# Chaotic Synchronizing Systems with Zero Time Delay and Free Couple via Iterative Learning Control
## 2 The Iterative Learning Control (ILC) Problem Formulation
### System Description
**_If the error dynamics in Equation (2) are convergent, then the iterative learning control law in Equation (6) is anon-increasing function._  
**Proof. ** From Equation (5) and Lemma 1, the error in Equation (2) can be rewritten as follows  
\[\mathbf{\Delta}^{(k)}=\left(\mathbf{x_{m}}-\hat{\mathbf{x}_{m}}\right)^{T}+~{}\left(\hat{ \mathbf{x}_{m}}-\mathbf{x_{s}^{(k)}}\right)^{T}=~{}\hat{\mathbf{\Delta}}_{\mathbf{m}}+\hat{ \mathbf{\Delta}}_{\mathbf{s}}^{(k)}, \tag{7}\]  
Suppose that \(\left(\mathbf{x_{m}}-\hat{\mathbf{x}_{m}}\right)^{T}\leq\varepsilon_{1}^{T}\) and \(\left(\hat{\mathbf{x}}_{m}-\mathbf{x_{s}^{(k)}}\right)^{T}\leq\varepsilon_{2}^{T}\) take the \(\varepsilon=\mathbf{Max}\left\{\varepsilon_{1}^{T},~{}\varepsilon_{2}^{T}\right\}\); Equation (7) can be rewritten in detail as \(\mathbf{\Delta}^{(k)}=\left(\mathbf{x_{m}}-\hat{\mathbf{x}_{m}}\right)^{T}+~{}\left(\hat{ \mathbf{x}}_{m}-\mathbf{x_{s}^{(k)}}\right)^{T}=~{}\hat{\mathbf{\Delta}}_{\mathbf{m}}+\hat{ \mathbf{\Delta}}_{\mathbf{s}}^{(k)}\leq 2\varepsilon\). The difference of iterative learning control law is \(\mathbf{u}^{(k)}(t)-\mathbf{B_{1}}\mathbf{u}^{(k-1)}(t)=\mathbf{B_{2}}\mathbf{\Delta}^{(k)}\leq\mathbf{ B_{2}}\mathbf{\Delta}^{(k)}\leq\mathbf{B_{2}}\mathbf{\Delta}^{(k)}=0\), which means the sequence \(\left\{\mathbf{u}^{(k)}(t)\right\}_{k=\mathbf{1,2,...}}\) is non-increasing as the error \(\mathbf{\Delta}^{(k)}\) is convergent to zero. The analytical approximation of the chaotic systems synchronizing the trajectoryin Equations (1a) and (1b) is to investigate the Lyapunov stability in [14; 15; 16; 17; 18]. The Lyapunov criterion introduced in Theorem 1 is a positive-definite function with no-time delay and couple free of the system (Equation (2)). **Theorem 1**. : _The iterative learning control law is chosen as Equation (6). The Lyapunov function can be defined as_  
\[\mathbf{V}^{(k)}(t)=\frac{1}{2}\Big{(}\mathbf{\Delta}^{(k)}\Big{)}^{T}\Big{(}\mathbf{\Delta} ^{(k)}\Big{)}+\mu\int\limits_{t_{k-1}}^{t_{k}}\Big{(}\mathbf{u}^{(k)}\Big{)}^{T} \Big{(}\mathbf{u}^{(k)}\Big{)}d\tau, \tag{8}\]  
1. [label=()]
2. When \(\mu=0,\mathbf{V}^{(k)}(t)=\frac{1}{2}\Big{(}\mathbf{\Delta}^{(k)}\Big{)}^{T}\Big{(}\bm {\Delta}^{(k)}\Big{)}\) is the Lyapunov function of an estimation system in the system (Equation (3)). 3. If \(\mathbf{V}^{(k)}(t)\) is Lyapunov function of the system (Equation (2)), then the system should bestable. Proof. : The proof of part (a) in the theorem has been proven. The derivative of the function \(\mathbf{V}^{(k)}(t)\) along the track of the system (Equation (2)) is in [28], the discussion and written as  
\[\dot{\mathbf{V}}^{(k)}(t)=\Big{(}\mathbf{\Delta}^{(k)}\Big{)}^{T}\bigg{(}\dot{\mathbf{ \Delta}}^{(k)}\bigg{)}, \tag{9}\]  
By using Lemma 1, Equation (9) has a trivial solution. The derivative of a Lyapunov function is equal to zero or negative, which implies that the system (Equation (2)) is stable. Next, the proof of part (b) is a general case of a chaotic system with no time-delay and couple free. The derivative of the function \(\mathbf{V}^{(k)}(t)\) along the track of the system (Equation (2)) is introduced in [14; 15; 16; 17; 18] and followed as:  
\[ \begin{split}\dot{\mathbf{V}}^{(k)}(t)=\Big{(}\mathbf{\Delta}^{(k)}\Big{)} ^{T}\bigg{(}\dot{\mathbf{\Delta}}^{(k)}\bigg{)}+\mathbf{\mu}\bigg{[}\Big{(}\mathbf{u}^{( k)}(t_{k})\Big{)}^{T}\Big{(}\mathbf{u}^{(k)}(t_{k})\Big{)}-\Big{(}\mathbf{u}^{(k)}(t_{k-1}) \Big{)}^{T}\Big{(}\mathbf{u}^{(k)}(t_{k-1})\Big{)}\bigg{]}\\ =\Big{(}\mathbf{\Delta}^{(k)}\Big{)}^{T}\bigg{(}\dot{\mathbf{\Delta}}^{(k )}\bigg{)}+\mathbf{\mu}\bigg{[}\Big{(}\mathbf{u}^{(k)}(t_{k})\Big{)}^{2}-\Big{(}\mathbf{u} ^{(k)}(t_{k-1})\Big{)}^{2}\bigg{]}.\end{split}  \tag{10}\]  
The first term in Equation (10) is proven in Equation (8), and the second term should be equal to zero or negative when the iterative learning control law is a non-increased function.