# Adaptive Fixed-Time Trajectory Tracking Control for Underactuated Hovercraft with Prescribed Performance in the Presence of Model Uncertainties

Authors: Mingyu Fu; Tan Zhang; Fuguang Ding; Duansong Wang
Journal: Complexity
 (2021)

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## Abstract

This paper develops an adaptive fixed-time trajectory tracking controller of an underactuated hovercraft with a prescribed performance in the presence of model uncertainties and unknown time-varying environment disturbances. It is the first time that the proposed method is applied to the motion control of the hovercraft. To begin with, based on the hovercraft's four degrees of freedom (DOF) model, the virtual control laws are designed using an error transforming function and the fixed-time stability theory to guarantee that the position tracking errors are constrained within the prescribed convergence rates and minimum overshoot. In addition, by combining the Lyapunov direct method and the adaptive radial basis function neural network (ARBFNN), the actual control laws are designed to ensure that the velocity tracking errors converge to a small region containing zero while handling model uncertainties and external disturbances effectively. Finally, all tracking errors of the closed-loop system are uniformly ultimately bounded and fixed-time convergent. Results from a comparative simulation study verify the effectiveness and advantage of the proposed method.


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## Body

## 1. Introduction

The underactuated hovercraft as shown in Figure1 possesses a flexible skirt around the bottom of it to seal the cushion air [1]. It can be lifted to the sail by a large enough air cushion force to obtain a unique amphibious performance. Therefore, the hovercraft has a higher speed than normal surface vessels. In recent years, because of its special amphibious performance, the hovercraft has been getting increasing attention in oceanic research work, for instance, oceanic resources exploration, scientific investigation, ocean expeditions, military missions, transportation, rescue, and other fields [2].Figure 1
Reference frame of the hovercraft.What is particularly noteworthy is that the hovercraft in navigation possesses very little contact with the sailing surface; therefore, it has smaller righting moments than normal surface vessels. It is easy to generate a large roll angle and large drift angle when turning the rudder so that the hovercraft runs in a dangerous situation. The air cushion force's lateral component under a large roll angle results in the hovercraft drifting sideways and losing course stability [3–5]. The roll motion occurs always when the hovercraft is turning and is more strenuous with the increase in the turn rate and drift angle. Regarding that the above dangerous sailing mechanism is closely relevant to the roll motion of the hovercraft; a four-DOF motion mathematical model of a hovercraft by taking the roll degree of freedom into account is established in this paper, which is closer to the sailing characteristics of the actual hovercraft than the three-DOF model.Usually, the hovercraft has the characteristics of the typical underactuated vessel. The major difficulty for motion control of the underactuated surface vessels is that the lateral motion is not controlled directly, namely, the number of DOF is more than the number of independent actuators in the underactuated system. The challenge to control underactuated vessels is how to apply two independent actuators to regulate three or more DOFs’ motions under unknown dynamic environments. Over the past two decades, for overcoming the above control problems, various control strategies have been presented by the endeavours of many research workers, and significant achievements have been achieved. In [6], an adaptive fuzzy stabilization control method was proposed to carry out trajectory tracking control of the underactuated surface vessel in the presence of unknown time-varying environment disturbances. Considering the robustness of the underactuated vessel's control, the literature [7] adopted the biologically inspired method to perform motion control of an underactuated surface vessel and used the single-layer neural network to handle the unknown dynamics. An adaptive dynamic sliding mode control method for trajectory tracking control of an underactuated underwater unmanned vehicle was proposed in [8], but the proposed adaptive sliding mode control method requires some assumptions, such as the existence of the first derivative of external disturbance and control input. In [9], the combination of neural network and the terminal sliding mode was used to design a finite-time trajectory tracking controller for an underactuated hovercraft. A H∞ observer-based fuzzy controller design for a vessel dynamic positioning system was investigated in [10], and the simulation results show that this method is effective for both known/unknown premise variable cases.Unfortunately, one important problem, the transient and steady error performance constraint playing a significant role in practice, is not addressed in the above papers. The transient performance constraints can guarantee that high-speed hovercraft avoids collision hazards and provides stability simultaneously. Recently, a performance constraint technique was proposed in [11–13]. This technique can guarantee that the transient errors converge at a designed exponential rate with a predesigned maximum overshoot. Now, the prescribed performance control strategy has been applied to many practical system controls, such as servo system [14], air vehicles [15, 16], and robots [17, 18]. With the further research of scientific researchers, a series of improved prescribed performance control technology has been proposed in [19–23] to constrain the transient and steady-state performance of the control system. An envelope-constraint-based tracking control method for air-breathing hypersonic vehicles with unknown nonaffine formulations is proposed in [19], and the desired transient performance is guaranteed for velocity and attitude tracking errors using envelope constraint. In [21], a novel performance function without the information of the initial error is proposed for limiting the tracking error of hypersonic flight vehicles into a prescribed range. A new prescribed performance control approach utilizing transformed errors instead of initial tracking errors is proposed for uncertain nonlinear dynamic systems in [22]. For the velocity dynamics of air-breathing hypersonic vehicles, an adaptive neural controller containing only one neural network is addressed using the prescribed performance control method in [23]. The highlights of the method in [23] are that the presented control method has a concise control architecture and low computational cost. Subsequently, the prescribed performance technique has been applied practically to the motion control of the underactuated vessels. For example, a robust adaptive controller was presented to perform the motion control of the underactuated ship with prescribed performance in the presence of model uncertainties and external disturbances [24]. In [25], the transient and steady-state performance was also featured for trajectory tracking control of underactuated underwater vessels.On the other hand, the controller designed in the practical engineering applications of the ship only guarantees that the stability of the system is not to make much sense. The system convergence speed is always an important index to evaluate the designed controller's performance. In the last few years, the control technique for specifying the system convergence time was proposed; for example, one type of delayed memristor-based fractional-order neural networks based on the finite-time stability problem was studied in [26]. A novel adaptive fixed-time controller was designed to perform output tracking problems of a class of multiinput multioutput nonlinear systems in the presence of system uncertainties [27]. A fixed-time sliding mode manifold was proposed [28], and the convergence time of the sliding mode manifold is shown to be independent of the initial conditions of the control system. In [29], a nonsingular fixed-time fast terminal sliding mode manifold with fixed-time convergence was presented, and the preestablished convergence time is also developed; unfortunately, the control law designed is discontinuous.Inspired by the abovementioned observations, a nonlinear trajectory tracking control method for an underactuated hovercraft is presented that ensures a prescribed performance, specifies the system convergence time, and handles the system uncertainties. Incorporating the prescribed performance and specified system convergence time simultaneously in the trajectory tracking control design of the underactuated hovercraft has been unprecedented. Accordingly, the main contributions of this paper are summarized as follows:(1)
Unlike the existing trajectory tracking control methods of the underactuated hovercraft in [9], the presented control strategy avoids the dangerous navigation situation when the hovercraft sails in a narrow channel. Compared with the barrier Lyapunov function method [30], one of the advantages of the prescribed performance method is that it can increase the space of the initial feasible solution of the system and ensure that the tracking error of the system can converge to any small region set in advance.(2)
To solve the unknown dynamic model uncertainties of the controlled system and external disturbances, an ARBFNN is applied to the trajectory tracking control scheme that is simple and easy to implement in practice.(3)
By selecting the control parameters appropriately, the convergence time that is independent of the initial values of the controlled system can be prespecified to guarantee a fast convergence performance of the nonlinear trajectory tracking system.(4)
It is proven that, by applying the presented controller, all tracking errors of the closed-loop system are uniformly ultimately bounded and uniformly fixed-time convergent despite the presence of system uncertainties.This paper is arranged as follows. The preliminaries and problem formulation are given in Section2. Section 3 is devoted to the design of the adaptive neural-based fixed-time trajectory tracking controller for the hovercraft with prescribed performance. Numerical simulation results are shown in Section 4. Section 5 concludes the work of this paper.


## 2. Preliminaries and Problem Formulation

### 2.1. Preliminaries

Notation. Throughout this paper, ⋅T denotes the transpose of a matrix ⋅, ⋅ represents the Euclidean norm of a vector, ⋅ signifies the absolute value of a scalar, λmin⋅ and λmax⋅ mean the minimum and maximum eigenvalues of a given matrix B∈Rn×n, respectively, U1∪U2 denotes the union of interval U1 and interval U2, and sign⋅ denotes the signum function.Lemma 1 (see [31]).
For any real numbersxi∈R, i=1,2,...,n, we can obtain the following inequalities:(1)∑i=1nxip≥∑i=1nxip,0<p≤1,n1−p∑i=1nxip,1<p<+∞.Lemma 2 (see [32]).
For any continuous unknown real functionfx:Rm⟶R, the radius basis function neural network (NN) can be employed to approximate it over the compact Ω⊆Rm as follows:(2)fx=w∗Thx+δx,where x∈Rm is the input vector of NN, w∗=w1∗,…,wn∗T∈Rn denotes the NN ideal bounded weight vector, n denotes the hidden note number of NN, and δx represents the ideal approximation error with satisfying δx≤δ¯, where δ¯ is a constant. The ideal weight value can be determined by the following equation:(3)w∗=argminw^supx∈Ωfx−w^Thx,where w^ is the estimation of the ideal weight value w∗, which is usually calculated by the adaptive updating law based on the Lyapunov stability theorem. The radius basis function vector is hx=h1x,…,hnxT:Ω⟶Rn, and hix are selected as the Gaussian function in the following form:(4)hix=exp−x−ϕi2εi2,i=1,2,…,n,where ϕi∈Rm and εi∈R denote the centre and width of the radius basis function, respectively.Definition 1.
Considering the following first-order system (5), if the system is finite-time stable and the upper bound of convergence time is independent of the initial values of the control system, the system is fixed-time stable:(5)x˙=fx+τ+d,where x∈R and τ∈R are the state and input of system (5), respectively, fx is a known continuous function, and d is an unknown continuous bounded function which signifies model uncertainties or/and external disturbances.Lemma 3 (see [33]).
Considering system (5), if there exists a Lyapunov function Vx which satisfies(6)V˙x≤−αVpx−βVqx,where α, p, β, and q are positive constants and satisfy 0<p<1<q, system (5) is fixed-time stable and the convergence time can be estimated as follows:(7)T≤1α1−p+1βq−1.Lemma 4 (see [34]).
In the light of Lemma3, if there is a positive bounded function Δ and a Lyapunov function Vx which satisfy(8)V˙x≤−αVpx−βVqx+Δ,where α, p, β, and q are positive constants and satisfy 0<p<1<q, the system state can converge to D=x:Vx≤minΔ/αθp1/p,Δ/βθq1/q within time T:(9)T≤1α1−θp1−p+1β1−θqq−1,where θp, θq∈0,1 are positive constants.


### 2.2. Hovercraft Model

According to [2], the following kinematic model and dynamic model are employed to characterize the four-DOF motions of the hovercraft in Figure 1:(10)x˙=ucosψ−vsinψcosϕ,y˙=usinψ+vcosψcosϕ,ϕ˙=p,ψ˙=rcosϕ,u˙=vr+FxD0m0+fu+τum0,v˙=−ur+FyD0m0+fv,p˙=MxD0Jx0+fp,r˙=MzD0Jz0+fr+τrJz0,with(11)fu=1m0−Δmu˙+Δmvr+ΔFxD+dutm0,fv=−1m0Δmv˙+ur+ΔFyD+dvtm0,fp=1Jx0−ΔJxp˙+ΔMxD+dptJx0,fr=1Jz0−ΔJzr˙+ΔMzD+drtJz0,where u and v denote velocities, p and r denote the angular velocities in the body-fixed frame, x and y represent positions, and ϕ and ψ represent attitudes with respect to the earth-fixed frame. The control inputs are represented by τu and τr. The known hull design mass and moment of inertia are signified by m0, Jx0, and Jz0. Δm, ΔJx, and ΔJz are viewed as the uncertainty of the ship's mass and moment of inertia. The known part of the current resistance models of the hovercraft, for example, air momentum force, skirt resistance, air resistance, and wave-making resistance, is denoted by FxD0, FyD0, MxD0, and MzD0, respectively, that can be determined through wind tunnel test and tank test. ΔFxD, ΔFyD, ΔMxD, and ΔMzD are viewed as the uncertainties of the model in hydrodynamic and aerodynamic drag caused by modelling errors. dut, dvt, dpt, and drt signify external environment resistance. fu, fv, fp, and fr contain the total model uncertainties and external environment resistances. FxD0, FyD0, MxD0, and MzD0 can be calculated by the following equations:(12)FxD0=Fxa+Fmcosβ+Fwmcosβ+Fskcosβ,FyD0=Fya+Fmsinβ+Fwmsinβ+Fsksinβ+Fc,MxD0=Mxa+Fmzmsinβ+Fwmzwmsinβ+Ghmtanϕ+Fskzsksinβ+Fczc,MzD0=Mza+Fmxmsinβ+Fmymcosβ+Fwmxwmsinβ+Fwmywmcosβ+Fskxsksinβ+Fskyskcosβ+Fcxc,where(13)Fxa=−0.5ρaVa2CxaSPP,Fya=−0.5ρaVa2CyaSLP,Mxa=−0.5ρaVa2CmxaSPPlc+Fyaza,Mza=−0.5ρaVa2CmzaSHPHhov+Fyaxa+Fxaya,Fm=ρaVaQ,Fwm=Cwmpc2Bc/ρwg,Fsk=0.5Va2Cskh/lsk−0.34lskSc0.5+2.8167pc/lc−0.259−1Fwm,Fc=2lcφpc0.5Bctanϕ+h0,MG=Ghmtanϕ,where Fxa, Fya, Mxa, and Mza denote air resistance, Fwm, Fm, Fc, and Fsk denote wave-making resistance, air momentum force, cushion resistance, and skirt resistance, respectively. Cxa, Cya, Cmxa, Cmza, Cwm, and Csk represent the relevant resistance parameters. MG means roll restoring moment. The length and width of the cushion are represented by lc and Bc, respectively. SLP, SPP, and SHP denote lateral, positive, and horizontal projection areas, respectively. Sc and pc are the area and pressure of the cushion, the average clearance for air leakage in static hovering mode is signified by h, hm denotes the metacentric height, h0 denotes the initial lifting height, the total length of the skirt is represented by lsk, the flow coefficient is signified by φ, Hhov means the hovercraft's height, the air density and water density are signified by ρa and ρw, za, zm, zwm, zsk, and zc denote heights of each force's acting point with respect to the mass centre of the hovercraft, and xa,ya, xm,ym, xwm,ywm, xsk,ysk, and xc,yc signify the coordinates of the force's acting points. β signifies slip angle, and Va denotes the relative wind speed which can be calculated by the following equations:(14)β=arctanvu,Va=u+Vωcosβω−ψ2+v+Vωsinβω−ψ2,βa=arctanv+Vωsinβω−ψu+Vωcosβω−ψ,where βa signifies the relative wind direction, Vω represents the absolute wind speed, and βω denotes the absolute wind direction.Assumption 1.
In the control process, we ignore the motion of the hovercraft's pitch and heave and set the pressure of each air chamber and the flow of the cushion fan to be constants.Assumption 2.
The same two air propellers and the same two air rudders are symmetrically mounted at the tail of the hovercraft. In addition, the air rudder provides the moment of steering and the propeller only provides the forward thrust.Assumption 3.
The model uncertaintiesfu, fv, fp, and fr are continuously bounded, namely, fi≤Timaxi=u,v,p,r with Timax being positive constants.Assumption 4.
The hovercraft's surge, sway velocities, and yaw angular velocity are bounded.Remark 1.
In practice, the velocity and yaw angular velocity of the hovercraft cannot be infinite because of the restraints of aerodynamic resistance, hydrodynamic damping term, and the ability of the actuator [35].


### 2.3. Problem Formulation

To expediently describe the control problem, we firstly define the reference trajectory in this paper. The hovercraft tracks that the desired path is generated by the virtual surface vessel:(15)x˙dy˙dψ˙d=cosψd−sinψd0sinψdcosψd0001udsetvdsetrdset.Assumption 5.
The virtual vessel statesxd, yd, and ψd and their first derivatives with respect to time x˙d, y˙d, and ψ˙d are bounded.
The position tracking errors can be defined as follows:(16)xe=x−xd,ye=y−yd.
By utilizing (10) and (15), the derivatives of the position error with respect to time can be calculated as follows:(17)x˙ey˙e=cosψ−sinψcosϕsinψcosψcosϕuv−x˙dy˙d.
Subsequently, we define the velocity and yaw angular velocity tracking errors as follows:(18)ue=u−αu,ve=v−αv,re=r−αr,where αu, αv, and αr are the virtual control laws to be designed later on, which are viewed as the desired velocities and the desired yaw angular velocity of the u,v, and r, respectively.
In terms of the equation of the hovercraft dynamics (10), the derivatives of formula (18) with respect to time can be calculated as follows:(19)u˙e=vr+FxD0m0+fu+τum0−α˙u,v˙e=−ur+FyD0m0+fv−α˙v,r˙e=MzD0Jz0+fr+τrJz0−α˙r.
To avoid the dangerous navigation situation when the hovercraft sails in a narrow channel, we consider the transient performance during the position tracking error convergence. The errors can be defined to satisfy the following prescribed performance [36]:(20)−α1ηt<xe<β1ηt,−α2ηt<ye<β2ηt,where ηt=η0−η∞e−at+η∞ is a smooth performance constraint function with parameters satisfying η0>η∞>0, αi>0, βi>0, i=1,2.Assumption 6.
Initial position tracking errors satisfy the strict conditions−α1η0<xe0<β1η0 and −α2η0<ye0<β2η0.
The control objective in this paper can be formulated as follows.
Considering the hovercraft model (10) subject to model uncertainties and external disturbances, an adaptive neural-based fixed-time trajectory tracking controller is designed to generate surge force τu, virtual control law αr, and yaw moment τr to guarantee velocity tracking errors ue and ve, and yaw angular velocity tracking error re converges to the small region containing zero within the fixed time. Then, the virtual control laws αu and αv are reasonably designed to ensure the position tracking errors remain within the given prescribed performance bounds while the desired trajectory is being tracked by the hovercraft.


## 2.1. Preliminaries

Notation. Throughout this paper, ⋅T denotes the transpose of a matrix ⋅, ⋅ represents the Euclidean norm of a vector, ⋅ signifies the absolute value of a scalar, λmin⋅ and λmax⋅ mean the minimum and maximum eigenvalues of a given matrix B∈Rn×n, respectively, U1∪U2 denotes the union of interval U1 and interval U2, and sign⋅ denotes the signum function.Lemma 1 (see [31]).
For any real numbersxi∈R, i=1,2,...,n, we can obtain the following inequalities:(1)∑i=1nxip≥∑i=1nxip,0<p≤1,n1−p∑i=1nxip,1<p<+∞.Lemma 2 (see [32]).
For any continuous unknown real functionfx:Rm⟶R, the radius basis function neural network (NN) can be employed to approximate it over the compact Ω⊆Rm as follows:(2)fx=w∗Thx+δx,where x∈Rm is the input vector of NN, w∗=w1∗,…,wn∗T∈Rn denotes the NN ideal bounded weight vector, n denotes the hidden note number of NN, and δx represents the ideal approximation error with satisfying δx≤δ¯, where δ¯ is a constant. The ideal weight value can be determined by the following equation:(3)w∗=argminw^supx∈Ωfx−w^Thx,where w^ is the estimation of the ideal weight value w∗, which is usually calculated by the adaptive updating law based on the Lyapunov stability theorem. The radius basis function vector is hx=h1x,…,hnxT:Ω⟶Rn, and hix are selected as the Gaussian function in the following form:(4)hix=exp−x−ϕi2εi2,i=1,2,…,n,where ϕi∈Rm and εi∈R denote the centre and width of the radius basis function, respectively.Definition 1.
Considering the following first-order system (5), if the system is finite-time stable and the upper bound of convergence time is independent of the initial values of the control system, the system is fixed-time stable:(5)x˙=fx+τ+d,where x∈R and τ∈R are the state and input of system (5), respectively, fx is a known continuous function, and d is an unknown continuous bounded function which signifies model uncertainties or/and external disturbances.Lemma 3 (see [33]).
Considering system (5), if there exists a Lyapunov function Vx which satisfies(6)V˙x≤−αVpx−βVqx,where α, p, β, and q are positive constants and satisfy 0<p<1<q, system (5) is fixed-time stable and the convergence time can be estimated as follows:(7)T≤1α1−p+1βq−1.Lemma 4 (see [34]).
In the light of Lemma3, if there is a positive bounded function Δ and a Lyapunov function Vx which satisfy(8)V˙x≤−αVpx−βVqx+Δ,where α, p, β, and q are positive constants and satisfy 0<p<1<q, the system state can converge to D=x:Vx≤minΔ/αθp1/p,Δ/βθq1/q within time T:(9)T≤1α1−θp1−p+1β1−θqq−1,where θp, θq∈0,1 are positive constants.


## 2.2. Hovercraft Model

According to [2], the following kinematic model and dynamic model are employed to characterize the four-DOF motions of the hovercraft in Figure 1:(10)x˙=ucosψ−vsinψcosϕ,y˙=usinψ+vcosψcosϕ,ϕ˙=p,ψ˙=rcosϕ,u˙=vr+FxD0m0+fu+τum0,v˙=−ur+FyD0m0+fv,p˙=MxD0Jx0+fp,r˙=MzD0Jz0+fr+τrJz0,with(11)fu=1m0−Δmu˙+Δmvr+ΔFxD+dutm0,fv=−1m0Δmv˙+ur+ΔFyD+dvtm0,fp=1Jx0−ΔJxp˙+ΔMxD+dptJx0,fr=1Jz0−ΔJzr˙+ΔMzD+drtJz0,where u and v denote velocities, p and r denote the angular velocities in the body-fixed frame, x and y represent positions, and ϕ and ψ represent attitudes with respect to the earth-fixed frame. The control inputs are represented by τu and τr. The known hull design mass and moment of inertia are signified by m0, Jx0, and Jz0. Δm, ΔJx, and ΔJz are viewed as the uncertainty of the ship's mass and moment of inertia. The known part of the current resistance models of the hovercraft, for example, air momentum force, skirt resistance, air resistance, and wave-making resistance, is denoted by FxD0, FyD0, MxD0, and MzD0, respectively, that can be determined through wind tunnel test and tank test. ΔFxD, ΔFyD, ΔMxD, and ΔMzD are viewed as the uncertainties of the model in hydrodynamic and aerodynamic drag caused by modelling errors. dut, dvt, dpt, and drt signify external environment resistance. fu, fv, fp, and fr contain the total model uncertainties and external environment resistances. FxD0, FyD0, MxD0, and MzD0 can be calculated by the following equations:(12)FxD0=Fxa+Fmcosβ+Fwmcosβ+Fskcosβ,FyD0=Fya+Fmsinβ+Fwmsinβ+Fsksinβ+Fc,MxD0=Mxa+Fmzmsinβ+Fwmzwmsinβ+Ghmtanϕ+Fskzsksinβ+Fczc,MzD0=Mza+Fmxmsinβ+Fmymcosβ+Fwmxwmsinβ+Fwmywmcosβ+Fskxsksinβ+Fskyskcosβ+Fcxc,where(13)Fxa=−0.5ρaVa2CxaSPP,Fya=−0.5ρaVa2CyaSLP,Mxa=−0.5ρaVa2CmxaSPPlc+Fyaza,Mza=−0.5ρaVa2CmzaSHPHhov+Fyaxa+Fxaya,Fm=ρaVaQ,Fwm=Cwmpc2Bc/ρwg,Fsk=0.5Va2Cskh/lsk−0.34lskSc0.5+2.8167pc/lc−0.259−1Fwm,Fc=2lcφpc0.5Bctanϕ+h0,MG=Ghmtanϕ,where Fxa, Fya, Mxa, and Mza denote air resistance, Fwm, Fm, Fc, and Fsk denote wave-making resistance, air momentum force, cushion resistance, and skirt resistance, respectively. Cxa, Cya, Cmxa, Cmza, Cwm, and Csk represent the relevant resistance parameters. MG means roll restoring moment. The length and width of the cushion are represented by lc and Bc, respectively. SLP, SPP, and SHP denote lateral, positive, and horizontal projection areas, respectively. Sc and pc are the area and pressure of the cushion, the average clearance for air leakage in static hovering mode is signified by h, hm denotes the metacentric height, h0 denotes the initial lifting height, the total length of the skirt is represented by lsk, the flow coefficient is signified by φ, Hhov means the hovercraft's height, the air density and water density are signified by ρa and ρw, za, zm, zwm, zsk, and zc denote heights of each force's acting point with respect to the mass centre of the hovercraft, and xa,ya, xm,ym, xwm,ywm, xsk,ysk, and xc,yc signify the coordinates of the force's acting points. β signifies slip angle, and Va denotes the relative wind speed which can be calculated by the following equations:(14)β=arctanvu,Va=u+Vωcosβω−ψ2+v+Vωsinβω−ψ2,βa=arctanv+Vωsinβω−ψu+Vωcosβω−ψ,where βa signifies the relative wind direction, Vω represents the absolute wind speed, and βω denotes the absolute wind direction.Assumption 1.
In the control process, we ignore the motion of the hovercraft's pitch and heave and set the pressure of each air chamber and the flow of the cushion fan to be constants.Assumption 2.
The same two air propellers and the same two air rudders are symmetrically mounted at the tail of the hovercraft. In addition, the air rudder provides the moment of steering and the propeller only provides the forward thrust.Assumption 3.
The model uncertaintiesfu, fv, fp, and fr are continuously bounded, namely, fi≤Timaxi=u,v,p,r with Timax being positive constants.Assumption 4.
The hovercraft's surge, sway velocities, and yaw angular velocity are bounded.Remark 1.
In practice, the velocity and yaw angular velocity of the hovercraft cannot be infinite because of the restraints of aerodynamic resistance, hydrodynamic damping term, and the ability of the actuator [35].


## 2.3. Problem Formulation

To expediently describe the control problem, we firstly define the reference trajectory in this paper. The hovercraft tracks that the desired path is generated by the virtual surface vessel:(15)x˙dy˙dψ˙d=cosψd−sinψd0sinψdcosψd0001udsetvdsetrdset.Assumption 5.
The virtual vessel statesxd, yd, and ψd and their first derivatives with respect to time x˙d, y˙d, and ψ˙d are bounded.
The position tracking errors can be defined as follows:(16)xe=x−xd,ye=y−yd.
By utilizing (10) and (15), the derivatives of the position error with respect to time can be calculated as follows:(17)x˙ey˙e=cosψ−sinψcosϕsinψcosψcosϕuv−x˙dy˙d.
Subsequently, we define the velocity and yaw angular velocity tracking errors as follows:(18)ue=u−αu,ve=v−αv,re=r−αr,where αu, αv, and αr are the virtual control laws to be designed later on, which are viewed as the desired velocities and the desired yaw angular velocity of the u,v, and r, respectively.
In terms of the equation of the hovercraft dynamics (10), the derivatives of formula (18) with respect to time can be calculated as follows:(19)u˙e=vr+FxD0m0+fu+τum0−α˙u,v˙e=−ur+FyD0m0+fv−α˙v,r˙e=MzD0Jz0+fr+τrJz0−α˙r.
To avoid the dangerous navigation situation when the hovercraft sails in a narrow channel, we consider the transient performance during the position tracking error convergence. The errors can be defined to satisfy the following prescribed performance [36]:(20)−α1ηt<xe<β1ηt,−α2ηt<ye<β2ηt,where ηt=η0−η∞e−at+η∞ is a smooth performance constraint function with parameters satisfying η0>η∞>0, αi>0, βi>0, i=1,2.Assumption 6.
Initial position tracking errors satisfy the strict conditions−α1η0<xe0<β1η0 and −α2η0<ye0<β2η0.
The control objective in this paper can be formulated as follows.
Considering the hovercraft model (10) subject to model uncertainties and external disturbances, an adaptive neural-based fixed-time trajectory tracking controller is designed to generate surge force τu, virtual control law αr, and yaw moment τr to guarantee velocity tracking errors ue and ve, and yaw angular velocity tracking error re converges to the small region containing zero within the fixed time. Then, the virtual control laws αu and αv are reasonably designed to ensure the position tracking errors remain within the given prescribed performance bounds while the desired trajectory is being tracked by the hovercraft.


## 3. Main Results

### 3.1. Error Transformation

To guarantee the position tracking errors always satisfy the prescribed performance (20), the following transformations are applied:(21)xeye=ηtSxεxSyεy,where Siεi,i=x,y are strictly monotonically increasing functions of εi. Select Siεi as follows:(22)SxεxSyεy=β1eεx+1/2lnα1/β1−α1e−εx+1/2lnα1/β1eεx+1/2lnα1/β1+e−εx+1/2lnα1/β1β2eεy+1/2lnα2/β2−α2e−εy+1/2lnα2/β2eεy+1/2lnα2/β2+e−εy+1/2lnα2/β2.Sinceηt≠0, the transformed errors εx and εy can be altered as follows:(23)εxεy=12lnxeηβ1+α1β1−12lnα1β1−xeηα112lnyeηβ2+α2β2−12lnα2β2−yeηα2.Theorem 1.
Regarding the position tracking errorsxe and ye and the transformed errors εx and εy, if εx and εy are bounded and the initial tracking errors satisfy −α1η0<xe0<β1η0, −α2η0<ye0<β2η0, then xe and ye will be ensured to stay within the prescribed performance (20) for all t≥0.Proof.
We can infer that (22) and (23) have the following properties:(1)
−α1<Sxεx<β1,−α2<Syεy<β2(2)
limεx⟶−∞Sxεx=−α1,limεx⟶∞Sxεx=β1,limεy⟶−∞Syεy=−α2,limεy⟶∞Syεy=β2(3)
Sx0=0,Sy0=0(4)
xe⟶0 when εx⟶0, ye⟶0 when εy⟶0
Accordingly, ifεx and εy are not infinite and the initial tracking errors xe and ye that are within the performance limits, we can deduce that the above properties (1), (2), (3), and (4) will guarantee always that xe and ye satisfy the prespecified performance (20). And when εx and εy converge to a small neighbourhood around zero, xe and ye can converge to a small neighbourhood around zero as well.


### 3.2. Design of the Virtual Control Laws

In this subsection, we apply the Lyapunov direct method to design the desired velocities as the virtual control law of position error. Firstly, differentiate the transformed errors (23) as follows:(24)ε˙xε˙y=ξx00ξycosψ−sinψcosϕsinψcosψcosϕuv−x˙d+xeη˙ηy˙d+yeη˙η,where(25)ξx=12η1xe/η+α1−1xe/η−β1,ξy=12η1ye/η+α2−1ye/η−β2.We can easily obtain thatξx≠0 and ξy≠0.According to the fixed-time stability lemma and Lyapunov direct method, the desired velocities are designed as follows:(26)αuαv=cosψsinψ−sinψcosϕcosψcosϕ×x˙d+xeη˙η−1ξxkp11signεxεxp1+kp12signεxεxq1+kxεxy˙d+yeη˙η−1ξykp21signεyεyp1+kp22signεyεyq1+kyεy,where kx, ky, kp11, kp12, kp21, kp22, p1, and q1 are positive constants with satisfying the inequality 0<p1<1<q1.Assumption 7.
The hovercraft's roll angleϕ satisfies ϕ<90°.Remark 2.
In the motion process of the hovercraft, the roll angle is impossible to reach±90° because of the effect of roll restoring moment. The control task of this paper will lose its significance when the environmental forces break this balanced relationship.
By substituting (26) into (24), we have the following:(27)ε˙xε˙y=ξx00ξycosψ−sinψcosϕsinψcosψcosϕue+αuve+αv−x˙d+xeη˙ηy˙d+yeη˙η=−kp11signεxεxp1+kp12signεxεxq1+kxεxkp21signεyεyp1+kp22signεyεyq1+kyεy+ueξxcosψ−veξxsinψcosϕueξysinψ+veξycosψcosϕ.
Consider the following candidate Lyapunov function,(28)Vp=12εx2+12εy2.
The time derivative of the Lyapunov function defined by (28) along (27) is as follows:(29)V˙p=εxε˙x+εyε˙y=−kp11εxp1+1−kp12εxq1+1−kxεx2−kp21εyp1+1−kp22εyq1+1−kyεy2+εxueξxcosψ−εxveξxsinψcosϕ+εyueξysinψ+εyveξycosψcosϕ.
The result in (29) will be applied to the stability analysis of the designed controller in Subsection 3.6.


### 3.3. Design of the Surge Control Law

The surge control law is designed to guarantee that the surge velocity tracking error converges to a small neighbourhood around zero within a fixed time. Consider the following candidate Lyapunov function:(30)Vu=12ue2+12w˜uTΓu−1w˜u,where Γu∈Rlu×lu is the positive definite diagonal matrix, lu is the hidden note number, and w˜u=wu∗−w^u is the estimation error of the weight value with w^u representing the estimation value of the ideal weight value wu∗.The time derivative of the Lyapunov function is defined by (30) along (19) yields the following:(31)V˙u=ueu˙e+w˜uTΓu−1−w^˙u=uevr+FxD0m0+fu+τum0−α˙u+w˜uTΓu−1−w^˙u.The ARBFNN is employed to deal with the model uncertaintyfu of (31). According to Lemma 2, we have the following:(32)fu=wu∗Thuzu+δuzu,f^u=w^uThuzu,where f^u is the estimation value of the fu, δuzu signifies the minimum approximation error that satisfies δuzu≤δ¯u with δ¯u being the designed positive constant, zu=u,u˙T∈Ωu denotes the input of the NN, and huzu∈Rlu is the Gaussian basis function.According to (31), we design the surge control law as follows:(33)τu=m0−vr−FxD0m0−f^u+α˙u−kuue−ku1signueuep1−ku2signueueq1−εxξxcosψ−εyξysinψ,where ku>1/2, ku1, and ku2 are positive constants and the updated law of the NN weight is designed as follows:(34)−w^˙u=−ueΓuhuzu+σuΓuw^u.By substituting (32) and (33) into (31), we can obtain the derivative of the Lyapunov function Vu as follows:(35)V˙u=uevr+FxD0m0+fu+τum0−α˙u+w˜uTΓu−1−w^˙u=uefu−f^u−kuue2−ku1uep1+1−ku2ueq1+1−ueεxξxcosψ−ueεyξysinψ+w˜uTΓu−1−ueΓuhuz+σuw^u=−kuue2−ku1uep1+1−ku2ueq1+1−ueεxξxcosψ−ueεyξysinψ+w˜uTσuw^u+ueδuzu≤−ku−12ue2−ku1uep1+1−ku2ueq1+1−ueεxξxcosψ−ueεyξysinψ+w˜uTσuw^u+12δ¯u2.Inequality (35) will be used to analyse the stability of the controlled system in Subsection 3.6.


### 3.4. Design of the Desired Angular Velocity in Yaw

The hovercraft is a kind of typical underactuated surface vessel because its lateral axis is not directly actuated. Thus, the virtual control law is designed to stabilize the sway velocity tracking error. Consider the following candidate Lyapunov function:(36)Vv=12ve2+12w˜vTΓv−1w˜v,where Γv∈Rlv×lv signifies the design positive definite diagonal matrix with lv being the hidden note number, w˜v=wv∗−w^v denotes the estimation error of the weight value, and the ideal weight value wv∗ is estimated by the w^v.The derivative of the Lyapunov functionVv with respect to time is given by the following:(37)V˙v=vev˙e+w˜vTΓv−1−w^˙v=ve−ur+FyD0m0+fv−α˙v+w˜vTΓv−1−w^˙v.The model uncertaintyfv of (37) is handled by applying the ARBFNN. According to Lemma 2, we can obtain the following:(38)fv=wv∗Thvzv+δvzv,f^v=w^vThvzv,where f^v is the estimation value of fv, the minimum approximation error is represented by δvzv and satisfies δvzv≤δ¯v with δ¯v being the design positive constant, zv=v,v˙T∈Ωv denotes the input of the NN, and hvzv∈Rlv is the Gaussian basis function.According to (37), the design desired yaw angular velocity as a stability function:(39)αr=1uFyD0m0−α˙v+f^v+kvve+kv1signvevep1+kv2signveveq1−εxξxsinψcosϕ+εyξycosψcosϕ,where kv>1/2, kv1, and kv2 are positive constants and the updated law of the NN weight is designed as follows:(40)−w^˙v=−veΓvhvz+σvΓvw^v.Remark 3.
In the practical motion control of the hovercraft, the pitch angle is always set to the positive value. Only in special cases, the negative pitch angle can be set, such as when the hovercraft enters and leaves the mother ship. Accordingly, the surge velocity is always set tou>0 in the motion control process of the hovercraft.
By substituting (38) and (39) into (40), we can obtain the derivative of the Lyapunov function Vv as follows:(41)V˙v=ve−ur+FyD0m0+fv−α˙v+w˜vTΓv−1−w^˙v=ve−kvve−kv1signvevep1−kv2signveveq1+fv−f^v+w˜vTΓv−1−w^˙v+εxveξxsinψcosϕ−εyveξycosψcosϕ=−uvere+veδvzv−kvve2−kv1vep1+1−kv2veq1+1+σvw˜vTw^v≤−kv−12ve2−kv1vep1+1−kv2veq1+1+σvw˜vTw^v+12δ¯v2−uvere+εxveξxsinψcosϕ−εyveξycosψcosϕ.
Inequality (41) will be used to perform the stability analysis of the nonlinear system in Subsection 3.6.


### 3.5. Design of the Yaw Control Law

Subsequently, we design the yaw control lawτr to guarantee that the tracking error of angular velocity in the yaw converges to a small region around zero within a fixed time. Select the Lyapunov candidate function as follows:(42)Vr=12re2+12w˜rTΓr−1w˜r,where Γr∈Rlr×lr is the designed positive definite diagonal matrix, lr is the hidden note number, and the estimation error of the weight value is represented by w˜r=wr∗−w^r with w^r denoting the estimation value of the ideal weight value wr∗.The time derivative of the Lyapunov function defined by (42) along (19) is calculated as follows:(43)V˙r=rer˙e+w˜rTΓr−1−w^˙r=reMzD0Jz0+fr+τrJz0−α˙r+w˜rTΓr−1−w^˙r.The model uncertaintyfr of (43) is tackled by applying the ARBFNN. Accordingly, in terms of Lemma 2, we have the following:(44)fr=wr∗Thrzr+δrzr,f^r=w^rThrzr,where f^r is the estimation value of the fr, the minimum approximation error is signified by δrzr and satisfies δrzr≤δ¯r with δ¯r being the designed positive constant, zr=r,r˙T∈Ωr denotes the input of the NN, and hrzr∈Rlr is the Gaussian basis function.According to (43), design the yaw control law as follows:(45)τr=Jz0α˙r−MzD0Jz0−f^r−krre−kr1signrerep1−kr2signrereq1+uve,where kr>1/2, kr1 and kr2 are positive constants, and the updated law of the NN weight is selected as follows:(46)−w^˙r=−reΓrhrzr+σrΓrw^r.Then, substituting (44) and (45) into (43), the derivative of the Lyapunov function is expressed as follows:(47)V˙r=refr−f^r−krre−kr1signrerep1−kr2signrereq1+uve+w˜rTΓr−1−reΓrhrzr+σrΓrw^r=rew˜rThrzr+δrzr−kr1rep1+1−kr2req1+1+uvere+w˜rTΓr−1−reΓrhrzr+σrΓrw^r−krre2=reδrzr−kr1rep1+1−kr2req1+1+uvere+σrw˜rTw^−krre2≤−kr−12re2−kr1rep1+1−kr2req1+1+uvere+σrw˜rTw^+12δ¯r2.The result in (47) will be applied to the stability analysis of the control system in the next subsection.


### 3.6. Stability Analysis

Theorem 2.
Consider the hovercraft trajectory tracking nonlinear system (10) under model uncertainty and external environment disturbance, and suppose that Assumptions 1–7 are satisfied. If the desired velocities αu and αv and desired angular velocity αr are calculated by (26) and (39), the adaptive fixed-time controllers τu and τr are obtained by (33) and (45), the model uncertainties and external disturbances are approximated by the ARBFNNs (32), (38), and (44), and the initial condition satisfies −α1η0<xe0<β1η0 and −α2η0<ye0<β2η0, then the position tracking errors can be guaranteed to remain within the given prescribed performance bounds (20); all tracking errors can converge to a small region containing zero within the fixed time and are uniformly ultimately bounded.Proof.
The proof process is divided into two steps. Firstly, we indicate that all error signals are uniformly ultimately bounded. Secondly, verify that all tracking errors will converge uniformly to a small neighbourhood around zero within a fixed time.Step 1.
Consider the Lyapunov candidate function of the system:(48)V=Vp+Vu+Vv+Vr=12εx2+12εy2+12ue2+12w˜uTΓu−1w˜u+12ve2+12w˜vTΓv−1w˜v+12re2+12w˜rTΓr−1w˜r.
With the help of (29), (35), (41), and (47), the time derivative of (48) satisfies the following:(49)V˙≤−kp11εxp1+1−kp12εxq1+1−kxεx2−kp21εyp1+1−kp22εyq1+1−kyεy2+εxueξxcosψ−εxveξxsinψcosϕ+εyueξysinψ+εyveξycosψcosϕ−ku−12ue2−ku1uep1+1−ku2ueq1+1−ueεxξxcosψ−ueεyξysinψ+w˜uTσuw^u+12δ¯u2−kv−12ve2−kv1vep1+1−kv2veq1+1+σvw˜vTw^v+12δ¯v2−uvere+εxveξxsinψcosϕ−εyveξycosψcosϕ−kr−12re2−kr1rep1+1−kr2req1+1+uvere+σrw˜rTw^+12δ¯r2=−kp11εxp1+1−kp12εxq1+1−kxεx2−kp21εyp1+1−kp22εyq1+1−kyεy2−ku−12ue2−ku1uep1+1−ku2ueq1+1−kv−12ve2−kv1vep1+1−kv2veq1+1−kr−12re2−kr1rep1+1−kr2req1+1−σuw˜uTw˜u+12δ¯u2−σvw˜vTw˜v−σrw˜rTw˜r+σuw˜uTwu∗+σvw˜vTwv∗+σrw˜rTwr∗+12δ¯v2+12δ¯r2.
In the light of Young’s inequality, the following inequalities can be obtained:(50)σuw˜uTwu∗≤σu2w˜uTw˜u+σu2wu∗Twu∗,σvw˜vTwv∗≤σv2w˜vTw˜v+σv2wv∗Twv∗,σrw˜rTwr∗≤σr2w˜rTw˜r+σr2wr∗Twr∗.
Thus, we can rewrite (49) as follows:(51)V˙≤−kp11εxp1+1−kp12εxq1+1−kxεx2−kp21εyp1+1−kp22εyq1+1−kyεy2−ku−12ue2−ku1uep1+1−ku2ueq1+1−σu2w˜uTw˜u−kv−12ve2−kv1vep1+1−kv2veq1+1−σv2w˜vTw˜v−kr−12re2−kr1rep1+1−kr2req1+1−σr2w˜rTw˜r+σu2wu∗Twu∗+σv2wv∗Twv∗+σr2wr∗Twr∗+12δ¯v2+12δ¯u2+12δ¯r2≤−kxεx2−kyεy2−ku−12ue2−σu2w˜uTw˜u−kv−12ve2−σv2w˜vTw˜v−kr−12re2−σr2w˜rTw˜r+σu2wu∗Twu∗+σv2wv∗Twv∗+σr2wr∗Twr∗+12δ¯v2+12δ¯u2+12δ¯r2≤−2ρV+C1,where(52)ρ=minkx,ky,ku−12,kv−12,kr−12,σu2λminΓu,σv2λminΓv,σr2λminΓr,C1=σu2wu∗Twu∗+σv2wv∗Twv∗+σr2wr∗Twr∗+12δ¯v2+12δ¯u2+12δ¯r2.
By integrating both sides of (51), the following inequality is obtained:(53)0≤V≤V0−C12ρe−2ρt+C12ρ.
Therefore, it is clear thatV is bounded, which implies that the tracking errors are uniformly ultimately bounded. Thus, in terms of Theorem 1, we can deduce that the position tracking errors xe and ye always satisfy their prescribed performance limits (20). Furthermore, we can reasonably suppose that there is always a positive constant ΔM such that w˜i≤ΔM,i=u,v,r.Step 2.
For indicating that the tracking errors of the controlled system are uniformly fixed-time stable, we alter inequality (51) as follows:(54)V˙≤−kp112p1+1/212εx2p1+1/2−kp212p1+1/212εy2p1+1/2−ku12p1+1/212ue2p1+1/2−kv12p1+1/212ve2p1+1/2−kr12p1+1/212re2p1+1/2−kp122q1+1/212εx2q1+1/2−kp222q1+1/212εy2q1+1/2−ku22q1+1/212ue2q1+1/2−kv22q1+1/212ve2q1+1/2−kr22q1+1/212re2q1+1/2+σu2wu∗Twu∗+σv2wv∗Twv∗+σr2wr∗Twr∗−σuλminΓup1+1/212w˜uTΓu−1w˜up1+1/2+σu2λminΓuw˜uTΓu−1w˜up1+1/2−σvλminΓvp1+1/212w˜vTΓv−1w˜vp1+1/2+σv2λminΓvw˜vTΓv−1w˜vp1+1/2−σrλminΓrp1+1/212w˜rTΓr−1w˜rp1+1/2+σr2λminΓrw˜rTΓr−1w˜rp1+1/2−σu2λminΓuw˜uTΓu−1w˜u−σv2λminΓvw˜vTΓv−1w˜v−σr2λminΓrw˜rTΓr−1w˜r−σuλminΓuq1+1/212w˜uTΓu−1w˜uq1+1/2+σu2λminΓuw˜uTΓu−1w˜uq1+1/2−σvλminΓvq1+1/212w˜vTΓv−1w˜vq1+1/2+σv2λminΓvw˜vTΓv−1w˜vq1+1/2−σrλminΓrq1+1/212w˜rTΓr−1w˜rq1+1/2+σr2λminΓrw˜rTΓr−1w˜rq1+1/2+12δ¯u2+12δ¯v2+12δ¯r2.
With the fact that for all real numbersx≥0, we can obtain the following inequality:(55)xp1+1/2−x≤p1+12p1+1/1−p1−p1+122/1−p1=ϑ0.
According to Lemma1 and (55), inequality (54) can be expressed as follows:(56)V˙≤−ρ1Vp1+1/2−81−q1/2ρ2Vq1+1/2+C2,where(57)ρ1=minkp112p1+1/2,kp212p1+1/2,ku12p1+1/2,σrλminΓrp1+1/2,kv12p1+1/2,kr12p1+1/2,σuλminΓup1+1/2,σvλminΓvp1+1/2,ρ2=minkp122q1+1/2,kp222q1+1/2,σrλminΓrq1+1/2,ku22q1+1/2,kv22q1+1/2,σuλminΓuq1+1/2,kr22q1+1/2,σvλminΓvq1+1/2,ϑ1=σu2λminΓuλmaxΓu−1q1+1/2+σv2λminΓvλmaxΓv−1q1+1/2+σr2λminΓrλmaxΓr−1q1+1/2,C2=12δ¯v2+12δ¯u2+12δ¯r2+3ϑ0+ϑ1ΔMq1+1+σu2wu∗Twu∗+σv2wv∗Twv∗+σr2wr∗Twr∗.
In terms of Lemma4, all tracking errors can converge to the residual set Dtotal=Dtotal_1∪Dtotal_2 within a fixed time Ttotal. Then, Dtotal and Ttotal can be formulated as follows:(58)Dtotal_1=x:x≤2Ξ,x=εx,εy,ue,ve,re,Dtotal_2=w˜i:w˜i≤2λmaxΓiΞ,i=u,v,r,Ξ=minC2ρ1θρ12/p1+1,C281−q1/2ρ2θρ22/q1+1,Ttotal≤2ρ11−θρ11−p1+281−q1/2ρ21−θρ2q1−1,where θρ1, θρ2∈0,1 are positive constants.
In the light of Theorem1 and the analyses of Steps 1 and 2, the boundedness of εx and εy ensures that the position tracking errors xe and ye always satisfy their prescribed performance restraints. Furthermore, in terms of (21), we can conclude that the position tracking errors xe and ye will converge to a small region Dp=Dp1∪Dp2 within a fixed time Ttotal, given by the following:(59)Dp1=xe:ηTtotalSx−2Ξ≤xe≤ηTtotalSx2Ξ,Dp2=ye:ηTtotalSy−2Ξ≤ye≤ηTtotalSy2Ξ,where ηTtotal=η0−η∞e−aTtotal+η∞.Remark 4.
In the light of (58), we know that the coefficients ρ1,p1,ρ2, and q1 determine the precision of the tracking errors, namely, the small enough tracking errors are obtained by selecting the suitable coefficients. If εx and εy approach zero within a fixed time Ttotal by choosing suitable coefficients, then the position tracking errors xe and ye approach zero within a fixed time based on (21) as well.Remark 5.
It is worth noting that the small tracking errors will bring about the large control energy so that the input saturation will occur in the control process. Accordingly, in practical applications, the coefficientsρ1,p1,ρ2, and q1 should be adjusted discreetly for obtaining superior control performance.Remark 6.
For dealing with uncertainty in the control system, many successful methods have been proposed by researchers. For example, the multiobjective optimal control method [37, 38], fuzzy approximation strategy [39], and neural approximation strategy [40] can effectively solve uncertainty in control problems and may yield low computational cost. However, an ARBFNN is used to approximate the model uncertainty in this paper to better combine with the fixed-time theory. In future work, we will combine the multiobjective optimal control method and fuzzy/neural approximation strategy to effectively deal with the uncertainty in the hovercraft's control system for obtaining the optimal control performance and lower computational cost in the real-time control process.Remark 7.
Through the efforts of researchers, the general prescribed performance constraint control method is improved positively such as [41, 42]. The improved performance function was designed to overcome the existing prescribed performance constraint control method's shortcoming that the initial errors have to be known in advance for control designs. In this paper, it is the first time that fixed-time stability theory is introduced into the general prescribed performance control method. Therefore, the proposed method improved the convergence speed of the transformation errors, so that the convergence speed of the position tracking errors is also accelerated. Accordingly, this paper only uses a general prescribed performance control method that is better introduced into fixed-time stability theory, and the improved prescribed performance control method [41, 42] will be introduced into future work.


## 3.1. Error Transformation

To guarantee the position tracking errors always satisfy the prescribed performance (20), the following transformations are applied:(21)xeye=ηtSxεxSyεy,where Siεi,i=x,y are strictly monotonically increasing functions of εi. Select Siεi as follows:(22)SxεxSyεy=β1eεx+1/2lnα1/β1−α1e−εx+1/2lnα1/β1eεx+1/2lnα1/β1+e−εx+1/2lnα1/β1β2eεy+1/2lnα2/β2−α2e−εy+1/2lnα2/β2eεy+1/2lnα2/β2+e−εy+1/2lnα2/β2.Sinceηt≠0, the transformed errors εx and εy can be altered as follows:(23)εxεy=12lnxeηβ1+α1β1−12lnα1β1−xeηα112lnyeηβ2+α2β2−12lnα2β2−yeηα2.Theorem 1.
Regarding the position tracking errorsxe and ye and the transformed errors εx and εy, if εx and εy are bounded and the initial tracking errors satisfy −α1η0<xe0<β1η0, −α2η0<ye0<β2η0, then xe and ye will be ensured to stay within the prescribed performance (20) for all t≥0.Proof.
We can infer that (22) and (23) have the following properties:(1)
−α1<Sxεx<β1,−α2<Syεy<β2(2)
limεx⟶−∞Sxεx=−α1,limεx⟶∞Sxεx=β1,limεy⟶−∞Syεy=−α2,limεy⟶∞Syεy=β2(3)
Sx0=0,Sy0=0(4)
xe⟶0 when εx⟶0, ye⟶0 when εy⟶0
Accordingly, ifεx and εy are not infinite and the initial tracking errors xe and ye that are within the performance limits, we can deduce that the above properties (1), (2), (3), and (4) will guarantee always that xe and ye satisfy the prespecified performance (20). And when εx and εy converge to a small neighbourhood around zero, xe and ye can converge to a small neighbourhood around zero as well.


## 3.2. Design of the Virtual Control Laws

In this subsection, we apply the Lyapunov direct method to design the desired velocities as the virtual control law of position error. Firstly, differentiate the transformed errors (23) as follows:(24)ε˙xε˙y=ξx00ξycosψ−sinψcosϕsinψcosψcosϕuv−x˙d+xeη˙ηy˙d+yeη˙η,where(25)ξx=12η1xe/η+α1−1xe/η−β1,ξy=12η1ye/η+α2−1ye/η−β2.We can easily obtain thatξx≠0 and ξy≠0.According to the fixed-time stability lemma and Lyapunov direct method, the desired velocities are designed as follows:(26)αuαv=cosψsinψ−sinψcosϕcosψcosϕ×x˙d+xeη˙η−1ξxkp11signεxεxp1+kp12signεxεxq1+kxεxy˙d+yeη˙η−1ξykp21signεyεyp1+kp22signεyεyq1+kyεy,where kx, ky, kp11, kp12, kp21, kp22, p1, and q1 are positive constants with satisfying the inequality 0<p1<1<q1.Assumption 7.
The hovercraft's roll angleϕ satisfies ϕ<90°.Remark 2.
In the motion process of the hovercraft, the roll angle is impossible to reach±90° because of the effect of roll restoring moment. The control task of this paper will lose its significance when the environmental forces break this balanced relationship.
By substituting (26) into (24), we have the following:(27)ε˙xε˙y=ξx00ξycosψ−sinψcosϕsinψcosψcosϕue+αuve+αv−x˙d+xeη˙ηy˙d+yeη˙η=−kp11signεxεxp1+kp12signεxεxq1+kxεxkp21signεyεyp1+kp22signεyεyq1+kyεy+ueξxcosψ−veξxsinψcosϕueξysinψ+veξycosψcosϕ.
Consider the following candidate Lyapunov function,(28)Vp=12εx2+12εy2.
The time derivative of the Lyapunov function defined by (28) along (27) is as follows:(29)V˙p=εxε˙x+εyε˙y=−kp11εxp1+1−kp12εxq1+1−kxεx2−kp21εyp1+1−kp22εyq1+1−kyεy2+εxueξxcosψ−εxveξxsinψcosϕ+εyueξysinψ+εyveξycosψcosϕ.
The result in (29) will be applied to the stability analysis of the designed controller in Subsection 3.6.


## 3.3. Design of the Surge Control Law

The surge control law is designed to guarantee that the surge velocity tracking error converges to a small neighbourhood around zero within a fixed time. Consider the following candidate Lyapunov function:(30)Vu=12ue2+12w˜uTΓu−1w˜u,where Γu∈Rlu×lu is the positive definite diagonal matrix, lu is the hidden note number, and w˜u=wu∗−w^u is the estimation error of the weight value with w^u representing the estimation value of the ideal weight value wu∗.The time derivative of the Lyapunov function is defined by (30) along (19) yields the following:(31)V˙u=ueu˙e+w˜uTΓu−1−w^˙u=uevr+FxD0m0+fu+τum0−α˙u+w˜uTΓu−1−w^˙u.The ARBFNN is employed to deal with the model uncertaintyfu of (31). According to Lemma 2, we have the following:(32)fu=wu∗Thuzu+δuzu,f^u=w^uThuzu,where f^u is the estimation value of the fu, δuzu signifies the minimum approximation error that satisfies δuzu≤δ¯u with δ¯u being the designed positive constant, zu=u,u˙T∈Ωu denotes the input of the NN, and huzu∈Rlu is the Gaussian basis function.According to (31), we design the surge control law as follows:(33)τu=m0−vr−FxD0m0−f^u+α˙u−kuue−ku1signueuep1−ku2signueueq1−εxξxcosψ−εyξysinψ,where ku>1/2, ku1, and ku2 are positive constants and the updated law of the NN weight is designed as follows:(34)−w^˙u=−ueΓuhuzu+σuΓuw^u.By substituting (32) and (33) into (31), we can obtain the derivative of the Lyapunov function Vu as follows:(35)V˙u=uevr+FxD0m0+fu+τum0−α˙u+w˜uTΓu−1−w^˙u=uefu−f^u−kuue2−ku1uep1+1−ku2ueq1+1−ueεxξxcosψ−ueεyξysinψ+w˜uTΓu−1−ueΓuhuz+σuw^u=−kuue2−ku1uep1+1−ku2ueq1+1−ueεxξxcosψ−ueεyξysinψ+w˜uTσuw^u+ueδuzu≤−ku−12ue2−ku1uep1+1−ku2ueq1+1−ueεxξxcosψ−ueεyξysinψ+w˜uTσuw^u+12δ¯u2.Inequality (35) will be used to analyse the stability of the controlled system in Subsection 3.6.


## 3.4. Design of the Desired Angular Velocity in Yaw

The hovercraft is a kind of typical underactuated surface vessel because its lateral axis is not directly actuated. Thus, the virtual control law is designed to stabilize the sway velocity tracking error. Consider the following candidate Lyapunov function:(36)Vv=12ve2+12w˜vTΓv−1w˜v,where Γv∈Rlv×lv signifies the design positive definite diagonal matrix with lv being the hidden note number, w˜v=wv∗−w^v denotes the estimation error of the weight value, and the ideal weight value wv∗ is estimated by the w^v.The derivative of the Lyapunov functionVv with respect to time is given by the following:(37)V˙v=vev˙e+w˜vTΓv−1−w^˙v=ve−ur+FyD0m0+fv−α˙v+w˜vTΓv−1−w^˙v.The model uncertaintyfv of (37) is handled by applying the ARBFNN. According to Lemma 2, we can obtain the following:(38)fv=wv∗Thvzv+δvzv,f^v=w^vThvzv,where f^v is the estimation value of fv, the minimum approximation error is represented by δvzv and satisfies δvzv≤δ¯v with δ¯v being the design positive constant, zv=v,v˙T∈Ωv denotes the input of the NN, and hvzv∈Rlv is the Gaussian basis function.According to (37), the design desired yaw angular velocity as a stability function:(39)αr=1uFyD0m0−α˙v+f^v+kvve+kv1signvevep1+kv2signveveq1−εxξxsinψcosϕ+εyξycosψcosϕ,where kv>1/2, kv1, and kv2 are positive constants and the updated law of the NN weight is designed as follows:(40)−w^˙v=−veΓvhvz+σvΓvw^v.Remark 3.
In the practical motion control of the hovercraft, the pitch angle is always set to the positive value. Only in special cases, the negative pitch angle can be set, such as when the hovercraft enters and leaves the mother ship. Accordingly, the surge velocity is always set tou>0 in the motion control process of the hovercraft.
By substituting (38) and (39) into (40), we can obtain the derivative of the Lyapunov function Vv as follows:(41)V˙v=ve−ur+FyD0m0+fv−α˙v+w˜vTΓv−1−w^˙v=ve−kvve−kv1signvevep1−kv2signveveq1+fv−f^v+w˜vTΓv−1−w^˙v+εxveξxsinψcosϕ−εyveξycosψcosϕ=−uvere+veδvzv−kvve2−kv1vep1+1−kv2veq1+1+σvw˜vTw^v≤−kv−12ve2−kv1vep1+1−kv2veq1+1+σvw˜vTw^v+12δ¯v2−uvere+εxveξxsinψcosϕ−εyveξycosψcosϕ.
Inequality (41) will be used to perform the stability analysis of the nonlinear system in Subsection 3.6.


## 3.5. Design of the Yaw Control Law

Subsequently, we design the yaw control lawτr to guarantee that the tracking error of angular velocity in the yaw converges to a small region around zero within a fixed time. Select the Lyapunov candidate function as follows:(42)Vr=12re2+12w˜rTΓr−1w˜r,where Γr∈Rlr×lr is the designed positive definite diagonal matrix, lr is the hidden note number, and the estimation error of the weight value is represented by w˜r=wr∗−w^r with w^r denoting the estimation value of the ideal weight value wr∗.The time derivative of the Lyapunov function defined by (42) along (19) is calculated as follows:(43)V˙r=rer˙e+w˜rTΓr−1−w^˙r=reMzD0Jz0+fr+τrJz0−α˙r+w˜rTΓr−1−w^˙r.The model uncertaintyfr of (43) is tackled by applying the ARBFNN. Accordingly, in terms of Lemma 2, we have the following:(44)fr=wr∗Thrzr+δrzr,f^r=w^rThrzr,where f^r is the estimation value of the fr, the minimum approximation error is signified by δrzr and satisfies δrzr≤δ¯r with δ¯r being the designed positive constant, zr=r,r˙T∈Ωr denotes the input of the NN, and hrzr∈Rlr is the Gaussian basis function.According to (43), design the yaw control law as follows:(45)τr=Jz0α˙r−MzD0Jz0−f^r−krre−kr1signrerep1−kr2signrereq1+uve,where kr>1/2, kr1 and kr2 are positive constants, and the updated law of the NN weight is selected as follows:(46)−w^˙r=−reΓrhrzr+σrΓrw^r.Then, substituting (44) and (45) into (43), the derivative of the Lyapunov function is expressed as follows:(47)V˙r=refr−f^r−krre−kr1signrerep1−kr2signrereq1+uve+w˜rTΓr−1−reΓrhrzr+σrΓrw^r=rew˜rThrzr+δrzr−kr1rep1+1−kr2req1+1+uvere+w˜rTΓr−1−reΓrhrzr+σrΓrw^r−krre2=reδrzr−kr1rep1+1−kr2req1+1+uvere+σrw˜rTw^−krre2≤−kr−12re2−kr1rep1+1−kr2req1+1+uvere+σrw˜rTw^+12δ¯r2.The result in (47) will be applied to the stability analysis of the control system in the next subsection.


## 3.6. Stability Analysis

Theorem 2.
Consider the hovercraft trajectory tracking nonlinear system (10) under model uncertainty and external environment disturbance, and suppose that Assumptions 1–7 are satisfied. If the desired velocities αu and αv and desired angular velocity αr are calculated by (26) and (39), the adaptive fixed-time controllers τu and τr are obtained by (33) and (45), the model uncertainties and external disturbances are approximated by the ARBFNNs (32), (38), and (44), and the initial condition satisfies −α1η0<xe0<β1η0 and −α2η0<ye0<β2η0, then the position tracking errors can be guaranteed to remain within the given prescribed performance bounds (20); all tracking errors can converge to a small region containing zero within the fixed time and are uniformly ultimately bounded.Proof.
The proof process is divided into two steps. Firstly, we indicate that all error signals are uniformly ultimately bounded. Secondly, verify that all tracking errors will converge uniformly to a small neighbourhood around zero within a fixed time.Step 1.
Consider the Lyapunov candidate function of the system:(48)V=Vp+Vu+Vv+Vr=12εx2+12εy2+12ue2+12w˜uTΓu−1w˜u+12ve2+12w˜vTΓv−1w˜v+12re2+12w˜rTΓr−1w˜r.
With the help of (29), (35), (41), and (47), the time derivative of (48) satisfies the following:(49)V˙≤−kp11εxp1+1−kp12εxq1+1−kxεx2−kp21εyp1+1−kp22εyq1+1−kyεy2+εxueξxcosψ−εxveξxsinψcosϕ+εyueξysinψ+εyveξycosψcosϕ−ku−12ue2−ku1uep1+1−ku2ueq1+1−ueεxξxcosψ−ueεyξysinψ+w˜uTσuw^u+12δ¯u2−kv−12ve2−kv1vep1+1−kv2veq1+1+σvw˜vTw^v+12δ¯v2−uvere+εxveξxsinψcosϕ−εyveξycosψcosϕ−kr−12re2−kr1rep1+1−kr2req1+1+uvere+σrw˜rTw^+12δ¯r2=−kp11εxp1+1−kp12εxq1+1−kxεx2−kp21εyp1+1−kp22εyq1+1−kyεy2−ku−12ue2−ku1uep1+1−ku2ueq1+1−kv−12ve2−kv1vep1+1−kv2veq1+1−kr−12re2−kr1rep1+1−kr2req1+1−σuw˜uTw˜u+12δ¯u2−σvw˜vTw˜v−σrw˜rTw˜r+σuw˜uTwu∗+σvw˜vTwv∗+σrw˜rTwr∗+12δ¯v2+12δ¯r2.
In the light of Young’s inequality, the following inequalities can be obtained:(50)σuw˜uTwu∗≤σu2w˜uTw˜u+σu2wu∗Twu∗,σvw˜vTwv∗≤σv2w˜vTw˜v+σv2wv∗Twv∗,σrw˜rTwr∗≤σr2w˜rTw˜r+σr2wr∗Twr∗.
Thus, we can rewrite (49) as follows:(51)V˙≤−kp11εxp1+1−kp12εxq1+1−kxεx2−kp21εyp1+1−kp22εyq1+1−kyεy2−ku−12ue2−ku1uep1+1−ku2ueq1+1−σu2w˜uTw˜u−kv−12ve2−kv1vep1+1−kv2veq1+1−σv2w˜vTw˜v−kr−12re2−kr1rep1+1−kr2req1+1−σr2w˜rTw˜r+σu2wu∗Twu∗+σv2wv∗Twv∗+σr2wr∗Twr∗+12δ¯v2+12δ¯u2+12δ¯r2≤−kxεx2−kyεy2−ku−12ue2−σu2w˜uTw˜u−kv−12ve2−σv2w˜vTw˜v−kr−12re2−σr2w˜rTw˜r+σu2wu∗Twu∗+σv2wv∗Twv∗+σr2wr∗Twr∗+12δ¯v2+12δ¯u2+12δ¯r2≤−2ρV+C1,where(52)ρ=minkx,ky,ku−12,kv−12,kr−12,σu2λminΓu,σv2λminΓv,σr2λminΓr,C1=σu2wu∗Twu∗+σv2wv∗Twv∗+σr2wr∗Twr∗+12δ¯v2+12δ¯u2+12δ¯r2.
By integrating both sides of (51), the following inequality is obtained:(53)0≤V≤V0−C12ρe−2ρt+C12ρ.
Therefore, it is clear thatV is bounded, which implies that the tracking errors are uniformly ultimately bounded. Thus, in terms of Theorem 1, we can deduce that the position tracking errors xe and ye always satisfy their prescribed performance limits (20). Furthermore, we can reasonably suppose that there is always a positive constant ΔM such that w˜i≤ΔM,i=u,v,r.Step 2.
For indicating that the tracking errors of the controlled system are uniformly fixed-time stable, we alter inequality (51) as follows:(54)V˙≤−kp112p1+1/212εx2p1+1/2−kp212p1+1/212εy2p1+1/2−ku12p1+1/212ue2p1+1/2−kv12p1+1/212ve2p1+1/2−kr12p1+1/212re2p1+1/2−kp122q1+1/212εx2q1+1/2−kp222q1+1/212εy2q1+1/2−ku22q1+1/212ue2q1+1/2−kv22q1+1/212ve2q1+1/2−kr22q1+1/212re2q1+1/2+σu2wu∗Twu∗+σv2wv∗Twv∗+σr2wr∗Twr∗−σuλminΓup1+1/212w˜uTΓu−1w˜up1+1/2+σu2λminΓuw˜uTΓu−1w˜up1+1/2−σvλminΓvp1+1/212w˜vTΓv−1w˜vp1+1/2+σv2λminΓvw˜vTΓv−1w˜vp1+1/2−σrλminΓrp1+1/212w˜rTΓr−1w˜rp1+1/2+σr2λminΓrw˜rTΓr−1w˜rp1+1/2−σu2λminΓuw˜uTΓu−1w˜u−σv2λminΓvw˜vTΓv−1w˜v−σr2λminΓrw˜rTΓr−1w˜r−σuλminΓuq1+1/212w˜uTΓu−1w˜uq1+1/2+σu2λminΓuw˜uTΓu−1w˜uq1+1/2−σvλminΓvq1+1/212w˜vTΓv−1w˜vq1+1/2+σv2λminΓvw˜vTΓv−1w˜vq1+1/2−σrλminΓrq1+1/212w˜rTΓr−1w˜rq1+1/2+σr2λminΓrw˜rTΓr−1w˜rq1+1/2+12δ¯u2+12δ¯v2+12δ¯r2.
With the fact that for all real numbersx≥0, we can obtain the following inequality:(55)xp1+1/2−x≤p1+12p1+1/1−p1−p1+122/1−p1=ϑ0.
According to Lemma1 and (55), inequality (54) can be expressed as follows:(56)V˙≤−ρ1Vp1+1/2−81−q1/2ρ2Vq1+1/2+C2,where(57)ρ1=minkp112p1+1/2,kp212p1+1/2,ku12p1+1/2,σrλminΓrp1+1/2,kv12p1+1/2,kr12p1+1/2,σuλminΓup1+1/2,σvλminΓvp1+1/2,ρ2=minkp122q1+1/2,kp222q1+1/2,σrλminΓrq1+1/2,ku22q1+1/2,kv22q1+1/2,σuλminΓuq1+1/2,kr22q1+1/2,σvλminΓvq1+1/2,ϑ1=σu2λminΓuλmaxΓu−1q1+1/2+σv2λminΓvλmaxΓv−1q1+1/2+σr2λminΓrλmaxΓr−1q1+1/2,C2=12δ¯v2+12δ¯u2+12δ¯r2+3ϑ0+ϑ1ΔMq1+1+σu2wu∗Twu∗+σv2wv∗Twv∗+σr2wr∗Twr∗.
In terms of Lemma4, all tracking errors can converge to the residual set Dtotal=Dtotal_1∪Dtotal_2 within a fixed time Ttotal. Then, Dtotal and Ttotal can be formulated as follows:(58)Dtotal_1=x:x≤2Ξ,x=εx,εy,ue,ve,re,Dtotal_2=w˜i:w˜i≤2λmaxΓiΞ,i=u,v,r,Ξ=minC2ρ1θρ12/p1+1,C281−q1/2ρ2θρ22/q1+1,Ttotal≤2ρ11−θρ11−p1+281−q1/2ρ21−θρ2q1−1,where θρ1, θρ2∈0,1 are positive constants.
In the light of Theorem1 and the analyses of Steps 1 and 2, the boundedness of εx and εy ensures that the position tracking errors xe and ye always satisfy their prescribed performance restraints. Furthermore, in terms of (21), we can conclude that the position tracking errors xe and ye will converge to a small region Dp=Dp1∪Dp2 within a fixed time Ttotal, given by the following:(59)Dp1=xe:ηTtotalSx−2Ξ≤xe≤ηTtotalSx2Ξ,Dp2=ye:ηTtotalSy−2Ξ≤ye≤ηTtotalSy2Ξ,where ηTtotal=η0−η∞e−aTtotal+η∞.Remark 4.
In the light of (58), we know that the coefficients ρ1,p1,ρ2, and q1 determine the precision of the tracking errors, namely, the small enough tracking errors are obtained by selecting the suitable coefficients. If εx and εy approach zero within a fixed time Ttotal by choosing suitable coefficients, then the position tracking errors xe and ye approach zero within a fixed time based on (21) as well.Remark 5.
It is worth noting that the small tracking errors will bring about the large control energy so that the input saturation will occur in the control process. Accordingly, in practical applications, the coefficientsρ1,p1,ρ2, and q1 should be adjusted discreetly for obtaining superior control performance.Remark 6.
For dealing with uncertainty in the control system, many successful methods have been proposed by researchers. For example, the multiobjective optimal control method [37, 38], fuzzy approximation strategy [39], and neural approximation strategy [40] can effectively solve uncertainty in control problems and may yield low computational cost. However, an ARBFNN is used to approximate the model uncertainty in this paper to better combine with the fixed-time theory. In future work, we will combine the multiobjective optimal control method and fuzzy/neural approximation strategy to effectively deal with the uncertainty in the hovercraft's control system for obtaining the optimal control performance and lower computational cost in the real-time control process.Remark 7.
Through the efforts of researchers, the general prescribed performance constraint control method is improved positively such as [41, 42]. The improved performance function was designed to overcome the existing prescribed performance constraint control method's shortcoming that the initial errors have to be known in advance for control designs. In this paper, it is the first time that fixed-time stability theory is introduced into the general prescribed performance control method. Therefore, the proposed method improved the convergence speed of the transformation errors, so that the convergence speed of the position tracking errors is also accelerated. Accordingly, this paper only uses a general prescribed performance control method that is better introduced into fixed-time stability theory, and the improved prescribed performance control method [41, 42] will be introduced into future work.


## 4. Simulation Results

In this section, we complete the computer simulation experiments by comparison with the finite-time terminal sliding mode control method in [9] to verify the effectiveness and robustness of the proposed control method. In simulations, the hovercraft's main particulars are shown in Table 1 [2, 9].Table 1
Main particulars of the hovercraft.
m0kg4000Jz0kg⋅m21.8×106Jx0kg⋅m22.5×105SPPm245SLPm293SHPm2260gm/s29.8Scm2212lskm65Bcm8.9βwdeg45hmm2.4lcm23.6hm1Hhovm5.9Vwknot5The control parameters are set askp11=0.01, kp12=0.02, kp21=0.001, kp22=0.001, kx=0.0005, ky=0.005, p1=0.7, q1=1.2, ku=1, kv=1.5, kr=2, ku1=0.1, ku2=0.4, kv1=1.2, kv2=0.15, kr2=3.5, kr1=0.02, α1=α2=1, β1=β2=1, η0=200, η∞=2.2, a=0.02, and δ¯u=δ¯v=δ¯r=0.1. Since ARBFNN is utilized to approximate the unknown terms fu, fv, and fr, the hidden node numbers for w^uThuzu, w^vThvzv, and w^rThrzr are set to n=13 and the centres ϕl1l1=1,…,n, ϕl2l2=1,…,n, and ϕl3l3=1,…,n are evenly spaced in 30,40, −2,2, and −1,1, respectively. The widths are chosen as εl1=2, εl2=1, and εl2=1, respectively.The hovercraft's initial values are set asx0=−100m, y0=100m, ϕ0=0, ψ0=30°, u0=30knots, v0=0, p0=0, and r0=0.The virtual surface vessel's parameters are set asxd0=0, yd0=0, ψd0=45°, udsett=35knots, vdsett=0, and rdsett=0.35°/s.The total uncertainties suffered by the controlled system are characterized by the following formula:(60)fufvfr=2usin0.05t10m0vcos0.03tm0102rcos0.02tJz0b1b2b3,where b=b1b2b3T∈R3 and b˙=−T−1b+Awn signifies the first-order Markov process with wn∈R3 representing the zero-mean Gaussian white noises’ vector, and the rest coefficients of the first-order Markov process can be expressed as follows:(61)b0=2×104,2×104,2×104T,T=diag103,103,103,A=diag1×104,1×104,1×104.The simulation results of the two different methods with and without prescribed performance constraints are shown in Figures2–7 and labelled as proposed controller and sliding mode controller, respectively. Figure 2 shows the hovercraft can follow the virtual surface vessel despite the presence of the system uncertainties. The desired trajectory is constructed by a circle, which can represent practical significance in the problem of path following or trajectory tracking. Furthermore, from Figure 3, it is obvious that the proposed method can ensure the prespecified transient and steady performance of the position tracking errors. However, the constraints on the position tracking errors are violated under the sliding mode controller.Figure 2
The actual and desired trajectory of the hovercraft.Figure 3
Position tracking errors.
(a)(b)Figure 4
Input of two methods.
(a)(b)Figure 5
Velocity and yaw angular velocity tracking errors.
(a)(b)(c)Figure 6
Roll angular and yaw angular states.
(a)(b)Figure 7
Velocity and yaw angular velocity states.
(a)(b)(c)Subsequently, from Figures3 and 5, it is observed that the controllers designed in this paper are effective and can ensure that the position tracking error, velocity tracking errors, and the yaw angular velocity tracking error converge to a very small region around zero. Further observation shows that the proposed controller has a faster convergence speed and higher tracking precision than the sliding mode controller. The curves of the roll angle, yaw angle, and velocity states evolved with respect to time are shown in Figures 6 and 7. These two controllers' control inputs are shown in Figure 4. All simulation results illustrate the superiority and robustness of the proposed method.


## 5. Conclusion

Driven by the practical requirements for the underactuated hovercraft to sail in confined channels, the Lyapunov direct method and fixed-time stability theory-based trajectory tracking control algorithm with a prescribed performance is proposed for the hovercraft's motion control while being subjected to model uncertainties and external disturbances. The prescribed performance bound technique and fixed-time stability theory, which are integrated into Lyapunov stability synthesis, successfully guarantee that the position tracking errors of the hovercraft remain within the prespecified performance limits. It is shown that with the proposed control strategy, the requirement of the prespecified performance is always not violated and all tracking errors are not only uniformly ultimately bounded but also fixed-time convergent to an arbitrarily small region containing zero. In the future, we will consider that the velocity states are constrained to guarantee the safe sailing of the high-speed hovercraft.



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*Source: 6677445-2021-04-12.xml*