}^{T}+\ \mathbf{B}\mathbf{u}^{(k)}(t)\ =\mathbf{A}\Big{(}x_{m}^{i}\Big{)}\mathbf{\Delta}^{(k)}+\mathbf{B}\mathbf{u}^{(k)}(t), \tag{2}\]  
The appropriate \(\mathbf{B}\mathbf{u}^{(k)}(t)\) is given and subtracted in Equation (2) to beabsorbed by \(\mathbf{B}\). The limitation of synchronization error must approach zero, that is, \(\lim\limits_{n\rightarrow\infty}\mathbf{\Delta}^{(k)}=\Big{(}\mathbf{x}_{\mathbf{s}}^{(k) }-\mathbf{x}_{\mathbf{m}}\Big{)}^{T}=0\), and the error dynamics should be less than or equal to zero, that is, \(\ \dot{\mathbf{\Delta}}\leqq 0\), when the iteration learning procedure is applied to the response system to track the drive system in the time interval [0, T] after a sufficiently large iterative number, \(k\). The characteristics between the drive and response systems are that the drive system is a reference system and the tracking system is a response system in the synchronization procedure, respectively. The response system traces the trajectory of the drive system by employing the output information of the drive system.