# Chaotic Synchronizing Systems with Zero Time Delay and Free Couple via Iterative Learning Control
## 2 The Iterative Learning Control (ILC) Problem Formulation
### Proposed Algorithm for Iterative Learning Control Law  
The iterative learning control algorithm is exhibited in Figure 1. The diagram contains three systems: drive system, the response system, and the estimated system, with three outputs, namely the output of the drive system, the output of the response system, and the output of error, respectively. The initial conditions of the drive system and the response system are different. The iterative learning control law of the first stage exhibits the error of initial conditions between the drive system and the response system. The estimation system in Equation (3) provides for the estimated state vectors as expressed in Equation (4), the drive and response systems, respectively. The drive system and the response system are closed-loop so that the feedback in the former is its own output of the drive system, and the feedback in the latter is the result of iterative learning control law as its own output of the response system.  
The algorithm in Figure 1 examines the learning control input \(\mathbf{u}^{(k)}\), which is bounded and convergent, satisfying the criteria of monotonically convergent conditions. The learning control input \(\mathbf{u}^{(k)}\) in Equation (6) is concerned with the ability to adjust the feedback error of the response system and track the trajectory of the drive system. Therefore, the iterative learning control law, \(\mathbf{u}^{(k)}\), must be bounded.  
**Corollary 1**.: _The learning control input \(\mathbf{u}^{(k)}\) in Equation (6) is a non-increasing and bounded function._  
Proof.: The learning control law in Equation (5) is an updated law to refresh the input of the system (Equation (1b)) proposed. The sequence \(\left\{\mathbf{u}^{(k)}(t)\right\}_{k=\mathbf{1},\mathbf{2},\...}\) is a non-increasing sequence such that the condition \(\mathbf{Max}\left\{\mathbf{u}^{(k)}(t)\right\}_{k=\mathbf{1},\mathbf{2},\...}=\mathbf{u}^{(0)}(t)\leq M\) is held with an upper bounded \(M\) of the real number in Lemma 2.  
The appropriate matrices \(\mathbf{B}\), \(\mathbf{B_{1}}\), and \(\mathbf{B_{2}}\) making the sequence \(\left\{\mathbf{u}^{(k)}(t)\right\}_{k=\mathbf{1},\mathbf{2},\...}\) being strictly decreasing are important. The ILC law \(\left\{\mathbf{u}^{(k)}(t)\right\}_{k=\mathbf{1},\mathbf{2},\...}\) can be expanded in the initial learning law \(\mathbf{u}^{(0)}(t)\) by induction as follows:  
\[\mathbf{u}^{(k)}(t)=(\mathbf{B_{1}})^{k}\mathbf{u}^{(0)}(t)+(\mathbf{B_{1}})^{k-1}\mathbf{B_{2}} \mathbf{\Delta}^{(1)}+(\mathbf{B_{1}})^{k-2}\mathbf{B_{2}}\mathbf{\Delta}^{(2)}+\cdots+\mathbf{B_{ 2}}\mathbf{\Delta}^{(k)}, \tag{11}\]  
The learning operator \(\mathbf{L}\) in this research follows the method of [PERSON] [13] as:  
\[\mathbf{L}\equiv\left(\mathbf{B}\mathbf{B_{1}^{k}}\right) \tag{12}\]  
\[||\mathbf{I}-\mathbf{L}\mathbf{B_{2}}||\leq\delta<1. \tag{13}\]  
The consequence is from the monotonically decreasing sequence \(\left\{\mathbf{u}^{(k)}(t)\right\}_{k=\mathbf{1},\mathbf{2},\...}\) of the ILC rule. The \(\mathbf{u}^{(0)}\) is the maximum in the monotonically decreasing sequence \(\left\{\mathbf{u}^{(k)}(t)\right\}_{k=1,\mathbf{2},\...}\) and \(||\mathbf{L}\mathbf{B_{2}}||\leq 1\) in Equation (13). Theses matrices can be found by the Linear Matrix Inequality (LMI) method, but were not the object in this research.