Universal controllers for stabilization and tracking of underactuated ships

doi:10.1016/S0167-6911(02)00214-1

Systems & Control Letters

Volume 47, Issue 4, 15 November 2002, Pages 299-317

https://doi.org/10.1016/S0167-6911(02)00214-1 Get rights and content

Abstract

We examine the problem of universal control for underactuated surface ships with only surge force and yaw moment. Namely, a single controller is to be designed to achieve stabilization and tracking simultaneously. We propose, in this paper, the first universal controller of which the synthesis is based on Lyapunov's direct method and backstepping technique. Our result is extendible to the input-saturation when the surge and yaw velocities are considered as the controls. Numerical simulations are provided to validate the effectiveness of the proposed controller and to demonstrate its sensitivity with respect to model parameters.

Introduction

Stabilization and tracking control of position (sway and surge) and orientation (yaw) of underactuated surface ships, have recently received considerable attention from the control community. The challenge of these problems is due to the fact that the motion of the underactuated ship in question possesses three degrees of freedom (yaw, sway and surge neglecting the motion in roll, pitch and heave) while there are only two available controls (surge force and yaw moment) under a nonintegrable second order nonholonomic constraint. By applying Brockett's theorem [3], it can be shown that any continuous time invariant feedback control law does not make the null solution of the underactuated ship dynamics asymptotically stable in the sense of Lyapunov. Furthermore as observed in [22], [24], [18] the underactuated surface ship system is not transformable into a chained system. Consequently, existing control schemes [10], [23] developed for standard chain-form systems cannot be applied directly. In the past few years, stabilization and tracking of underactuated ships have been studied separately from different viewpoints.

In [22], a discontinuous state feedback control law was proposed by using σ-process to exponentially stabilize the underactuated ship at the origin. In [24], the authors developed a discontinuous time-varying feedback stabilizer for a nonholonomic system and applied to underactuated ships. Some local exponential stabilization results were reported in [18] based on time-varying homogeneous control approach. An application of averaging and homogeneous system theory to underactuated ships was reported in [19]. By transforming the underactuated ship kinematics and dynamics into the so-called skew form, some dynamic feedback results on stabilization were given in [15], [2].

In [6], the authors used a continuous time-invariant state feedback controller to achieve global exponential position tracking under an assumption that the reference surge velocity is always positive. Unfortunately, the orientation of the ship was not controlled. An application of the recursive technique proposed in [10] for the standard chain form systems yields a high-gain based local tracking result [12]. By applying the cascaded approach, a global tracking result was obtained in [14]. The stability analysis relies on the stability theory of linear time-varying systems. Based on Lyapunov's direct method and passivity approach, two constructive tracking solutions of an underactuated surface ship were proposed in [7]. A very restrictive assumption made in [21], [14], [7] is that the yaw reference velocity does not converge to zero and satisfies PE (persistent excitation) conditions of various kinds. Consequently, a straight-line reference trajectory cannot be tracked. In a recent paper [4], we have removed the above assumption. However in [4], the case of regulation/stabilization and parking was not covered.

In [15], [2], the authors developed a high-gain dynamic feedback control law to achieve global ultimate regulation and tracking of underactuated ships. The control development was based on a transformation of the ship tracking system into the so-called skew form. The dynamics of closed loop system is increased due to the use of some exponentially stable oscillator. It is worth mentioning that in [13], the authors proposed a time-varying velocity feedback controller to achieve both stabilization and tracking of unicycle mobile robots at the kinematics level motivated from the work in [23]. However this controller cannot be extended directly to the case of underactuated ships due to the nonintegrable second order constraint. From the above discussion, an open challenging problem in controlling underactuated surface ships is to find a single controller that can solve the problem of stabilization and trajectory tracking for the ship with only two propellers.

Motivated by our recent work [4] and the idea of controlling mobile robots [23], [13], [9] this paper provides a positive answer to the above challenging problem. Our control development is based on Lyapunov's direct method [11] and the backstepping technique [12]. The proposed controller guarantees the global asymptotical convergence of the regulation and tracking error to the origin. In particular, we allow the reference trajectory to be a curve including a straight-line, a path approaching to the origin, or just a set-point. Since our proposed control design procedure generates explicit control Lyapunov function, it is straightforward to extend our results to the case with actuator dynamics and/or unmeasured thruster dynamics. It is also of interest to note that in the situation where the yaw moment is absent instead of sway force can be directly dealt with by our proposed control design. More importantly, our proposed control laws can be directly extended to the case of input saturation, when the surge and yaw velocities are considered as the controls, see Remark 5. Simulations with a monohull ship with a length of $32 m$ and a mass of $118×10^{3} kg$ illustrate the effectiveness of our proposed controller and to demonstrate its sensitivity with respect to model parameters.

The rest of this paper is organized as follows. The problem formulation is stated in the next section. In Section 3, the controller is designed and the main result is stated. In Section 4, proof of the main result is given. Sensitivity analysis of our proposed controller to model parameters is presented in Section 5. Simulations for illustrating the effectiveness of our proposed controller are given in Section 6. Some conclusions are drawn in Section 7.

Section snippets

Problem formulation

The simplified underactuated ship dynamics studied in [18], [19], [21], [7], [4] can be derived from the mathematical model of a surface ship moving in six degrees of freedom [5] as $x ̇ =u cos (ψ)−v sin (ψ), y ̇ =u sin (ψ)+v cos (ψ), ψ ̇ =r, u ̇ = m_{22} m_{11} vr− d_{11} m_{11} u+ 1 m_{11} τ_{u}, v ̇ =− m_{11} m_{22} ur− d_{22} m_{22} v, r ̇ = (m_{11} −m_{22}) m_{33} uv− d_{33} m_{33} r+ 1 m_{33} τ_{r},$ where x,y and ψ are the surge displacement, sway displacement and yaw angle in the earth fixed frame, u,v and r denote surge, sway and yaw velocities respectively. The positive constant terms d_jj

Control design

Motivated by our recent work [4], we define the following coordinate: $z_{e} =ψ_{e} + arcsin k(t)y_{e} 1+x_{e}^{2} +y_{e}^{2}, with k(t)=λ_{1} (u_{d} +λ_{2} cos (λ_{3} t))$ where the constants $λ_{i}, 1⩽i⩽3$ are such that sup_t⩾0|k(t)|<1. They will be specified in the stability analysis in Section 4. Notice that (5) is well defined and that convergence of z_e and y_e implies the convergence of ψ_e.

Remark 3

By using the nonlinear coordinate (5) instead of z_e=ψ_e+k(t)y_e, we avoid the ship whirling around for large y_e. It is also different from the ones in [13], [9]

Proof of Theorem 1

As discussed in Section 2, we only need to show that the transformed tracking errors, (x_e,y_e,z_e,v_e) are globally asymptotically stable at the origin. Under Assumption 1, boundedness of the controls τ_u and τ_r follows readily. Substituting , into (6) yields the closed loop system $x ̇_{e} =−k_{1} x_{e} +k_{2} r_{d} y_{e} +u_{d} (ϖ_{2} −ϖ_{1})ϖ_{2}^{−1} + k(t)v_{d} ϖ_{2}^{−1} y_{e} +r_{1e}^{d} y_{e} +r_{d} y_{e} + r_{2e}^{d} y_{e} + u ̃_{e} + r ̃_{e} y_{e} +p_{x}, y ̇_{e} =v_{e} +v_{d} (ϖ_{2} −ϖ_{1})ϖ_{2}^{−1} −k(t)u_{d} ϖ_{2}^{−1} y_{e} −r_{1e}^{d} x_{e} − r_{d} x_{e} −r_{2e}^{d} x_{e} − r ̃_{e} x_{e} +p_{y}, v ̇_{e} =− d_{22} m_{22} v_{e} − m_{11} m_{22} u_{e}^{d} r_{d} − m_{11} m_{22} (u_{e} +u_{d})r_{1e}^{d} − m_{11} m_{22} (r_{d} +r_{1e}^{d}) u ̃_{e} − m_{11} m_{22} (u ̃_{e} +u_{e}^{d} +u_{d})(r$

Sensitivity analysis

The control law (14) has been designed under the assumptions that the system parameters are precisely known and there are no environmental disturbances. Indeed, these assumptions are unrealistic in practice. The aim of this section is to discuss the sensitivity of our proposed controller in relation to the inaccurate knowledge of the ship parameters. Discussion related to environmental disturbances can be carried out similarly.

From Section 3, the control law (14) can be easily modified to

Simulations

This section validates the control law (14) by simulation with a monohull ship with a length of $32 m$ , a mass of $118×10^{3} kg$ and other parameters calculated by using VERES—a program that calculates the added mass and damping matrices for surface ships as follows: $m_{11} =120×10^{3} kg, m_{22} =172.9×10^{3} kg, m_{33} =636×10^{5} kg m^{2}, d_{11} =215×10^{2} kg s^{−1}, d_{22} =97×10^{3} kg s^{−1}, d_{33} =802×10^{4} kg m^{2} s^{−1}$ . The reference trajectory is generated by the virtual ship with the initial conditions as $[x_{d} (0),y_{d} (0),ψ_{d} (0),v_{d} (0)]=[0 m,0 m,0 rad,0 ms^{−1}]$ and

Conclusions

A universal controller has been obtained in this paper to solve simultaneously both regulation and tracking problems of underactuated surface ships with only surge force and yaw moment available. The proposed controller is able to globally asymptotically force the underactuated ships to follow any reference trajectory generated by a suitable virtual ship in a frame attached to the ship body. When the yaw reference velocity satisfies a PE condition, the global K-exponential stability of the

Acknowledgements

We would like to thank Professor I. Mareels for his helpful comments. The study of the first author has been supported by IPRS and UPA from the University of Western Australia.

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^☆: The work of the first and second authors has been supported in part by the National Science Foundation under Grants ECS-0093176 and INT-9987317.

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Universal controllers for stabilization and tracking of underactuated ships☆

Abstract

Introduction

Section snippets

Problem formulation

Control design

Proof of Theorem 1

Sensitivity analysis

Simulations

Conclusions

Acknowledgements

Systems Control Lett.

Automatica

Automatica

Automatica

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IEEE Trans. Automat. Control

Asymptotic stability and feedback stabilization

Guidance and Control of Ocean Vehicles

Nonlinear adaptive backstepping designs for tracking control of ships

Internat. J. Adaptive Control Signal Process.

A small-gain control method for nonlinear cascaded systems with dynamic uncertainties

IEEE Trans. Automat. Control

Tracking control of mobile robotsa case study in backstepping

Automatica

A recursive technique for tracking control of nonholonomic systems in chained form

IEEE Trans. Automat. Control