Research Article  Open Access
Barrier Lyapunov FunctionBased Adaptive Control of an Uncertain Hovercraft with Position and Velocity Constraints
Abstract
This paper considers the problem of constrained path following control for an underactuated hovercraft subject to parametric uncertainties and external disturbances. A fourdegreeoffreedom hovercraft model with unknown curvefitted coefficients is first rewritten into a parameterized form. By introducing a barrier Lyapunov function into the lineofsight guidance, the specific transient tracking performance in terms of position error is guaranteed. A novel constrained yaw rate controller is proposed to ensure timevarying 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 highperformance 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 wavemaking 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 timevarying 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 lineofsight (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 [13â€“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 modelbased controllers might not achieve the desired control objective in practical applications. Furthermore, a curvefitted hovercraft model was obtained in [1] by replacing the complex hydrodynamic and aerodynamic force and moment functions with curvefitted approximations; by neglecting the heave and pitch motion, a fourdegreeoffreedom (DOF) controloriented 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 strictfeedback 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 backsteppingbased adaptive control scheme to estimate the unknown parameters, and the tuningfunction 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 [23â€“25]. To improve the tracking performance of a hovercraft under these system uncertainties, the above 4DOF 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 semiglobally stable. To compensate for the vehicleâ€™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 timevarying sideslip angle. Reference [32] presented a compound LOS guidance by combining the time delay control and ESO technique to estimate the timevarying 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 authorâ€™s knowledge, the problem of tracking errorconstrained path following control for an uncertain hovercraft with a timevarying 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 strictfeedback systems with output constraints [39], purefeedback 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 tantype 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 tantype BLF to design an errorconstrained LOS path following controller for a 3DOF conventional surface vessel. To break the static constraint limitations in the abovementioned works, a timevarying output constraint was handled in [46] by using a timevarying 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 BLFbased adaptive path following control scheme is developed for a 4DOF underactuated hovercraft subject to parametric uncertainties and external disturbances. The main contributions of this paper can be summarized as follows:
A novel positionconstrained 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 timevarying BLF, an attitude controller is proposed to ensure the yaw rate timevarying 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 amplitudeconstrained 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 firstorder 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.
For motion control of the hovercraft, we first define the earthfixed frame and the bodyfixed frame . The origin of the bodyfixed frame is located at the center of gravity (CG). By neglecting the pitch and heave motion and making some basic assumptions, the 4DOF curvefitted hovercraft dynamics are derived in [1] and are formulated as follows:where is the mass of the hovercraft; and are the moments of inertia of the hovercraft about the axis and axis, respectively; and are the surge and sway linear velocities, respectively; and and represent the roll and yaw angular velocities, respectively. The right side of (3) consists of various forces and moments acting on the hovercraft, in which the suffix represents the aerodynamic force, denotes the hydrodynamic force, denotes the cushion force, denotes the air momentum force, denotes the thrust, denotes the rudder moment, and denotes the restoring moment during heeling of the hovercraft. The curvefitted aerodynamic forces and moments in (3) are written asIn addition, the curvefitted hydrodynamic forces and moments in (3) are written aswhere denotes the relative wind speed. , , and are the projection areas of the hovercraft along the three axes. The average length of the hovercraft is denoted by , the vertical distance from the center of the lateral area is denoted by , the velocity of the hovercraft is denoted by , the lateral area of the skirts in contact with the water surface is denoted by , the cushion length is denoted by , and the height of CG is defined as . , , and represent the rudder angle, heeling angle, and sideslip angle of the hovercraft, respectively. represents the curvefitted coefficient. The expressions of other component forces and moments can be found in [1] (Chapter 6).
In this paper, we assume that , , , , and in (4) and (5) are unknown constants. Thus, (4) and (5) can be parameterized into the following form:By substituting (6) and other component forces and moments into (3), a 4DOF parameterized hovercraft model can be given aswhere the control inputs in (7) are and . The dynamics , , , and are the integration of other known component forces and moments in (3), which are defined as , , , and . In addition, , , , and in (7) denote the bounded disturbances, including the unmodeled dynamics and external disturbances induced by wind and waves. See Appendix B for the detailed expressions of the parameters in (7).
The desired parameterized path is defined by , where denotes the path variable. A pathfixed reference frame is defined with its origin at . The pathtangential angle is calculated by where and . For a hovercraft located at , the geometry of LOS guidance is illustrated in Figure 2.
The alongtrack error and the crosstrack error in the pathfixed frame are written asAccording to the kinematic equations of surface vessels given in [48], the 4DOF kinematics of the hovercraft is expressed aswhere and are the coordinates of the hovercraftâ€™s CG in the earthfixed frame. is the heading angle of the hovercraft, and is the heeling angle of the hovercraft.
The time derivative of the tracking errors in (9) is written asSubstituting (8) and (10) into (11) yieldsSetting , (12) can finally be written aswhere is the projection of the lateral velocity onto the horizontal plane. is the hovercraft speed. If the roll motion is not considered, we obtain . The obtained equation (13) is similar to the previous results [8, 31] for 3DOF surface vessels.
Suppose that , , and are positive constant constraints and is the symmetric timevarying yaw rate constraint, where the time derivatives of , , , and are bounded. The control objective of this study is to design a constrained path following controller for a hovercraft in the presence of parametric uncertainties and external disturbances that can guarantee the prescribed tracking performance: and . The yaw rate satisfies the constraint , and the surge velocity satisfies the constraint for all , where .
Remark 4. In the previous path following literature [8, 10, 29, 31], the proposed LOS guidance algorithms only regulate the steadystate tracking errors to be zero or within the neighborhood of zero, while the transient tracking performances are not considered. Reference [45] also proposed a LOS guidance to limit the position tracking errors. However, only the output constraints of the vessel were guaranteed; the yaw rate and velocity constraints were not guaranteed. In this paper, both the state and the output constraints of vessel system can be strictly guaranteed by the proposed control scheme. In addition, the yaw rate of a hovercraft is of particular concern to sailors during maneuver. Because the hovercraft usually hovers over the water with a high forward velocity, a large yaw rate or an excessive rudder angle can easily influence the navigation safety of a hovercraft as compared to a conventional displacement ship. Reference [33] shows that the stability boundary of the yaw rate varies with the surge speed of the hovercraft, which leads to the conversion of our design task into a timevarying state constrained problem.
To facilitate the control design, we make the following assumptions.
Assumption 5. The disturbances , , , and are bounded. There exist unknown positive constants , , , and such that , , . and hold for all .
Assumption 6. The initial tracking errors , , and satisfy the prescribed constraints , , and , and the yaw rate tracking error satisfies .
Assumption 7. For any and , the first and second time derivatives of the desired path are both bounded.
Assumption 8. Due to underactuated configuration of the hovercraft, a constraint on sway dynamics without control input is required that the sway velocity of the hovercraft is passive bounded.
3. Control System Design
3.1. PositionConstrained LOS Guidance
In this subsection, the barrier Lyapunov function will be introduced into the LOS guidance to guarantee the position constraints. The Lyapunov function is defined as follows:The time derivative of is calculated aswhere and .
Substituting (13) into (15), we obtain Taking the time derivative of yieldsInvoking (17) in (16), we havewithDefining the tracking error as and substituting (20) into (18) yieldwith and .
From (21), we can obtain the update law for :Note that is the denominator of (22); to avoid , we can take when , where is a small positive constant.
The desired heading angle of the hovercraft is chosen to bewhere are positive design constants and is the lookahead distance.
By invoking (22) and (23), (21) becomeswhere with .
Remark 9. In (25), we have and . By using Lâ€™Hopitalâ€™s rule, we obtain and . Thus, singularity will not occur in (25).
3.2. Attitude Control
In this subsection, the attitude kinematic controller for the hovercraft will be designed to stabilize the yaw angle tracking error . Subsequently, the yaw rate controller will be developed via BLFbased adaptive control, while the timevarying constraint on the yaw rate, , will be strictly guaranteed.
Suppose that the virtual yaw rate satisfieswhere and . The yaw rate tracking error is defined as . If a yaw rate controller exists that can ensure the inequalitywhere and then the timevarying constraint on yaw rate can be guaranteed.
Remark 10. If the designed yaw rate controller can make , then we have and . Using (26), we obtain .
However, one problem is how to ensure the inequality in (26) because there is no effective mechanism to guarantee it. Motivated by [49], the command filter is employed to constrain the magnitude of the virtual yaw rate by the timevarying constraint . The command filter is shown in Figure 3.
Here, the nominal virtual yaw rate is filtered to provide the magnitude and bandwidth limited virtual yaw rate and its derivative . In addition, an auxiliary system will be designed to compensate for the estimation error .
Step 1 (kinematics controller). Choose the following candidate Lyapunov function :Taking the time derivative of (28), we obtain To avoid the complex derivative computations of included in the nominal virtual control law , a secondorder trackingdifferentiator [50, 51] will be used for the estimation of , which is described aswhere , , and are positive design constants; is the input signal of (30); and denote the respective estimations of and . Let and define the estimation error asAccording to Theorem 2 presented in [50], we have , which means that can be sufficiently small by choosing a large . Hence, it can be concluded that there exists arbitrarily small positive constant such that .
As such, the nominal virtual control law for (29) is given bywhere and denote the design constants. . is the state of the auxiliary system and compensates for the constraint effect . The auxiliary system is designed aswhere and . is a small positive design constant that satisfies the performance requirement. is given by
Remark 11. Note that, in (33), will exponentially converge onto the small set if . If the state in (33) satisfies , then saturation effects occur again and the initial value of the auxiliary system should be properly reset to ensure that can respond to the case .
Consider the candidate Lyapunov function Invoking (32) and (33), the time derivative of is given byNote that, in (36),Then, we have where .
Step 2 (dynamics controller). Consider the following timevarying symmetric BLF:where . Recalling Remark 10, the constrained yaw rate controller will be proposed to guarantee the constraint . Define the new variable as ; (39) can be rewritten asIt can be seen that is positive definite and continuously differentiable in the set . The time derivative of is calculated as with .
Substituting the third equation of (7) into (41), we haveThe yaw rate control law for (42) becomeswhere , with a small constant. and are design constants. is directly obtained from the output of the command filter. , , and are adaptive estimations of , , and , respectively.
The adaptation laws for , , and are designed as follows:where , , and are positive design constants.
Next, we construct the candidate Lyapunov function as follows:Taking the time derivative of (45) and using (42) and (44), we obtainNote that andSubstituting the inequalities (47)(49) and (2) into (46), in the set, we can obtain thatwhere , with a positive constant defined as
3.3. Velocity Tracking Control
In this subsection, the designed control force should enforce the following constraint on the velocity: . In addition, the constraint should also be strictly guaranteed. Using the first equation in (7), the time derivative of is calculated asChoosing the barrier Lyapunov functionand differentiating (53) yieldwhere .
We design the surge velocity tracking controller for as follows:where and are design constants.
The adaptation laws for , , and are given aswhere , , and are positive design constants.
Construct the Lyapunov function asTaking the time derivative of (57) and invoking (54)(56), we obtain Note also that , , and .
We then have where and is a positive constant defined as
Remark 12. In (55), the desired surge speed is required to be continuously differentiable. In previous works [10, 11, 45], the desired surge speed was chosen as a constant for all . However, in practice, the desired surge speed of a hovercraft is often directly switched to another value, which will lead to the nonexistence of . Thus, to ensure that the desired surge speed is differentiable when changes from to between any time interval and , we give a speed smoother as follows: The scalar function is defined aswhere . if , and if , with and a positive constant. The speed smoother is shown in Figure 4.
3.4. Stability Analysis
Theorem 13. Consider the parameterized hovercraft models (7) and (10) in the presence of parametric uncertainties and external disturbances and assume that Assumptions 5â€“8 are satisfied. If the update law for the path variable is chosen as (22), the desired heading angle of the hovercraft is calculated by (23), the auxiliary system is designed as (33), the adaptation laws for unknown parameters and external disturbances are given by (44) and (56), and the controllers are obtained from (43) and (55); then the following properties hold:
(i) All the error signals in the closedloop system are uniformly ultimately bounded (UUB).
(ii) The static constraints on position and surge velocity tracking errors and the timevarying constraints on the yaw rate are never violated, i.e., , , and hold for all .
Proof. (i) Construct the following candidate Lyapunov function:By virtue of (24), (38), (50), and (59), the time derivative of is given bywhere and .
If the initial conditions satisfy Assumption 6, then the constructed Lyapunov function satisfies the inequalityFrom (65), we know that . Therefore, all the error signals in the closedloop system are uniformly ultimately bounded. Noting the expressions of and , the ultimate bounds can be made arbitrarily small by adjusting the design parameters.
(ii) , the BLF (14), (40), and (53) satisfy the following inequalities:where .
Then, from (66), we obtainIn (67), from , , we know that . Recalling Remark 10, we conclude that . Therefore, all the prescribed constraints of the hovercraft system can be guaranteed for . This completes the proof.
4. Simulation
In this section, numerical simulations will be implemented to demonstrate the effectiveness and safety of the proposed control scheme for hovercraft navigation and to compare our scheme with the results presented in [11, 45]. A general curvilinear path simulation scenario is considered. In simulations, the uncertainty parameters in (7), including , , , , and , have 20% maximum relative uncertainties according to their nominal values; i.e., their real values are chosen according towhere is randomly selected from . The nominal parameters of the hovercraft are shown in Appendix A, Table 1. For numerical simulations, the timevarying bounded disturbances in (7) are generated by using the firstorder GaussMarkov process [52]:where are zeromean Gaussian white noise processes and are constants.
