# Chaotic Synchronizing Systems with Zero Time Delay and Free Couple via Iterative Learning Control
## 2 The Iterative Learning Control (ILC) Problem Formulation
### System Description
To achieve the goal of synchronization and search a system whose trajectory is close to the drive system, it is necessary to find an estimated system similar to the drive system. The estimated system in [21] can be defined by the measuring system of the drive system as shown in Equation (3). \[\dot{\mathbf{x}}_{\mathbf{m}}(t)=\mathbf{H}(\mathbf{x},t)+\mathbf{\varepsilon}=\mathbf{A}\Big{(}x_{m }^{i}\Big{)}\dot{\mathbf{x}}_{\mathbf{m}}(t)+\mathbf{\varepsilon}, \tag{3}\]  
The nonlinear problem has no general solution. The perturbation and linearization techniques will be applied to Equation (3). The least square linear estimation is familiar to minimize the errors in measurement processes. The error criteria of Equation (3) is defined as  
\[\mathbf{E}(\mathbf{\varepsilon})=\mathbf{\varepsilon}^{T}\mathbf{\varepsilon}=\Big{(}\dot\mathbf{{x}}_{\mathbf{m}}(t)-\mathbf{A}\Big{(}x_{m}^{i}\Big{)}\dot{\mathbf{x}}_{\mathbf{m}}(t)\Big{)}^{ T}\Big{(}\dot{\mathbf{x}}_{\mathbf{m}}(t)-\mathbf{A}  
The design of the Lyapunov function reaches the synchronization of the chaotic system in which the manifold \(\mathbf{x_{m}}=\mathbf{x_{s}^{(k)}}\) must be stable [14; 15; 16; 17; 18].This fact indicates the system's synchronization of Equations (1a) and (1b) through the ILC procedure so that the error dynamics Equation (2) are stable by the Lyapunov criteria and the local Lipschitz condition is satisfied during the period of the system.  
**Lemma 1.**_Equation (2) has a trivial solution by the ILC procedure \(f(\mathbf{x_{m}},t)\) Equation (1a) and \(f\left(\mathbf{x_{s}^{(k)}},t\right)\) in Equation (1b) are satisfied by the local Lipschitz condition in the interval \([0,\)_T_.  
**Proof.** By the ILC procedure, there is a trivial solution \(\mathbf{\Delta}^{(k)}=\left(\mathbf{x_{m}}-\mathbf{x_{s}^{(k)}}\right)^{T}=0\), which implies \(\delta>0\) for all \(\varepsilon>0\) and \(k=0\), \(1\), \(2\), \(\dots\) such that \(||f(\mathbf{x_{m}},t)-f\left(\mathbf{x_{s}^{(k)}},t\right)||<\varepsilon\) as \(||\mathbf{x_{m}}-\mathbf{x_{s}^{(k)}}||<\delta\), which means the condition \(||f(\mathbf{x_{m}},t)-f\left(\mathbf{x_{s}^{(k)}},t\right)||<\varepsilon||\mathbf{x_{m}}- \mathbf{x_{s}^{(k)}}||\) isheld. The \(f(\mathbf{x_{m}},t)\) isconvergent to \(f\left(\mathbf{x_{s}^{(k)}},t\right)\) when \(\mathbf{\Delta}^{(k)}=\left(\mathbf{x_{m}}-\mathbf{x_{s}^{(k)}}\right)^{T}\) approaches zero. The consequence of this lemma implies that \(||f(\mathbf{x_{m}},t)-f\left(\mathbf{x_{s}^{(k)}},t\right)||<\varepsilon\delta\to 0\). \(\square\)  
The convergence of the synchronization error, \(\mathbf{\Delta}^{(k)}\), indicates the error dynamics \(\dot{\mathbf{\Delta}}^{(k)}\) as non-increasing, that is \(\dot{\mathbf{\Delta}}^{(k)}\leq 0\), and is dependent on the iterative learning control law, \(\mathbf{u}^{(k)}(t)\) in Equation (1a) is chosen as:  
\[\mathbf{u}^{(k)}(t)=\mathbf{B_{1}}\mathbf{u}^{(k-1)}(t)+\mathbf{B_{2}}\Big{(}\hat{\mathbf{\Delta} }_{m}+\hat{\mathbf{\Delta}}_{s}^{(k)}\Big{)}\!\!=\mathbf{B_{1}}\mathbf{u}^{(k-1)}(t)+\mathbf{B _{2}}\mathbf{\Delta}^{(k)}, \tag{6}\]  
in which \(\mathbf{B_{1}}\) and \(\mathbf{B_{2}}\) are appropriate constant matrices and symmetry. The learning law \(\mathbf{u}^{(k-1)}\) is prior to \(\mathbf{u}^{(k)}.\hat{\mathbf{\Delta}}_{\mathbf{m}}=\left(\mathbf{x_{m}}-\mathbf{x_{m}^{(k)}} \right)^{T}\), and \(\hat{\mathbf{\Delta}}_{\mathbf{s}}^{(k)}=\left(\hat{\mathbf{x}}_{\mathbf{m}}-\mathbf{x_{s}^{(k)}} \right)^{T}\) are the errors between the estimated state vectors and the state vectors of \(\mathbf{x_{m}}\) and \(\mathbf{x_{s}^{(k)}}\), respectively. The sum of errors of Equation (6) is \(\hat{\mathbf{\Delta}}_{\mathbf{m}}+\hat{\mathbf{\Delta}}_{\mathbf{s}}^{(k)}=~{}\mathbf{\Delta}^{(k)}\). If the error dynamics in Equation (2) is convergent then the iterative learning control law \(\mathbf{u}^{(k)}(t)\) is decremented. The completed proof is exhibited in Lemma 2. **Lemma 2.