# Chaotic Synchronizing Systems with Zero Time Delay and Free Couple via Iterative Learning Control

^{*}

## Abstract

**:**

## 1. Introduction

## 2. The Iterative Learning Control (ILC) Problem Formulation

#### 2.1. System Description

_{m}, C

_{s}, and B = B

^{T}are non-singular constant matrices with appropriated dimensions and some entries of the nonsingular polynomial matrix $\mathit{A}\left({x}_{m}^{i}\right)$ are replaced by the i-th component of ${\mathit{x}}_{\mathit{m}}\left(t\right)$,which is the factor of system synchronization, in which i = 1, 2, 3. The input signal sequence ${\left\{{\mathit{u}}^{\left(k\right)}\left(t\right)\right\}}_{k=\mathbf{1},\mathbf{2},\dots}$ in ${\mathit{R}}^{m}$ is the control learning law after the k-th iterative learning for the response system synchronizing the drive system.

**B**. The limitation of synchronization error must approach zero, that is, $\underset{\mathrm{n}\to \infty}{\mathrm{lim}}{\mathbf{\Delta}}^{\left(k\right)}={\left({\mathit{x}}_{\mathit{s}}^{\left(k\right)}-{\mathit{x}}_{\mathit{m}}\right)}^{T}=0$, and the error dynamics should be less than or equal to zero, that is, $\dot{\Delta}\leqq 0$, 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.

**E**is to differentiate

**E**with respect to the state vector ${\widehat{\mathit{x}}}_{\mathit{m}}$ and equal to the results:

**E**.

**Lemma**

**1.**

**Proof.**

**Lemma**

**2.**

**Proof.**

**Theorem**

**1.**

- (a)
- When $\mu =0,{\mathit{V}}^{\left(k\right)}\left(t\right)=\frac{1}{2}{\left({\mathbf{\Delta}}^{\left(k\right)}\right)}^{T}\left({\mathbf{\Delta}}^{\left(k\right)}\right)$ is the Lyapunov function of an estimation system in the system (Equation (3)).
- (b)
- If${\mathit{V}}^{\left(k\right)}\left(t\right)$is Lyapunov function of the system (Equation (2)), then the system should bestable.

**Proof.**

**u**

^{(k)}(t), in the Lyapunov function applied in a more complex system. The decision was the most appropriate for iterative learning control law and parameters

**B**and

_{1}**B**to reduce the divergence of non-linear systems; this should be discussed and studied for the synchronization of non-linear systems.

_{2}#### 2.2. Proposed Algorithm for Iterative Learning Control Law

**u**

^{(k)}, which is bounded and convergent, satisfying the criteria of monotonically convergent conditions. The learning control input

**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,

**u**

^{(k)}, must be bounded.

**Corollary**

**1.**

**u**

^{(k)}in Equation (6) is a non-increasing and bounded function.

**Proof.**

**B**,

**B**, and

_{1}**B**making the sequence ${\left\{{\mathit{u}}^{\left(k\right)}\left(t\right)\right\}}_{\mathit{k}=\mathbf{1},\mathbf{2},\cdots}$, being strictly decreasing are important. The ILC law${\left\{{\mathit{u}}^{\left(k\right)}\left(t\right)\right\}}_{\mathit{k}=\mathbf{1},\mathbf{2},\cdots}$ can be expanded in the initial learning law${\mathit{u}}^{\left(0\right)}\left(t\right)$by induction as follows:

_{2}**L**in this research follows the method of Hauser [13] as:

**u**

^{(0)}is the maximum in the monotonically decreasing sequence ${\left\{{\mathit{u}}^{\left(k\right)}\left(t\right)\right\}}_{k=1,2,\cdots}$ and $||\mathit{L}{\mathit{B}}_{2}||\le 1$in Equation (13). Theses matrices can be found by the Linear Matrix Inequality (LMI) method, but were not the object in this research.□

## 3. Example Illustration and Demonstrated Results

#### 3.1. The Example of Iterative Learning Algorithmto Decide Learning Law

**R**

^{3}. The parameters are explained as ${\mathit{x}}_{\mathit{m}}={\left({x}_{m1},{x}_{m2},{x}_{m3}\right)}^{T}$, ${\mathit{x}}_{\mathit{s}}^{\left(k\right)}={\left({x}_{s1}^{\left(k\right)},{x}_{s2}^{\left(k\right)},{x}_{s3}^{\left(k\right)}\right)}^{T},$ and ${\dot{\mathbf{\Delta}}}^{\left(k\right)}={\left({\dot{\mathit{x}}}_{\mathit{m}}\left(t\right)-{\dot{\mathit{x}}}_{\mathit{s}}^{\left(k\right)}\left(t\right)\right)}^{T}$. The ${x}_{m}^{1}$ in polynomial matrix$\mathit{A}\left({x}_{m}^{1}\right)$ is the first component of the state vector in the drive system. The state vector in the estimation system is${\widehat{\mathit{x}}}_{\mathit{m}}\left(t\right)={\left(\mathit{A}{\left({x}_{m}^{1}\right)}^{T}\mathit{A}\left({x}_{m}^{1}\right)\right)}^{-1}\mathit{A}{\left({x}_{m}^{1}\right)}^{T}{\dot{\mathit{x}}}_{\mathit{m}}{}^{T}$and the iterative learning law with the initial condition${\mathit{u}}^{\left(0\right)}\left(t=0\right)={\mathbf{\Delta}}^{\left(0\right)}\left(t=0\right)={\left({\mathit{x}}_{\mathit{m}\mathbf{0}}-{\mathit{x}}_{\mathit{s}\mathbf{0}}^{\left(\mathbf{0}\right)}\right)}^{T}$. The${\mathit{u}}^{\left(k\right)}\left(t\right)$in Equation (6) and the Lyapunov equation are respectively shown as Equations (6) and (8).The matrices

**B**and

_{1}**B**in the ILC rule of Equation (6) are:

_{2}**B**

_{2}is from the decomposition of the coefficient matrix of the system in (Equation (14)) and the matrix ${\mathit{B}}_{1}={\left(\mathit{M}\right)}^{k}{\left({\mathit{B}}_{2}\right)}^{-2}$.The parameters $0\le k\le 1$ and the matrix$\mathit{M}$ is in [33]. The matrix

**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. The drive system used the ode45 function in Simulink of MATLAB, and the response system used the Euler method with the estimated state vectors in Equation (5).The relevant simulation results are shown in the next section.

#### 3.2. Simulation Results and Discussion

**B**in Equation (6) was chosen as the identity matrix, and

_{1}**B**in Figure 5b as a diagonal matrix [0.1, 0.1, 0.92]. In both Figure 5a,b, one of the components rose sharply and continued to vibrate in a bounded interval. Although the other components gradually decreased in the previous ILC rule, the ILC rule still makes the response system under an unpredictable situation.

_{1}## 4. Conclusions

## Author Contributions

## Conflicts of Interest

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**Figure 3.**The simulation of each component of the error dynamics system ${\dot{\mathbf{\Delta}}}^{\left(\mathrm{k}\right)}$.

**Figure 5.**The different behaviors of IL Care chosen as the worst learning law and listed as (

**a**)

**B**is identity matrix; and (

_{1}**b**)

**B**= Dig [0.1, 0.1, and 0.92].

_{1}Value of μ | [−1, 1] | (−∞, −1] | [1, ∞] | Total | Average |
---|---|---|---|---|---|

1 | 1945 | 4811 | 5345 | 12001 | −2.5314 |

10 | 396 | 5473 | 6132 | 12001 | –19.896 |

50 | 93 | 5623 | 6285 | 12001 | –97.0546 |

–10 | 393 | 6149 | 5459 | 12001 | 18.635 |

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**MDPI and ACS Style**

Cheng, C.-K.; Chao, P.C.-P. Chaotic Synchronizing Systems with Zero Time Delay and Free Couple via Iterative Learning Control. *Appl. Sci.* **2018**, *8*, 177.
https://doi.org/10.3390/app8020177

**AMA Style**

Cheng C-K, Chao PC-P. Chaotic Synchronizing Systems with Zero Time Delay and Free Couple via Iterative Learning Control. *Applied Sciences*. 2018; 8(2):177.
https://doi.org/10.3390/app8020177

**Chicago/Turabian Style**

Cheng, Chun-Kai, and Paul C. -P. Chao. 2018. "Chaotic Synchronizing Systems with Zero Time Delay and Free Couple via Iterative Learning Control" *Applied Sciences* 8, no. 2: 177.
https://doi.org/10.3390/app8020177