## 3 Model and Research
### Adaptive Control Navigation System  
As was mentioned before, the FIXAR-007 aircraft undergoes no mechanical transition between the \"position hold\" and \"airplane flight\" controls. Theoretically, this means that the UAV under study can be fully controlled at any steady state required for the operation. The system designed above is unable to achieve that due to the sliding transition at the low-airspeed state. This section will investigate the application of adaptive controllers to achieve the goal of a universal control system capable of controlling an aircraft in any steady state required. [PERSON] in his paper [24] presents the most comprehensive study of the techniques used to control a multirotor system. In particular, he notes the positive aspects of the feedback linearization method also presented in [25]. Nevertheless, the most promising method for VTOL with a wide range of nonlinear perturbations is model reference adaptive control. Its variation, known as L1 adaptive controller, was well discussed in [26] and was the basis for this paper. Figure 5 illustrates the adaptive L1 controller's block diagram. The disturbance can represent any type of non-linear disturbance, such as changes in the plant caused by changes in parameter values or actuator failures. The insertion of the low-pass filter C(s)  
Figure 4: Sliding control systems: (**a**) Plane flight controllers; (**b**) Copter position hold controllers. performs two critical functions. First and foremost, it restricts the bandwidth of the control signal before it is sent to the plant. This prevents the commanding of high-bandwidth oscillating control signals, which are commonly used by fast-adapting MRAC controllers. Second, the reference model M(s) only provides the high-frequency portion of the input signal [27]. This signal generally corresponds to the portion of the disturbance d(s) that cannot be canceled given the confined actuator bandwidth. As it is supplied into the reference model M(s) along with the reference signal, this implies that the output of M(s) is the achievable system output, a realistic aim that the system should be able to fulfill given its bandwidth restrictions [28]. The resulting control system block diagram is presented in Figure 6. The system inputs are the target position and heading (Xtgt, Vtgt, and Yawtgt). They are used to calculate the position and heading error and for the dynamic limiter block, which is used to limit target velocity to maintain cruise airspeed for efficient flight. Velocity control is provided using L1 adaptive controllers described below. L1 controller and dynamic limits subsystems are presented in Figure 7.  
Figure 5: L1 adaptive controller block diagram. Figure 6: Control system design. There, \(X_{tgt}\) is desired position, \(X_{e}\) is current position, \(E_{bx}\), \(E_{by}\) and \(E_{z}\) are current position error in body axis calculated from position error \(P_{err}\). \(V_{x}\), \(V_{y}\) and \(V_{z}\) are the axis velocities and the targets are limited, using dynamic limits for \(x\) and \(y\) axis. Adaptive L1 controller is presented for every axis and shown at the Figure 7a. Dynamic limits block is described at the Figure 7b and is using distance to the point, current airspeed As and current ground speed \(V_{c}\) as inputs. The controller outputs are combined to the system output \(C=[C_{T}C_{R}C_{P}C_{Y}]^{T}\), which is transferred to the low-level stabilization system. At the Figure 7a the L1 controller model in Simulink is presented. The Tgt input is used as controller command input, same as r(t) at the Figure 5. Val input is used for current system state, named as x(t) at the Figure 5. Command output to the object u(t) is presented at the Out block. Figure 7b presents the dynamic limits block, which is used to set cruise speed \(As_{tgt}\) from simulation environment. The airspeed integral controller calculates the X speed which is applied according to the drone ground speed. This results the drone flying at the cruise speed, controlled using both airspeed and ground speeds.