# Trajectory Tracking Control for Reaction–Diffusion System with Time Delay Using P-Type Iterative Learning Method

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## Abstract

**:**

## 1. Introduction

## 2. Problem Formulation and Preliminaries

#### 2.1. Problem Formulation

**Remark**

**1.**

**Remark**

**2.**

#### 2.2. Preliminaries

**Assumption**

**1.**

**Lemma**

**1**

**.**For any scalar function $z(\xb7,t)\in \mathcal{H},x\in [0,L]$, we have

**Lemma**

**2**

**.**Let ${\psi}_{1}(x)$ and ${\psi}_{2}(x)$ be any two real integrable functions in $[a,b]$, then the following inequality holds

**Lemma**

**3**

**.**If $a\ge 0$ and $b\ge 0$ are nonnegative real numbers, then the following inequality holds for any real number $\u03f5>0$

**Definition**

**1**

**.**For a vector function $f(\xb7,t):[0,T]\to {\mathbb{R}}^{n}$, λ-norm is defined as follows:

## 3. Iterative Learning Control Design

#### 3.1. Open-Loop P-Type Iterative Learning Control Design

#### 3.2. Closed-Loop P-Type Iterative Learning Control Design

## 4. Convergence Analysis

#### 4.1. Open-Loop ILC Convergence Analysis

**Theorem**

**1.**

**Proof.**

#### 4.2. Closed-Loop ILC Convergence Analysis

**Theorem**

**2.**

**Proof.**

## 5. Numerical Simulation

#### 5.1. Open-Loop ILC Simulation

#### 5.2. Closed-Loop ILC Simulation

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

ILC | Iterative Learning Control |

PDE | Partial Differential Equation |

D-PDE | Delay Partial Differential Equation |

MIMO | Multiple Inputs and Multiple Outputs |

RMS | Root Mean Square |

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**Figure 1.**Abstract structure diagram of the distributions of actuators in one-dimensional spatial domain.

**Figure 2.**Trajectories of outputs ${y}_{k,i}(t)$ at the specified k-th iteration using open-loop iterative learning schemes (13).

**Figure 3.**Trajectories of ${z}_{k}(x,t)$ at several specified iterations using open-loop iterative learning schemes (13).

**Figure 4.**Trajectories of inputs $|{u}_{k,i}(t)|$ and output errors $|{e}_{k,i}(t)|$ using open-loop iterative learning schemes (13).

**Figure 5.**Trajectories of outputs ${y}_{k,i}(t)$ at the specified k-th iteration using closed-loop iterative learning schemes (20).

**Figure 6.**Trajectories of inputs $|{u}_{k,i}(t)|$ and output errors $|{e}_{k,i}(t)|$ using closed-loop iterative learning schemes (20).

Symbol | Explanation | Value |
---|---|---|

L | Spatial Length | 1 |

T | Finite Time Interval | 0.4 |

$\alpha $ | Model Parameter | 4 |

$\beta $ | Delay-Model Parameter | 1 |

$\tau $ | Time Delay Parameter | 0.1 |

$\u03f5$ | Scalar Parameter | 1 |

$\gamma $ | Coefficient Parameter | 0.01 |

p | Lyapunov Parameter | 0.6272 |

$\eta $ | Scalar Parameter | 0.5824 |

${t}_{s}$ | Sampling Time | 0.001s |

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**MDPI and ACS Style**

Liu, Y.; Li, J.; Jin, Z.
Trajectory Tracking Control for Reaction–Diffusion System with Time Delay Using P-Type Iterative Learning Method. *Actuators* **2021**, *10*, 186.
https://doi.org/10.3390/act10080186

**AMA Style**

Liu Y, Li J, Jin Z.
Trajectory Tracking Control for Reaction–Diffusion System with Time Delay Using P-Type Iterative Learning Method. *Actuators*. 2021; 10(8):186.
https://doi.org/10.3390/act10080186

**Chicago/Turabian Style**

Liu, Yaqiang, Jianzhong Li, and Zengwang Jin.
2021. "Trajectory Tracking Control for Reaction–Diffusion System with Time Delay Using P-Type Iterative Learning Method" *Actuators* 10, no. 8: 186.
https://doi.org/10.3390/act10080186