# Robust Backstepping Control of Wing Rock Using Disturbance Observer
## 1 Introduction  
Wing rock motion is an undesired roll-dominated oscillatory coupled with yaw oscillation for the aircraft flying at subsonic speed and high angle of attack (AOA). When an aircraft suffers the wing rock motion, the large amplitude and high frequency oscillatory of the rolling and yawing angle will severely limit the maneuverability, tracking accuracy, and operational safety of the aircraft. Hence, the wing rock phenomenon should be paid more attention, and the corresponding wing rock controller should be further studied. In past decades, many researches have been done to analyze the mechanism of the wing rock. Mathematically, three possible nonlinear factors were considered as the inducements of the wing rock phenomenon, which are the variation of damping in roll with angle of sideslip, cubic variation of lateral derivatives with roll rate and sideslip, and the aerodynamic hysteresis in steady-state rolling moment [1; 2]. Meanwhile, different wing-rock models have been developed [3; 4; 5; 6]. Based on these simplified models, many control schemes were developed to suppress the influence of wing rock. Adaptive feedback linearization technique was employed for the wing rock in [7; 8]. An \(L_{1}\) adaptive controller was designed for the wing-body rock motion [9]. Fuzzy adaptive approaches have been employed in the wing rock during the last decade [10; 11; 12]. The neural network technique and backstepping approach have been also extensively studied [13; 14; 15]. In [16; 17; 18], simplified single degree-of-freedom wing-rock models were studied, and the disturbance observeror traditional extended state observer (ESO) was adopted in the design of robust wing rock control law. However, the yaw oscillation was often ignored in the design of wing rock controller for the convenience of research. Considering the serious coupling effect between the roll and yaw motion, multi-degree-of-freedom wing rock models should be further studied. In this paper, an uncertain multi-input and multi-output (MIMO) nonlinear wing rock model is studied, and the coupling effect between the roll oscillation and yaw oscillation will be fully considered in the design of the wing rock controller. Meanwhile, to enhance the system robustness, the ESO and radial basis function neural network (RBFNN) techniques are combined to tackle the external disturbances and system uncertainties, respectively. Up to now, more and more researchers have paid attention to the study of the ESO or RBFNN based control techniques[19; 20; 21]. In [22], an adaptive neural network based control law was developed for an uncertain MIMO system with unknown control coefficient matrices and input nonlinearities. In [23], the observer and adaptive neural network (NN) techniques were combined to tackle the uncertainties. To tackle the state constraint problem, the adaptive neural network technique was employed for an uncertain robot system [24]. And the ESO technique was firstly proposed in the active disturbance rejection control (ADRC) [25]. It has been successfully used in many challenging engineering problems [26; 27]. The main advantage of the traditional ESO is that it can handle various disturbances with little model information, and it takes both structured and unstructured uncertainties as generalized disturbances. Hence, considering the satisfactory uncertainty estimation performance, the ESO and RBFNN are fully combined in the design of the robust wing-rock control law. Meanwhile, except for uncertainties, the input saturation problem will also cause adverse effects on the performance of the wing rock control. Input saturation is a challenging problem in the wing rock control. With the increase of AOA, the aircraft will lose the control efficiency gradually, which severely degrades the closed-loop system performance. During the past decades, there exist extensive researches on the control of mechanical systems with input saturation [28; 29; 30].