**1. Introduction**

Autonomous underwater vehicles (AUVs) have been widely used in marine scientific research, underwater resource exploration, underwater oil and gas pipeline and structure overhaul, seabed hydrothermal research, and military fields [1,2]. When AUVs perform underwater tasks, they usually need to complete path-tracking tasks [3].

The 6-DOF motion of AUV in three-dimensional underwater space is coupled and nonlinear, and the parameters of the model are often difficult to obtain precisely. In modelbased control methods, the control performance will suffer from parametric uncertainties [4]. Moreover, external disturbances caused by ocean currents will also degrade the control performance [5,6]. Therefore, it is a challenge to enhance the robustness against external disturbances and parametric uncertainties in model-based control methods [7]. Until now, researchers have applied strategies for improving the robustness of model-based control methods such as the model predictive control (MPC) technique [8,9] and sliding mode control (SMC) technology [10] in the path-tracking control of AUVs. Note that MPC can easily handle the physical constraints of the AUV when formulating the optimal control problem. It is also well-known that MPC technology can provide some assistance for the disturbance rejection [11]. In other words, the MPC technology itself is robust against disturbance. Therefore, MPC is widely used in the path-tracking control of AUVs [12,13].

Zhang proposed a 3D path-tracking control method for AUVs using a linear model predictive control (LMPC) [13]. The LMPC controller is used to calculate the speed control law. Then, the control inputs of the AUV were directly calculated based on the dynamics model, where the physical constraints on the control input failed to be considered. In [14,15], the speed control law was generated by the kinematics LMPC, and the control inputs were

**Citation:** 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

Academic Editor: Giuseppe Prencipe

Received: 24 August 2023 Revised: 19 September 2023 Accepted: 3 October 2023 Published: 13 October 2023

**Copyright:** © 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

generated by the dynamic LMPC. These physical constraints on the control input can be considered when formulating the optimal control problem. Compared with [13], the method in [14,15] can also enhance the robust performance against disturbances, by the robustness of the nominal MPC technology itself. However, there is no direct disturbance rejection strategy, such as disturbance estimation [16,17] or robust MPC technology [18]. The robustness of the nominal MPC technology itself is limited. These disturbance rejection strategies can significantly improve the robustness performance, compared with the nominal MPC technology. Therefore, a direct disturbance rejection strategy can be introduced in the nominal MPC technology to improve the tracking control performance.

The extended Kalman filter technology is used to estimate external current disturbances [17]. Based on the 12-dimensional kinematic model and kinetic model, a NMPC controller is proposed to calculate the optimal control law using these results of disturbance estimation. However, the disturbance estimation will bring extra dimensions, which may lead to poor real-time performance. To overcome the challenge, disturbance estimation is only based on the 5-dimensional kinematic model using MPC, which can save online optimization computing time [10]. Note that control inputs are calculated using adaptive sliding mode control technology, which is sensitive to the noise in the actual control system. The control performance may suffer from the chattering problem in the practical application [19].

Tube MPC, as a disturbance rejection strategy, was first proposed by Blanchini [20]. Compared with the disturbance estimation, the robustness improvement is achieved by its own relatively stable mechanism. Suffering from external disturbance and parametric uncertainties, there is a model mismatch between the nominal model and the actual model. A robust positively invariant (RPI) set is proposed to measure the boundedness of the mismatch [21]. In the tube MPC scheme, the tight constraint is calculated by tightening the constraints of the actual system by an RPI set. The control law of the tube MPC scheme consists of a nominal optimal control law and a feedback control law. The nominal control law is obtained by solving a receding horizon optimal control problem with a tight constraint. The feedback control law is used to address the deviation of the nominal and actual states due to the model mismatch. The traditional tube MPC scheme [21,22] is proposed for AUV's path tracking [18]. Note that the RPI set is obtained based on the assumed disturbance upper bound. Hence, the corresponding tight constraints may become too conservative to degrade the path-tracking performance. Based on the coupled 6-dimensional AUV model, both the RPI set and the terminal feasible set are easy to have no solution. Moreover, online calculating tight constraints of the nominal model brings too much computing time, which will also lead to poor real-time performance.

Since the inherent robustness of the nominal MPC to address the model mismatch is limited, the tube MPC has the potential to improve robustness against model mismatches. However, the control performance suffers from poor real-time performance and no solution for the RPI set. Our motivation is to apply the tube MPC to enhance the robustness of AUV's path tracking, with these issues addressed. This study proposes a tube-based eventtriggered path-tracking strategy, which consists of a kinematics LMPC controller and a tube MPC controller. To converge the nominal path-tracking deviation, the kinematics LMPC controller is used to calculate the optimal speed control law. The tube MPC controller is used to compute the control input of the AUV to track the speed control law. Compared with the tube MPC technology used in [18], to avoid no solution to the RPI, the coupled kinetic model is decoupled into three Lipschitz nonlinear models [23]: a surge speed control model, a heading control model, and a depth control model. With the corresponding Lipschitz constant obtained, nonlinear properties of these models when formulating a linear matrix inequality (LMI) are used to calculate the RPI set and the feedback matrix. The terminal feasible set is obtained based on linear differential inclusion (LDI) technology. In order to achieve good real-time performance, constraints on the nominal model and the feedback matrix are all calculated offline. Note that the hydrodynamic force of the AUV is related to the surge speed. The mismatch may depend on the surge speed change command. These offline calculated invariant constraints are too conservative to achieve better control performance. Then, an event-triggering mechanism is used. When the surge speed change command does not exceed the upper bound, two decision variables are introduced to formulate a flexible tube. Then, adaptive constraints on the nominal model are obtained to address the mismatch. When the surge speed change command exceeds the upper bound, the offline tight constraints will be used. The main contributions of this work are as follows:


The remainder of this paper is organized as follows. In Section 2, preliminaries are given. In Section 3, the AUV's motion model and the path-tracking problem are given. In Section 4, the detail design of the tube-based event-triggered path-tracking strategy is given. In Section 5, the numerical simulation analysis is shown.

## **2. Preliminaries**

The actual nonlinear continuous-time dynamics is described as a Lipschitz nonlinear system [23]: .

$$
\dot{\mathbf{x}} = f(\mathbf{x}, \boldsymbol{\mu}, \boldsymbol{\omega}) = A\mathbf{x} + B\boldsymbol{\mu} + \mathbf{g}(\mathbf{x}) + B\_{\boldsymbol{\omega}}\boldsymbol{\omega} \tag{1}
$$

with *<sup>x</sup>* <sup>∈</sup> <sup>R</sup>*n*×<sup>1</sup> and *<sup>u</sup>* <sup>∈</sup> <sup>R</sup>*m*×1. *<sup>ω</sup>* <sup>∈</sup> <sup>=</sup> - *<sup>ω</sup>* <sup>∈</sup> <sup>R</sup>*n*×<sup>1</sup> : *ω*<sup>∞</sup> <sup>&</sup>lt; *<sup>c</sup><sup>ω</sup>* denotes the bounded external disturbance. Positive constant *c<sup>ω</sup>* is the disturbance upper bound. System (1) is also subject to state and control input constraints:

$$
\alpha \in \mathcal{X} \subset \mathbb{R}^{n \times 1}, \mu \in \mathcal{U} \subset \mathbb{R}^{m \times 1} \tag{2}
$$

where U is a compact set and X is bounded. Here *g*(*x*) is a Lipschitz nonlinear function with a Lipschitz constant *L* > 0 such that:

$$\|\|\mathbf{g}(\mathbf{x}\_1) - \mathbf{g}(\mathbf{x}\_2)\|\| \le L \|\|\mathbf{x}\_1 - \mathbf{x}\_2\|\| \, \forall \mathbf{x}\_1, \mathbf{x}\_2 \in \mathcal{X} \tag{3}$$

The overline format of a variable denotes its nominal value, e.g., *x* denotes the nominal value of *x*. The continuous-time nominal model is given by:

$$
\dot{\overline{\pi}} = f(\overline{\pi}, \overline{\pi}, 0) \tag{4}
$$

and the corresponding discrete-time system models are given by:

$$\mathbf{x}\_{t+1} = f\_d(\mathbf{x}\_t, \mathbf{u}\_t, \omega\_t) \tag{5}$$

$$
\overline{\mathfrak{X}}\_{t+1} = f\_d(\overline{\mathfrak{X}}\_t \, \overline{\mathfrak{u}}\_t, 0) \tag{6}
$$

Define <sup>K</sup>*N*1:*N*<sup>2</sup> :<sup>=</sup> {*N*1, *<sup>N</sup>*<sup>1</sup> <sup>+</sup> 1, ··· , *<sup>N</sup>*<sup>2</sup> <sup>−</sup> 1, *<sup>N</sup>*2}. The nominal cost function of predicted state sequence, *xk*|*t*, *<sup>k</sup>* <sup>∈</sup> <sup>K</sup>0:*NT* , and control input sequence, *uk*|*t*, *<sup>k</sup>* <sup>∈</sup> <sup>K</sup>0:*NT*−1, is given as:

$$J = \sum\_{k=0}^{N\_T - 1} l(\overline{\mathfrak{x}}\_{k|t'} \overline{\mathfrak{u}}\_{k|t}) + V\_f(\overline{\mathfrak{x}}\_{N\_T|t}) \tag{7}$$

where *NT* is the predictive horizon. *l* is the positive definite stage cost and *Vf* is the terminal cost:

$$I(\overline{\boldsymbol{x}}\_{k|t'}\overline{\boldsymbol{u}}\_{k|t}) = \left\|\overline{\boldsymbol{x}}\_{k|t}\right\|\_{Q\boldsymbol{r}}^2 + \left\|\overline{\boldsymbol{u}}\_{k|t}\right\|\_{R\_T}^2, \\ V\_f(\overline{\boldsymbol{x}}\_{Nt}|t) = \left\|\overline{\boldsymbol{x}}\_{Nt}\right\|\_{P\_T}^2 \tag{8}$$

The state deviation between the actual system and the nominal actual is denoted by *z* = *x* − *x*. The deviation system is given as:

$$
\dot{z} = \dot{\mathbf{x}} - \dot{\overline{\mathbf{x}}} = f(\mathbf{x}, \boldsymbol{\mu}, \omega) - f(\overline{\mathbf{x}}, \overline{\mathbf{u}}, 0) \tag{9}
$$

In a tube MPC controller, the control law consists of a nominal MPC control law *u* and a state feedback control law *κ*(*x*, *x*):

$$
\mu := \overline{\mathfrak{u}} + \kappa(\overline{\mathfrak{x}}, \mathfrak{x}) \tag{10}
$$

where *u* is obtained by solving an optimal control problem, and *κ*(*x*, *x*) is used to converge the state deviation *z*.

**Definition 1.** *(Robust positively invariant (RPI) set): A set* Ω ⊂ X *is the RPI set of deviation system (9), if there exists a feedback control law κ*(*x*, *x*) ∈ U, *such that for all zt*<sup>0</sup> ∈ Ω *and ω* ∈ *, it holds that zt* ∈ Ω *for all t* ≥ *t*0.

Then the constraints of nominal system (6) are given with an RPI set Ω as:

$$\mathfrak{X} \in \mathcal{X} := \mathcal{X} \odot \Omega, \mathfrak{X} \in \mathcal{U} := \{ \mathfrak{T} | \mathfrak{T} + \mathfrak{x}(\mathfrak{T}, \mathfrak{x}) \in \mathcal{U} \}\tag{11}$$

where X and U are tight constraint sets, which can be expressed as:

$$\mathcal{R}(\overline{\mathfrak{x}}, \overline{\mathfrak{u}}) \in M := \left\{ (\overline{\mathfrak{x}}, \overline{\mathfrak{u}}) \in \mathbb{R}^{(n+m)\times 1} \Big| h\_j(\overline{\mathfrak{x}}, \overline{\mathfrak{u}}) \le 0, j = 1, 2, \cdots, \,\,\mu \right\} \tag{12}$$

Considering linear constraints, these constraints can also be expressed as a polytope:

$$M = \left\{ \begin{bmatrix} \overline{\mathfrak{X}} \\ \overline{\mathfrak{u}} \end{bmatrix} \in R^{(\mathfrak{u}+m)\times 1} : c\_j \mathfrak{x} + d\_j \mathfrak{u} \le 1, j = 1, 2, \dots, p \right\} \tag{13}$$

Considering tight constraints (11) and nominal system dynamics (6), the following optimal control problem is formulated to calculate the nominal MPC control law:

$$\min\_{\overline{\omega}\_{k\mid t}, k \in \mathbb{N}\_{0:N\_T-1}} J \tag{14}$$

$$\begin{aligned} \overline{\mathfrak{X}}\_{0|t} &= \overline{\mathfrak{X}}\_{0}, \overline{\mathfrak{u}}\_{0|t} = \overline{\mathfrak{u}}\_{0} \\ \overline{\mathfrak{X}}\_{k+1|t} &= f\_{d} \left( \overline{\mathfrak{X}}\_{k|t}, \overline{\mathfrak{u}}\_{k|t}, 0 \right) \\ h\_{j} \left( \overline{\mathfrak{X}}\_{k|t}, \overline{\mathfrak{u}}\_{k|t} \right) &\leq 0, j = 1, 2, \cdots, p, \\ \overline{\mathfrak{X}}\_{N\_{\Upsilon}|t} &\in X\_{f} \end{aligned}$$

where *Xf* is the terminal feasible set. The optimal control input sequence *u*<sup>∗</sup> *k*|*t* , *<sup>k</sup>* <sup>∈</sup> <sup>K</sup>0:*NT*−1, is the solution to optimal control problem (14), and the nominal optimal MPC control law *u* is obtained by:

$$
\overline{u} = \overline{u}\_{0|t}^\* \tag{15}
$$

#### **3. AUV Motion Model and Problem Formulation**

In this section, the kinematics model and kinetic model of the AUV are given, where both external disturbances and parametric uncertainty are considered in the kinetic model. In the proposed tube-based event-triggered path-tracking strategy, described in Section 4, based on the kinematics model, a speed control law is designed to converge the nominal path-tracking deviation. Then, based on the kinetic model, the control input of the AUV is calculated to track the speed control law. Correspondingly, two problems treated in this study are formalized.
