Mathematical Problems in Engineering

Volume 2019, Article ID 1940784, 16 pages

https://doi.org/10.1155/2019/1940784

## Barrier Lyapunov Function-Based Adaptive Control of an Uncertain Hovercraft with Position and Velocity Constraints

College of Automation, Harbin Engineering University, Harbin 150001, China

Correspondence should be addressed to Taiqi Wang; nc.ude.uebrh@iqiatgnaw

Received 20 November 2018; Accepted 27 January 2019; Published 12 February 2019

Academic Editor: Sergey Dashkovskiy

Copyright © 2019 Mingyu Fu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

This paper considers the problem of constrained path following control for an underactuated hovercraft subject to parametric uncertainties and external disturbances. A four-degree-of-freedom hovercraft model with unknown curve-fitted coefficients is first rewritten into a parameterized form. By introducing a barrier Lyapunov function into the line-of-sight guidance, the specific transient tracking performance in terms of position error is guaranteed. A novel constrained yaw rate controller is proposed to ensure time-varying yaw rate constraint satisfaction, in which the yaw rate barrier is required to vary with the speed of the hovercraft. Moreover, a command filter is incorporated into the control design to generate the desired virtual controls and its time derivatives. Theoretical analyses show that, under the proposed controller, the position tracking error constraints and the yaw rate constraint can be strictly guaranteed. Finally, numerical simulations illustrate the effectiveness and advantages of the proposed control scheme.

#### 1. Introduction

As a high-performance amphibious marine craft, a hovercraft utilizes a flexible skirt system around its periphery such that the hull is totally supported by a pressurized air cushion. The hovercraft has attracted increasing attention in both military and civil fields because of its superior high speed and amphibious characteristics [1]. Due to its complex wave-making resistance and skirt drag caused by the cushion system, the dynamics of a hovercraft are very uncertain, nonlinear, and coupled [2]. Moreover, a hovercraft is underactuated because the actuators are equipped for surge and yaw motion only [3]. In addition, a hovercraft is essentially hovering over the water surface, which causes less water friction drag than conventional displacement ships. Therefore, a hovercraft can slip considerably and undergo great heeling during fast turning, requiring the yaw rate of a hovercraft to be regulated within specific safe ranges during maneuvering. These requirements make controller design of a hovercraft a challenging task.

Path following, which requires a hovercraft to follow a geometric path that is time independent, is one of the typical control scenarios for a marine surface vessel (MSV). In [4], a global path following controller was designed for MSVs based on a cascaded approach, and the stability was proved by using the linear time-varying theory. Reference [5] utilized a backstepping technique to develop a nonlinear path following controller for MSVs, in which the control design was based on feedback dominance instead of feedback linearization. Reference [6, 7] presented a model predictive control scheme with line-of-sight (LOS) guidance to improve the path following performance. Reference [8] proposed an adaptive path following controller to estimate the ocean currents, while a new integral LOS guidance law was obtained based on adaptive control. Neural network (NN) control is usually used to deal with the uncertain nonlinear system [9], which has also been widely introduced into path following control to cope with the uncertain dynamics of surface vessels. In [10, 11], the NN control approach together with the integral LOS guidance was proposed for underactuated surface vessels with parameter uncertainties. Reference [12] developed a saturated path following controller for surface vessels, and the uncertainties and disturbances were approximated by using NN.

One of the greatest challenges for hovercraft control is the inexact dynamics, which are due to the intricate interactions between the cushion system and the water surface; these interactions lead to an unclear hydrodynamic structure and parametric uncertainties for control design. An exceedingly simplified hovercraft model was derived in [13–15], in which the hydrodynamic damping coefficients were assumed to be zero. The same hovercraft model was adopted in [16, 17]. However, because the hydrodynamic force and moment are not included in this model, the proposed model-based controllers might not achieve the desired control objective in practical applications. Furthermore, a curve-fitted hovercraft model was obtained in [1] by replacing the complex hydrodynamic and aerodynamic force and moment functions with curve-fitted approximations; by neglecting the heave and pitch motion, a four-degree-of-freedom (DOF) control-oriented hovercraft model was obtained for control design. However, the effects of parametric uncertainties and external disturbances were not considered in [1]. The uncertainties and disturbances always exist in a practical control system, and some works have investigated the feedback control schemes for different form uncertain nonlinear systems, such as triangular form systems [18, 19] and strict-feedback systems [20]. In addition, the adaptive control method is regarded as a powerful method to cope with such system uncertainties [21]. For example, [22] presented a backstepping-based adaptive control scheme to estimate the unknown parameters, and the tuning-function based approach was proposed to avoid the overparametrization in backstepping design. The adaptive backstepping technique has been widely applied in path following control of vessels [23–25]. To improve the tracking performance of a hovercraft under these system uncertainties, the above 4-DOF hovercraft dynamics [1] are rewritten into a parameterized form in this paper, and the adaptive control method is adopted to estimate the uncertain parameters and external disturbances.

LOS guidance, which has been widely explored for surface vessels due to its simplicity and small computational burden, is an effective guidance algorithm for path following control. In [26, 27], it was shown that the equilibrium point of the proportional LOS guidance law is uniformly and exponentially semi-globally stable. To compensate for the vehicle’s constant sideslip due to environmental disturbances, an integral LOS (ILOS) approach was proposed in [28, 29] by adding an additive integral action into the conventional LOS guidance. In [10, 30], an adaptive LOS (ALOS)-based controller was developed with a parameter adaption law to eliminate the effects of system uncertainties. In [31], the authors discussed the drawbacks of the ILOS and ALOS. A modified extended state observer- (ESO-) based LOS guidance approach was proposed to identify the time-varying sideslip angle. Reference [32] presented a compound LOS guidance by combining the time delay control and ESO technique to estimate the time-varying ocean currents. Note that all the aforementioned LOS guidance algorithms only regulate the steady state of position tracking errors to be zero, while the transient tracking performance cannot be guaranteed. However, it is important to ensure the prescribed transient tracking performance of a hovercraft in practice. Moreover, the yaw rate of a hovercraft should be limited to below the stability boundary for safe navigation [33]. If the maximum yaw rate of a hovercraft exceeds the corresponding stability boundary for a period of time, the hovercraft will become unstable, especially at high speeds, which can even lead to a capsizing hazard. As shown in [33], the stability boundary of the yaw rate varies with the surge speed of a hovercraft. Thus, the yaw rate constraint should also be time varying because the desired speed command for a hovercraft can change in practice. To the author’s knowledge, the problem of tracking error-constrained path following control for an uncertain hovercraft with a time-varying yaw rate constraint has rarely been considered.

To guarantee the state or output constraints of the system, some control methods have been included in model predictive control [34], nonovershooting control [35], and prescribed performance control [36]. However, in these methods, the state constraints remain difficult to guarantee in practical applications. More recently, a barrier Lyapunov function (BLF) has been proposed for nonlinear systems to ensure the state and output constraints [37, 38], which has been applied to strict-feedback systems with output constraints [39], pure-feedback systems [40], switched systems [41], and practical applications for hypersonic flight vehicles [42] or missile guidance [43]. In addition to the conventional logarithmic function form, a modified tan-type BLF was proposed in [44], which is a general method for systems with state constraints because it works even if the state constraints are removed. In [45], the authors utilized the tan-type BLF to design an error-constrained LOS path following controller for a 3-DOF conventional surface vessel. To break the static constraint limitations in the abovementioned works, a time-varying output constraint was handled in [46] by using a time-varying BLF, which allowed the output constraints to be both time varying and asymmetric. Moreover, to facilitate the backstepping design for the attitude subsystem of a hovercraft, a command filter was introduced to avoid the analytical computation of the virtual control laws [47].

Motivated by the above considerations, a BLF-based adaptive path following control scheme is developed for a 4-DOF underactuated hovercraft subject to parametric uncertainties and external disturbances. The main contributions of this paper can be summarized as follows:

A novel position-constrained LOS guidance algorithm is proposed by introducing BLF into the classical LOS guidance design procedure. The proposed LOS guidance can guarantee the prescribed tracking performance in terms of position tracking errors.

By virtue of the time-varying BLF, an attitude controller is proposed to ensure the yaw rate time-varying constraints of a hovercraft. The yaw rate of a hovercraft will not exceed the stability boundary, which has practical significance for the safe navigation of a hovercraft.

A command filter is integrated into the control design to generate the amplitude-constrained virtual control laws, which avoid the analytic computation of the virtual control law derivative. In addition, a novel auxiliary system is developed to compensate for the filtering error.

The remainder of this paper is organized as follows. The problem formulation is introduced in Section 2. Section 3 is devoted to the LOS guidance, altitude subsystem, and velocity subsystem control design. Numerical simulation examples are presented in Section 4. Section 5 concludes the paper.

#### 2. Problem Formulation and Preliminaries

##### 2.1. Preliminaries

*Notation*. Throughout this paper, denotes the transpose of a matrix ; represents the absolute value of a scalar; represents the Euclidean norm of a vector; denotes the estimate of ; the estimation error is defined as ; and denotes the set of nonnegative real numbers.

Lemma 1 (see [44]). *For any and , the following inequality always holds: *

Lemma 2 (see [37]). *For any positive constant , positive integer , and satisfying , there exists *

*Definition 3 (see [39]). *A barrier Lyapunov function is a scalar function defined with respect to the system on an open region containing the origin. It is continuous, is positive definite, has continuous first-order partial derivatives at every point of , has the property as approaches the boundary of , and satisfies along the solution of for and some positive constant .

##### 2.2. Dynamic Model of a Hovercraft

The typical configuration of the hovercraft is shown in Figure 1, in which two sets of ducted air propellers are mounted at the stern and a pair of rudders is mounted behind the duct flaps to create turning moments.