## 1 Introduction
To simplify the controller design process with prescribed performance, many researchers focus on the backstepping control strategy [17; 18; 19; 20; 21; 22; 23; 24; 25]. A novel backstepping control method was developed in [17] for a class of uncertain system, and the stability of the system was examined under the impact of artificial delays. In [18], an adaptive backstepping approach was proposed for the nonlinear systems with quantized states to solve the discontinuity problem brought by state quantization. However, the above backstepping methods inevitably have the problem of differential expansion, which will damage the control performance and affect the stability of the system, especially for high-order systems. In addition, the differential signals are easily affected by high-frequency disturbances or uncertainties. In order to address this problem, [PERSON] et al. [20; 21] utilized a compensated tracking error signal to introduce the concept of the command filter so that the problem can be addressed. In [25], modified error compensation signals were proposed along with new virtual control signals so as to ensure the finite-time tracking performance for a class of nonlinear systems. Besides, the backstepping control was applied to practical engineering by combining other advanced control methods like sliding mode control for better control performance, which has been investigated by researchers [19; 22]. In [19], the proposed backstepping control strategy was incorporated with a predictive model control method to maintain the DC-bus voltage in the photovoltaic (PV) grid-connected system. [PERSON] et al. [22] integrated adaptive sliding mode control with the command filter-based backstepping control strategy for the linear induction motor with end effects, uncertain parameters and external disturbances to solve its speed control problem. However, they have no assurance of the control performance and range of tracking error. Motivated by the aforementioned studies, an adaptive command filter-based backstepping controller (CBC) is presented for the ship roll stabilization in this paper. The dynamic performance and the steady-state of the compensated tracking error is guaranteed through the implementation of the prescribed performance control strategy. To offset the impact caused by external disturbances, a nonlinear disturbance observer is constructed and the stability of the whole system is examined via Lyapunov stability criterion. Due to the time-delay in the estimation, a disturbance estimation error may exist in the design of the proposed command-filter based backstepping controller. To eliminate the impact caused by this error, an adaptive control method is integrated with the design of the prescribed performance controller. The main contributions of this study are as follows:  
* To the best of the author's knowledge, the result in this paper is the first attempt to incorporate a prescribed performance control strategy in the ship rolling system control with adaptive sliding mode control. * Compared to some existing methods, the impact caused by the external disturbances and the system uncertainty is taken into account, even with the estimation delay problem that is eliminated by the adaptive control strategy. This paper is organized as follows: In Section 2, the system dynamics, command filter and the background knowledge for the prescribed performance control strategy are presented. In Section 3, the command-filter based controller with prescribed performance is constructed and the effectiveness of the disturbance observer is analyzed. Section 4 demonstrates the simulation results. Conclusions and future work are presented in Section 5.