Path following control for marine surface vessel with uncertainties and input saturation

Abstract

This paper investigates the path following control problem for an unmanned marine surface vessel (MSV) in the presence of uncertainties and input saturation. The backstepping technique augmented by a robust adaptive radial basis function neural network (RBFNN) and an auxiliary design system is employed as the main control framework. Based on the dynamic model of the MSV, an improved adaptive integral line-of-sight (LOS) guidance law is first proposed, which is suitable for any parametric paths and can deal with time-varying ocean currents. The guidance law calculates the desired yaw angle and estimates the currents. Then the controller is extended to cope with the MSV yaw tracking and velocity control by resorting to the augmented backstepping technique. The uncertainties of dynamics are compensated by the robust RBFNNs. Each robust RBFNN utilizes an nth-order smooth switching function to combine a conventional RBFNN with a robust control. The auxiliary design system is presented to analyze the effect of input saturation, and states of the auxiliary design system are used to develop the controller. This systematic design methodology is proved to achieve ultimate boundedness of the closed-loop MSV system. Simulations validate the effectiveness of the proposed control approach.

Introduction

Three types of control technologies play a crucial role in developing marine surface vessel (MSV) to achieve its specified tasks automatically [1]: setpoint control, trajectory tracking control, and path following control. Setpoint control [2], [3], [4], [5], [6] is important for dynamic positioning of vessels in fixed target operations such as autonomous docking. Trajectory tracking control [7], [8], [9], [10], [11], [12] enables the ship to track the desired time-referenced trajectory or virtual objects. In path following control scenario [13], [14], [15], MSV is required to follow a path at a certain speed that is specified without temporal constraint. Typical trajectory tracking and path following control applications are way-point navigation, reconnaissance, and surveillance.

Considering their low maneuverability character, it is more practical to study the path following control problem for MSVs. A popular and effective way to achieve convergence to the desired path is to implement a lookahead-based line-of-sight (LOS) guidance law mimicking an experienced sailor [16], [17], [18]. This method exploits the geometry of the problem and generates the desired yaw angle, which is fed into the attitude tracking subsystem, and it has been applied to the flight control field [19], [20]. Guided LOS motion control of AUVs using sliding mode control for stabilizing the combined speed, steering and diving responses was addressed in [21]. Uniform global asymptotic stability (UGAS) and uniform local exponential stability of the LOS guidance law was proved by Pettersen and Lefeber [22], and it was extended to a more complete vehicle model in [23], [24]. Furthermore, [18] presented a uniform semiglobal exponential stability proof for the LOS guidance laws, which was slightly weaker than global exponential stability. The above-mentioned LOS method is actually a kind of proportional guidance law, it has limitations when the vehicle is exposed to unknown disturbances and uncertainties. Therefore, it is necessary to modify the LOS guidance law to include integral action. A modified LOS guidance law with integral action was proposed in [25], [26], [27], [28], which guarantees global asymptotic path following of straight-line paths in the presence of constant ocean currents. An adaptive integral LOS controller for path following of marine craft was presented in [29], [30], and global convergence of the cross-track error was proved.

The stability analyses of aforementioned LOS methods are based on the cascaded system stability theorem [31], which can be written as UGAS+UGAS+UGB $\iff$ UGAS in the simplified form, where UGB stands for uniform global boundedness. This means that the subsystem is required to be UGAS. However, for the subsystem in the presence of time-varying currents, uncertainties and disturbances, the controller has to include a sign function to offset these uncertainties to get the UGAS result of the closed loop system [30], [32]. This induces system chattering inevitably and the control variable is discontinuous. Instead, MSV is applicable to the backstepping technique, since its model can be decomposed into an approximate strict-feedback format. In addition, the backstepping technique can be combined with other approaches to handle uncertain parameters, unmodeled dynamics, and external disturbances. References [6], [7], [33] utilized the backstepping augmented by practical control, neural network, and adaptive algorithms to realize the stable tracking for MSVs.

Neural networks, with their strong approximate capacity, attract more and more attention. It has been employed to design robust adaptive control combining with backstepping technique for the uncertain MIMO nonlinear systems [34], [35]. Thus, the neural network can be introduced to handle the uncertainty and disturbance of MSVs [33], [36], [37]. The parameter update laws in the above neural networks all evolved from Lyapunov-based approaches. Nevertheless, it is worth noting that each neural network above is valid only over an active region, while the tracking performance, even the tracking stability may be deteriorated outside this region.

Every input of a system is bounded by physical restriction of actuators. Input saturation, usually ignored to simplify the design of a control system, causes performance degradation, lag, overshoot, undershoot as well as instability in the closed-loop response of practical systems. Therefore, input saturation must be taken into account in the control design [38]. Analysis and design of control systems with input saturation have been reported in [34] for tracking control of uncertain MIMO nonlinear systems, in [39] for position control of a MSV, and in [40] for global tracking control of underactuated ships. An adaptive steering law for asymptotically stable ship with saturation limits was proposed in [41], combined with a linear quadratic controller and a Riccati based anti-windup compensator. Reference [36] considered the cooperative path following problem of multiple MSVs subject to input saturation, unknown dynamical uncertainty and unstructured ocean disturbances.

In this paper, we present an improved adaptive integral LOS guidance law by integrating the results of [16], [17], [30]. The improved method is suitable for any parametric paths, and is applied to the path following control for a MSV in the presence of unknown time-varying currents. The guidance law calculates the desired yaw and estimates the currents. Then the controller is extended to cope with the MSV yaw tracking and velocity control by resorting to the backstepping technique [32]. The uncertainties of dynamics are compensated by using a robust adaptive radial basis function neural network (RBFNN) [42], [43]. Each robust RBFNN utilizes an nth-order smooth switching function to combine a conventional RBFNN with a robust control. The conventional RBFNN dominates in the neural active region, while the robust control retrieves the transient outside the active region, so that the stability range can be widened [43]. The auxiliary design system is presented to analyze the effect of input saturation, and states of the auxiliary design system are utilized to develop the controller. It is proved that, with the proposed control approach, the tracking errors of the controlled closed-loop system are ultimately bounded. Simulations verify the theoretical analysis.

The contributions of this paper are summarized as follows. (1) Adaptive integral LOS is improved to calculate the desired yaw and estimate the unknown time-varying currents in a uniform form for all kinds of parametric paths. (2) Robust RBFNNs are integrated to estimate and compensate for the unmodeled dynamics so that the tracking performance is guaranteed. (3) The input saturation is incorporated into the path following design to cope with physical constraints.

This paper is organized as follows. Some useful preliminaries are introduced in Section 2. The model of the MSV with currents and uncertainties is presented in Section 3. Section 4 is devoted to designing the adaptive integral LOS guidance and robust adaptive RBFNN augmenting backstepping control algorithm on the MSV. Section 5 simulates the proposed control approach, and finally, we conclude this paper and propose some further work in Section 6.