# Tube-Based Event-Triggered Path Tracking for AUV against Disturbances and Parametric Uncertainties

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## Abstract

**:**

## 1. Introduction

- A tube-based event-triggered path-tracking strategy, which consists of a LMPC controller and a tube MPC controller, is proposed to enhance the robustness against disturbances and parametric uncertainties. The LMPC controller is used to calculate the speed control law to converge the path-tracking deviation, and the tube MPC controller is used to track the speed control law.
- In the tube MPC controller, with nonlinear characteristics of AUV hydrodynamic force considered, tight constraints in the nominal control law and the feedback matrix in the feedback control law are obtained by formulating two LMIs. To achieve real-time performance, these linear matrix inequalities are all calculated offline.
- To overcome control performance degradation brought by conservative tight constraints calculated offline, an event-triggering mechanism is used to dynamically adjust these constraints in the nominal control law according to the surge speed change command. Compared with conservative tight constraints, better path tracking can be achieved, and the real-time performance is also satisfied.

## 2. Preliminaries

**Definition 1.**

## 3. AUV Motion Model and Problem Formulation

#### 3.1. AUV Motion Model

#### 3.2. Model Decoupling

- 1.
- Surge speed nominal control model:

- 2.
- Heading nominal control model:

- 3.
- Depth nominal control model:

#### 3.3. Problem Statement

**Problem 1.**

**Problem 2.**

## 4. Methodology

#### 4.1. LMPC Controller

#### 4.2. Tube MPC Controller

**Assumption 1.**

**Lemma**

**1**

**.**For a Lipschitz nonlinear system (1), there exists a positive definite matrix $X\in {R}^{n\times n}$, a matrix $Y\in {R}^{m\times n}$, and scalars ${\lambda}_{0}>\lambda >0$ and $\mu >0$ such that:

**Lemma**

**2**

**.**Suppose the LDI of nominal system (6) is given by (33), and the constraints of nominal system (6) are obtained by (11) and (31). There exist matrices $0<{W}_{1}\in {R}^{n\times n}$ and ${W}_{2}\in {R}^{m\times n}$ such that:

#### 4.3. Implementation of the Proposed Strategy

Algorithm 1 Offline strategy |

1. Define nominal cost function (7); choose state and control input constraints (2) |

2. Choose appropriate parameters $\lambda \text{}$and$\text{}L$ to solve LMI (31) |

3. Obtain feedback matrix $K$ and RPI set $\mathsf{\Omega}$ (32) |

4. Calculate invariant rigid tube (11) |

5. Choose appropriate weight matrices $Q\text{}\mathrm{a}\mathrm{n}\mathrm{d}\text{}R$ to solve optimal control problem (37) 6. Obtain terminal feasible set ${X}_{f}$ |

Algorithm 2 Online AUV path-tracking algorithm |

1. Measure AUV’s actual state ${\eta}_{t},{\nu}_{t}$, and nominal state ${\overline{\eta}}_{t},{\overline{\nu}}_{t}$ |

2. Solve optimal control problem (29) to obtain the speed control law ${\overline{\nu}}_{{d}_{t}}$ |

3. If $\u2206{{u}_{s}}_{t}\le \text{}\u2206{\overline{u}}_{s}$: 4. Based on these decoupling models (19–20,22), separately formulate optimal control problem (38) to obtain nominal control vector ${\overline{\tau}}_{t}={\left({{\overline{F}}_{x}}_{t},{{\overline{\delta}}_{r}}_{t},{{\overline{\delta}}_{s}}_{t}\right)}^{T}$ 5. Otherwise: 6. Based on these decoupling models (19–20,22), separately formulate optimal control problem (14) to obtain nominal control vector ${\overline{\tau}}_{t}={\left({{\overline{F}}_{x}}_{t},{{\overline{\delta}}_{r}}_{t},{{\overline{\delta}}_{s}}_{t}\right)}^{T}$ 7. End |

8. Calculate the AUV’s control vector ${\tau}_{t}={\overline{\tau}}_{t}+K\left({\nu}_{t}-{\overline{\nu}}_{t}\right)$ |

9. Set $t=t+1$, and go back to 1 |

## 5. Numerical Simulation

#### 5.1. Parameters Set

#### 5.2. Analysis and Discussion

## 6. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Nomenclature

$\u229d$ | Pontryagin difference, $\mathit{A}\u229d\mathit{B}=\left\{\mathit{x}|\mathit{x}+\mathit{y}\mathit{\u03f5}\mathit{A},\mathit{y}\mathit{\u03f5}\mathit{B}\right\}$ | ${\mathit{I}}_{\mathit{n}}$ | n-dimensional identity matrix |

${\mathit{\alpha}}_{\mathbf{m}\mathbf{i}\mathbf{n}}\left(\xb7\right)$(${\mathit{\alpha}}_{\mathbf{m}\mathbf{a}\mathbf{x}}\left(\xb7\right)$) | the smallest (largest) real part of eigenvalues of a matrix | ${\mathit{R}}^{\mathit{m}\times \mathit{n}}$ | A matrix with m rows and n columns |

$\mathbb{w}$ | bounded external disturbance | ${\mathit{c}}_{\mathit{\omega}}$ | disturbance upper bound |

$\mathit{Q},\mathit{R},\mathit{P}$ | positive weight matrix | ${\Vert \xb7\Vert}_{\mathit{Q}}^{2}$ | quadratic norm of a vector with positive weight matrix $\mathit{Q}$ |

$\mathit{g}\left(\xb7\right)$ | Lipschitz nonlinear function | $\mathit{L}$ | Lipschitz constant |

$\overline{\mathit{x}},\overline{\mathit{u}}$ | nominal state and control input | $\mathit{x},\mathit{u}$ | actual state and control input |

${\mathit{J}}_{\mathit{N}},{\mathit{J}}_{1}$ | cost function | $\mathit{K}$ | feedback matrix |

${\mathit{X}}_{\mathit{f}}$ | terminal feasible set | $\mathsf{\Omega}$ | robust positively invariant (RPI) set |

$\mathit{h}\left(\xb7\right)<0$ | inequality constraint | $\mathit{M}$ | constraint set |

${\mathit{f}}_{\mathit{d}}\left(\xb7\right)$ | state transition function | ${\mathit{N}}_{\mathit{T}},{\mathit{N}}_{\mathit{l}}$ | predictive horizon |

$\mathsf{\Theta}\left(\xb7\right)$ | linear differential inclusion function | $\mathbf{C}\mathbf{o}\mathsf{\Theta}\left(\xb7\right)$ | minimum convex polytope |

$\mathbf{d}\mathbf{e}\mathbf{t}\left(\xb7\right)$ | determinant calculation | ${\mathit{\alpha}}_{\mathit{w}}\left(\xb7\right)$ | polynomial function |

${\mathbb{K}}_{{\mathit{N}}_{1}:{\mathit{N}}_{2}}$ | set $\left\{{\mathit{N}}_{1},{\mathit{N}}_{1}+1,\cdots ,{\mathit{N}}_{2}-1,{\mathit{N}}_{2}\right\}$ |

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**Figure 1.**AUV coordinate system [24].

Parameter | Value | Parameter | Value | Parameter | Value |
---|---|---|---|---|---|

${Q}_{\eta}$ | $\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{g}\{\mathrm{4.4,19.2,5.2},$ $\mathrm{20.5,25.5}\}$ | ${P}_{\eta}$ | $\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{g}\{\mathrm{4.4,19.2,5.2}$ $,\mathrm{20.5,25.5}\}$ | ${\overline{\nu}}_{min}$ | $\begin{array}{c}-[0;0.06;0.01;\\ 0.03;0.08]\end{array}$ |

${Q}_{\nu}$ | $25.5$ | ${P}_{\nu}$ | $25.5$ | ${N}_{l}$ | 4 |

${R}_{\nu}$ | $\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{g}\{\mathrm{2,0.3,5},\mathrm{2,0.1}\}$ | ${{\overline{u}}_{l}}_{max}$ | $\begin{array}{c}[0.2;0.01;0.01;\\ 0.03;0.05]\end{array}$ | ${\overline{\nu}}_{max}$ | $\begin{array}{c}[1.2;0.06;0.01;\\ 0.03;0.08]\end{array}$ |

Parameter | Value | Parameter | Value | Parameter | Value | Parameter | Value |
---|---|---|---|---|---|---|---|

${Q}_{T}$ | $20.5$ | ${R}_{T}$ | $50.5$ | ${P}_{T}$ | $138.8$ | ${N}_{T}$ | $5$ |

$T$ | $0.05$ | ${P}_{\mathit{v}}$ | $25.5$ | $\text{}\lambda $ | $2.7$ | ${P}_{R}$ | $2.3$ |

$\mathcal{X}$ | $\left\{{u}_{s}|0\le {u}_{s}\le 1.2\right\}$ | $\mathcal{U}$ | $\left\{\left({F}_{x},\u2206{F}_{x}\right)|\left|{F}_{x}\right|\le 15,\left|\u2206{F}_{x}\right|\le 2\right\}$ | ${c}_{\omega}$ | $0.12$ | $\mu $ | $1.7$ |

$K$ | $-182.38$ | $\overline{w}$ | $5$ | ${c}_{\delta ,u}$ | $0.01$ | $\rho $ | $0.5$ |

$l$ | $3$ | ${a}_{1}$ | $0.2$ | ${a}_{2}$ | $0.1$ | ${a}_{3}$ | $0.05$ |

Parameter | Value | Parameter | Value | Parameter | Value | Parameter | Value |
---|---|---|---|---|---|---|---|

${Q}_{T}$ | $\begin{array}{c}diag\{190.5,\\ 180.5\}\end{array}$ | ${R}_{T}$ | $50$ | ${P}_{T}$ | $[\mathrm{6462.1,215.8};$ $\mathrm{215.8,3688.3}]$ | ${N}_{T}$ | $9$ |

$T$ | $0.05$ | $\overline{s}$ | $5\xb0$ | $\text{}\lambda $ | $0.6$ | ${P}_{R}$ | $\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{g}\{0.91,$ $0.61$ |

$\mathcal{X}$ | $\left\{(\nu ,r)|\left|\nu \right|\le 0.01,\left|r\right|\le 0.05\right\}$ | $\mathcal{U}$ | $\left\{\left({\delta}_{r},\u2206{\delta}_{r}\right)|\left|{\delta}_{r}\right|\le 20\xb0,\left|\u2206{\delta}_{r}\right|\le 5\xb0\right\}$ | ${c}_{\omega}$ | $0.07$ | $\mu $ | $2.6$ |

$K$ | $[-28.29;11.54]$ | $\overline{w}$ | $10\xb0$ | ${c}_{\delta ,u}$ | $0.01$ | $\rho $ | $0.5$ |

$l$ | $3$ | ${a}_{1}$ | $0.2$ | ${a}_{2}$ | $0.1$ | ${a}_{3}$ | $0.05$ |

Parameter | Value | Parameter | Value | Parameter | Value | Parameter | Value |
---|---|---|---|---|---|---|---|

${Q}_{T}$ | $\begin{array}{c}diag\{2.5,\\ 5.5\}\end{array}$ | ${R}_{T}$ | $5$ | ${P}_{T}$ | $[\mathrm{199.5,25.6};$ $\mathrm{25.6,103.9}]$ | ${N}_{T}$ | $9$ |

$T$ | $0.05$ | $\overline{s}$ | $5\xb0$ | $\text{}\lambda $ | $1.8$ | ${P}_{R}$ | $\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{g}\{0.29,$ $0.59$ |

$\mathcal{X}$ | $\left\{(w,q)|\left|w\right|\le 0.02,\left|q\right|\le 0.07\right\}$ | $\mathcal{U}$ | $\left\{\left({\delta}_{s},\u2206{\delta}_{s}\right)|\left|{\delta}_{s}\right|\le 14\xb0,\left|\u2206{\delta}_{s}\right|\le 5\xb0\right\}$ | ${c}_{\omega}$ | $0.05$ | $\mu $ | $2.9$ |

$K$ | $[-28.29;11.54]$ | $\overline{w}$ | $10\xb0$ | ${c}_{\delta ,u}$ | $0.01$ | $\rho $ | $0.5$ |

$l$ | $3$ | ${a}_{1}$ | $0.2$ | ${a}_{2}$ | $0.1$ | ${a}_{3}$ | $0.05$ |

Method | Max Position Deviation (m) | Max Pitch Angle Deviation (°) | Max Yaw Angle Deviation (°) |
---|---|---|---|

MPC | 0.38 | 3.45 | 3.45 |

RTMPC | 0.12 | 0.58 | 1.07 |

ATMPC | 0.04 | 0.57 | 0.45 |

Method | Max Time Consumption (ms) | Average Time Consumption (ms) |
---|---|---|

MPC | 7.25 | 7.27 |

RTMPC | 7.88 | 7.34 |

ATMPC | 11.35 | 10.25 |

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## Share and Cite

**MDPI and ACS Style**

Chen, Y.; Bian, Y.
Tube-Based Event-Triggered Path Tracking for AUV against Disturbances and Parametric Uncertainties. *Electronics* **2023**, *12*, 4248.
https://doi.org/10.3390/electronics12204248

**AMA Style**

Chen Y, Bian Y.
Tube-Based Event-Triggered Path Tracking for AUV against Disturbances and Parametric Uncertainties. *Electronics*. 2023; 12(20):4248.
https://doi.org/10.3390/electronics12204248

**Chicago/Turabian Style**

Chen, Yuheng, and Yougang Bian.
2023. "Tube-Based Event-Triggered Path Tracking for AUV against Disturbances and Parametric Uncertainties" *Electronics* 12, no. 20: 4248.
https://doi.org/10.3390/electronics12204248