# Adaptive Event-Triggered Neural Network Fast Finite-Time Control for Uncertain Robotic Systems

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## Abstract

**:**

## 1. Introduction

- 1.
- Adaptive event-triggered control is designed by using the logarithm-type BLF and the ETM, which effectively reduces the update frequency of the transmitted information between the controller and the actuator while ensuring that the outputs of the robotic systems remain within the predefined constraint interval.
- 2.
- Using fast finite-time stability theory, an adaptive NN control approach is proposed. In particular, while compensating for system uncertainties, the system’s convergence speed is also accelerated, so that the tracking error quickly converges to a bounded and adjustable compact set within a finite time to improve the system’s robustness.

## 2. Problem Description and Preliminaries

#### 2.1. System Exposition

**Property**

**1**

**Assumption**

**1**

**.**The desired signal vector ${y}_{s}={\left[{y}_{s1},{y}_{s2},\dots ,{y}_{sn}\right]}^{T}\in {\mathcal{R}}^{n}$ and its first-order derivative ${\dot{y}}_{s}={\left[{\dot{y}}_{s1},{\dot{y}}_{s2},\dots ,{\dot{y}}_{sn}\right]}^{T}\in {\mathcal{R}}^{n}$ are both continuous and bounded. There exist positive constant vectors ${S}_{0}$ and ${S}_{1}$ such that $\left|{y}_{s}\right|\le {S}_{0}\le {\overline{k}}_{v1}$ and $\left|{\dot{y}}_{s}\right|\le {S}_{1}$.

**Assumption**

**2**

**.**Assume that for any $t\in [0,\infty )$, the constrained force $Z\left(t\right)$ is uniformly bounded, and there exists $\overline{Z}>0$, which satisfies $\left|Z\left(t\right)\right|\le \overline{Z}$.

**Lemma**

**1**

**.**Considering a general system $\dot{x}=f(x,w)$, if there exists a continuous function $V(x,t)$, ${\gamma}_{1}>0$, ${\gamma}_{2}>0$, $\varrho \in (0,\infty )$ and $h\in (0,1)$ such that

**Lemma**

**2**

**.**For $\forall {\vartheta}_{j}\in \mathcal{R}$, $j=1,2,\dots ,n$ and $a\in [0,1]$, the following inequality holds

**Lemma**

**3**

**.**For any δ, $\mu \in \mathcal{R}$, one has

**Lemma**

**4**

**.**For any ${\kappa}_{1}>0$, ${\kappa}_{2}>0$, ${\kappa}_{3}>0$, ${\psi}_{1}>0$, ${\psi}_{2}>0$ and ${\psi}_{3}>0$, the following inequality holds

#### 2.2. Radial Basis Function Neural Networks

## 3. Adaptive Event-Triggered Fast Finite-Time Control Design

#### 3.1. Control Design

**Remark**

**1.**

**Remark**

**2.**

#### 3.2. Stability Analysis

**Theorem**

**1.**

- 1.
- All the closed-loop system signals are bounded, and the tracking errors ${\xi}_{1i}$ rapidly converge to the bounded and adjustable compact set within a finite time.
- 2.
- The system output does not exceed the predefined constraint interval, and the Zeno phenomenon does not occur successfully.

**Proof of Theorem**

**1.**

**Remark**

**3.**

Algorithm 1: The proposed control method |

Choose the following controller parameters: The parameters of ${\alpha}_{1i}$: ${z}_{1i}$, ${k}_{1i}$, ${s}_{1i}$, h; The parameters of ${\alpha}_{2i}$ and ${\zeta}_{i}$: ${z}_{2i}$, ${b}_{2i}$, ${r}_{i}$, ${a}_{i}$, ${\rho}_{i}$; Choose the following controller parameters: Reference output: ${y}_{s}$; Initializing: ${x}_{i}\left(0\right)$, ${\widehat{\zeta}}_{i}\left(0\right)$; State feedback: ${x}_{i}\left(t\right)$. FOR EACH t 1. Update the system states by solving (1); 2. ${\alpha}_{1i}$ is computed by solving (16); 3. ${\alpha}_{2i}$ and ${\widehat{\zeta}}_{i}$ are computed by solving (30) and (33), respectively; 4. The control input ${\varpi}_{i}\left(t\right)$ is calculated by solving (25); 5. Update the control input ${\phi}_{i}\left(t\right)$ according to the following rules: IF $\left|{m}_{i}\left(t\right)\right|\ge {\lambda}_{i}\left|{\phi}_{i}\left(t\right)\right|+{o}_{i}$ ${\phi}_{i}\left(t\right)={\varpi}_{i}\left(t\right)$; ELSE ${\phi}_{i}\left(t\right)$ keeps the value of the previous moment; END 6. ${\phi}_{i}\left(t\right)$ is applied to the system (2); END FOR System output: y. |

## 4. Simulation

#### 4.1. Example A: Two-DOF Rigid Robotic System

#### 4.2. Example B: A Three-DOF Rigid Robotic System

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Table 1.**Definition of elements in $H\left({x}_{1}\right)$, $K({x}_{1},{x}_{2})$, $D\left({x}_{1}\right)$ and $\mathcal{P}\left({x}_{1}\right)$.

Elements | Definition | Elements | Definition |
---|---|---|---|

${H}_{11}$ | ${m}_{1}{L}_{c1}^{2}+{m}_{2}({L}_{1}^{2}+{L}_{c2}^{2}$ | ${K}_{22}$ | 0 |

$+2{L}_{1}{L}_{c2}cos\left({w}_{2}\right))+{I}_{1}+{I}_{2}$ | ${D}_{11}$ | $({m}_{1}{L}_{c2}+{m}_{2}{L}_{1})gcos\left({w}_{1}\right)$ | |

${H}_{21}$ | ${m}_{2}({L}_{c2}^{2}+{L}_{1}{L}_{c2}cos\left({w}_{2}\right))+{I}_{2}$ | $+{m}_{2}{L}_{c2}gcos({w}_{1}+{w}_{2})$ | |

${H}_{12}$ | ${m}_{2}({L}_{c2}^{2}+{L}_{1}{L}_{c2}cos\left({w}_{2}\right))+{I}_{2}$ | ${D}_{21}$ | ${m}_{2}{L}_{c2}gcos({w}_{1}+{w}_{2})$ |

${H}_{22}$ | ${m}_{2}{L}_{c2}^{2}+{I}_{2}$ | ${\mathcal{P}}_{11}$ | $-{L}_{1}sin\left({w}_{1}\right)-{L}_{2}sin({w}_{1}+{w}_{2})$ |

${K}_{11}$ | $-{m}_{2}{L}_{1}{L}_{c2}{\dot{w}}_{2}sin\left({w}_{2}\right)$ | ${\mathcal{P}}_{21}$ | ${L}_{1}cos\left({w}_{1}\right)+{L}_{2}cos({w}_{1}+{w}_{2})$ |

${K}_{21}$ | ${m}_{2}{L}_{1}{L}_{c2}{\dot{w}}_{1}sin\left({w}_{2}\right)$ | ${\mathcal{P}}_{12}$ | $-{L}_{2}sin({w}_{1}+{w}_{2})$ |

${K}_{12}$ | $-{m}_{2}{L}_{1}{L}_{c2}({\dot{w}}_{1}+{\dot{w}}_{2})sin\left({w}_{2}\right)$ | ${\mathcal{P}}_{22}$ | ${L}_{2}cos({w}_{1}+{w}_{2})$ |

Control Design | Overshoot (rad) | Settling Time (s) |
---|---|---|

PD | $2.36s$ | 0.03 |

LASC | $0.87s$ | 0.04 |

Proposed control method | $0.87s$ | 0.1 |

**Table 3.**Definition of elements in $H\left({x}_{1}\right)$, $K({x}_{1},{x}_{2})$ and $D\left({x}_{1}\right)$.

Elements | Definition | Elements | Definition |
---|---|---|---|

${H}_{11}$ | ${m}_{3}{p}_{3}^{2}si{n}^{2}\left({w}_{2}\right)+{m}_{3}{p}_{1}^{2}+{m}_{2}{p}_{1}^{2}+{I}_{1}$ | ${K}_{13}$ | ${m}_{3}{w}_{3}si{n}^{2}\left({w}_{2}\right){\dot{w}}_{1}$ |

${H}_{12}$ | ${m}_{3}{w}_{3}{p}_{1}cos\left({w}_{2}\right)$ | $-{m}_{3}{p}_{1}{w}_{3}sin\left({w}_{2}\right){\dot{w}}_{2}$ | |

${H}_{13}$ | ${m}_{3}{p}_{1}sin\left({w}_{2}\right)$ | ${K}_{21}$ | $-{m}_{3}{w}_{3}^{2}sin\left({w}_{2}\right)cos\left({w}_{2}\right){\dot{w}}_{1}$ |

${H}_{21}$ | ${m}_{3}{w}_{3}{p}_{1}cos\left({w}_{2}\right)$ | ${K}_{22}$ | ${m}_{3}{w}_{3}{\dot{w}}_{3}$ |

${H}_{22}$ | ${m}_{3}{w}_{3}^{2}+{I}_{2}$ | ${K}_{23}$ | ${m}_{3}{p}_{1}cos\left({w}_{2}\right){\dot{w}}_{1}-{m}_{3}{w}_{3}{\dot{w}}_{2}$ |

${H}_{23}$ | 0 | ${K}_{31}$ | $-{m}_{3}{w}_{3}si{n}^{2}\left({w}_{2}\right){\dot{w}}_{1}$ |

${H}_{31}$ | ${m}_{3}{p}_{1}sin\left({w}_{2}\right)$ | $+{m}_{3}{p}_{1}cos\left({w}_{2}\right){\dot{w}}_{2}$ | |

${H}_{32}$ | 0 | ${K}_{32}$ | $-{m}_{3}{w}_{3}{\dot{w}}_{2}+{m}_{3}{p}_{1}cos\left({w}_{2}\right){\dot{w}}_{2}$ |

${H}_{33}$ | ${m}_{3}$ | ${K}_{33}$ | 0 |

${K}_{11}$ | ${m}_{3}{w}_{3}^{2}sin\left({w}_{2}\right)cos\left({w}_{2}\right){\dot{w}}_{2}$ | ${D}_{11}$ | 0 |

$+{m}_{3}{w}_{3}^{2}si{n}^{2}\left({w}_{2}\right){\dot{w}}_{3}$ | ${D}_{21}$ | $-{m}_{3}g{w}_{3}cos\left({w}_{2}\right)$ | |

${K}_{12}$ | ${m}_{3}{w}_{3}^{2}sin\left({w}_{2}\right)cos\left({w}_{2}\right){\dot{w}}_{1}$ | ${D}_{31}$ | $-{m}_{3}gsin\left({w}_{2}\right)$ |

$-{m}_{3}{p}_{1}{w}_{3}sin\left({w}_{2}\right)({\dot{w}}_{1}+{\dot{w}}_{2})$ |

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## Share and Cite

**MDPI and ACS Style**

Wang, J.; Du, Y.; Zhang, Y.; Gu, Y.; Chen, K.
Adaptive Event-Triggered Neural Network Fast Finite-Time Control for Uncertain Robotic Systems. *Mathematics* **2023**, *11*, 4841.
https://doi.org/10.3390/math11234841

**AMA Style**

Wang J, Du Y, Zhang Y, Gu Y, Chen K.
Adaptive Event-Triggered Neural Network Fast Finite-Time Control for Uncertain Robotic Systems. *Mathematics*. 2023; 11(23):4841.
https://doi.org/10.3390/math11234841

**Chicago/Turabian Style**

Wang, Jianhui, Yongping Du, Yuanqing Zhang, Yixiang Gu, and Kairui Chen.
2023. "Adaptive Event-Triggered Neural Network Fast Finite-Time Control for Uncertain Robotic Systems" *Mathematics* 11, no. 23: 4841.
https://doi.org/10.3390/math11234841