# Adaptive Optimal Control of UAV Brake Based on Parameter Estimation
## 1 Introduction  
With the rapid development of unmanned aerial vehicles (UAVs), the safety and efficiency of braking systems have drawn significant attention. Due to the coupling mechanism of vertical and longitudinal torques, the highly nonlinear aircraft dynamics model, and time-varying friction, it is challenging to establish an accurate mathematical model [1]. The most common control strategy for brake systems is slip rate control, which obtains the maximum tire-road friction by regulating the slip rate. Several methods have been proposed, such as dynamic surface control [2], adaptive control [3], and extreme value search control [4]. These methods focus on adjusting the slip rate in real time to track the expected slip rate but only partially consider an excellent dynamic performance and the system's robustness. The ideal brake control should ensure that the brake operating point is always in the stable region. A rule-based control strategy has been studied in [5; 6] to avoid the brake operating point being in an unstable region, which requires high model accuracy and has a complex controller structure. In [6], an adaptive radial basis function (RBF) neural network (NN) is used to predict the upper bound of the system's complex external interference to improve the brake control's robustness. The RBFNN and error backpropagation NN were used, respectively, to approximate the nonlinear interference of reduction gear deformation and switched reluctance motor in an Electro Mechanical Actuator (EMA) [7; 8]. However, these models need to be trained offline in advance. The friction coefficient affecting the binding force between the tire and runway in brake systems is influenced by external environments and various factors, making it challenging to measure directly or calculate accurately. Previous studies [9; 10] proposed a fitting function between aircraft speed and slip rate. Still, this method requires a large amount of data for offline fitting and may not reflect the complex and changeable runway environment in real time and reduce control accuracy. Ref. [11] proposed the estimation of aircraftspeed and ground-tire joint force based on the Kalman filter, but this requires additional sensors or algorithmic complexity, leading to practical difficulties. In addition, uncertain nonlinear systems with unmodeled dynamics can also be approximated by neural networks (NNs) [12; 13; 14] and fuzzy logic systems (FLSs) [15; 16; 17]. In these methods, the unknown friction is considered part of the nonlinearity of the unknown system and approximated by NN or FLS. Ref. [18] proposes an adaptive feedforward NN control scheme to deal with the pandybat friction effect, assuming that the friction model parameters are known. Moreover, traditional adaptive control design often adopts methods such as gradient [19], least squares [20], and the modified parameter estimation method [21]. However, none of the mentioned achieve convergence of the estimated parameters and tracking errors simultaneously. A new parameter estimation method [22] is proposed to ensure the convergence of parameters. In the controller design, the adaptive rate based on the parameter estimation method ensures that the unknown parameter converges to the true value and the tracking error converges to zero. Moreover, as the balance criterion of resource control and control effect plays an increasingly important role in the design process, optimal control has attracted wide attention [23; 24; 25; 26; 27]. At present, the optimal control theory related to the design of the controller has been a significant research result. It can meet the tracking tasks or control objectives, while minimizing some performance indicators. In the classical optimal control method, the derived Riccati or Hamilton-Jacobi-Bellman (HJB) equation is usually solved offline iteratively [28]. In recent years, adaptive dynamic programming (ADP) has been proven to be a feasible method to solve the optimal control solution online [29].