# Multivariable Adaptive Super-Twisting Guidance Law Based on Barrier Function
## 1 Introduction  
Precision guidance has always been a key technology in missile control field with the development of flight vehicles. Due to the convenience of implementation and the accuracy of interception, the proportional navigation guidance (PNG) law and its variants [1, 2, 3, 4] have been widely used in various engineering practices. PNG methods are designed to keep the line-of-sight (LOS) angular rate converge to zero to intercept targets. However, in practice, the maneuvers of the target, desired constraints and the disturbances of measurement may lead to poor results of PNG. Hence, a series of modern guidance laws based on advanced control theory have been proposed to address the problems. References [5, 6] introduced optimal control theory into waypoint-following guidance problems and calculated the guidance law by solving optimal control problems. On the basis of H-infinite theory, reference [7, 8] proposed robust guidance laws by regarding unpredictable target maneuvers as bounded unknowns. Reference [9] disposed of the constraints and the performance index by using the convex optimization method to obtain guidance gains. Utilizing model predictive control, reference [10] designed a suboptimal terminal angle constrained guidance law. At the same time, due to the inherent properties of strong robustness against external disturbances and uncertainties of systematic parameters, the sliding mode control (SMC) [11, 12, 13, 14] theory was applied to guidance law design, especially in the scenarios with constraints. To obtain a global stable guidance law, the LOS angles are utilized to construct the sliding manifold so that the sliding manifold could converge to zero or theneighborhood of zero. In [15], a three-dimensional adaptive finite-time guidance law that can accelerate the convergent rate was presented. After that, reference [16] presented an analytical solution with input saturation for [15]. Reference [17] proposed an adaptive reentry guidance law for the hypersonic vehicle with a specified impact angle. By employing a second-order sling mode observer, reference [18] proposed a nonsingular terminal sliding mode guidance law with autopilot lag consideration. The integral sliding mode guidance law presented in [19] addressed the problem of steady-state error of the traditional SMC. Reference [20] implemented an optimization design by using the neural network, which improved the fuzzy variable structure of the sliding mode. Moreover, ref. [21] designed an adaptive anti-saturation integral sliding mode guidance law by introducing an auxiliary system to solve the problem of input saturation. However, to keep the sliding surface on zero or in the neighborhood of zero, the control commands of sliding mode controllers are always discontinuous, sometimes even chatter distinctly. The phenomenon of chattering may lead to undesirable implementation of the actuators [22]. To eliminate the chattering of sliding mode guidance law, the super-twisting algorithm (STA) [23] controller became popular. Denoting the target accelerations as the disturbances, the STA controller can reach the sliding mode by designing the gains of the super-twisting controller to counteract the disturbances. The dynamical adaptation gains are expected to be as small as possible meanwhile sufficient to offset the disturbances. Some super-twisting methods are based on the usage of the filtered value of the equivalent control as an estimation of the disturbance [24; 25; 26]. Moreover, some are based on increasing gains [27; 28]. Additionally, super-twisting methods are applied in the area of guidance law design. Reference [29] designed an STA-like guidance law considering actuator faults based on the equivalent control value. References [30; 31] designed adaptive STA guidance laws which could be convergent in finite time based on the increasing gains. Introducing a faster convergence error form, ref. [32] proposed a modified adaptive STA guidance law considering seeker delays.