A Data-Driven Adaptive Method for Attitude Control of Fixed-Wing Unmanned Aerial Vehicles

In this paper, a real-time online data-driven adaptive method is developed to deal with uncertainties such as high nonlinearity, strong coupling, parameter perturbation and external disturbances in attitude control of fixed-wing unmanned aerial vehicles (UAVs). Firstly, a model-free adaptive control (MFAC) method requiring only input/output (I/O) data and no model information is adopted for control scheme design of angular velocity subsystem which contains all model information and up-mentioned uncertainties. Secondly, the internal model control (IMC) method featured with less tuning parameters and convenient tuning process is adopted for control scheme design of the certain Euler angle subsystem. Simulation results show that, the method developed is obviously superior to the cascade PID (CPID) method and the nonlinear dynamic inversion (NDI) method.


Introduction
The performance of the attitude controller of a fixed-wing UAV determines the quality of its autonomous flight.Some accurate mathematical model-based methods were proposed for the attitude control, for example, PID and LQR methods (linearized model based) [1] [2] [3], adaptive control method [4], feedback linearization method [5] [6] and nonlinear dynamic inversion method [7] [8].
For some other methods, perturbation within a small range is allowed [9] [10] or which satisfies the following two assumptions: is smooth; and its continuous and bounded partial derivatives ( ) , 0 A2: The system Σ satisfies generalized Lipschitz condition, that is, for two real numbers ( ) ≥ , and a positive number b, the system Σ satisfies: ( ) ( ) ( ) ( ) In above, ( ) ( ) and where, y L and u L ( ) are integers called pseudo order (PO).
Theorem [13]: For the nonlinear system Σ mentioned above that satisfies the A1 and A2, at a given condition of 0 , there must be a time-varying vector called pseudo gradient (PG) , which can transform the system Σ into the following FFDL model: and for an arbitrary time k, ( Proof: See [13].

Control Law Design
Denote ( ) d y k as reference signal.The cost function of input signal is selected as: where, 0 λ > is a weighting factor.
According to Equations ( 5) and ( 6) and let , the control law can be derived as: where, ( ] In Equation (7), only PG is unknown and needs to be updated online.
To obtain PG, the following cost function is adopted: where, 0 µ > is a weighting factor.
The reset conditions for PG are: ( ) ( ) where, ε is a small positive number.

Model for Unmanned Aerial Vehicles
The model described here is only used to generate flight data for simulations.
The nonlinear model of a UAV [21] used in the research as a studying case is listed as follows.
Rotational equations: 1 sin tan cos tan 0 cos sin 0 sin sec cos sec , , , , , , In which 1 v , 2 v and 3 v are virtual inputs which are used in the following of the design of Euler angle control law.And Translational equations:  Advances in Aerospace Science and Technology and In  Appendix E of Reference [21] also gives specific values of the above symbols.

Brief Introduction of Internal Model Control
The general structure of an IMC-based control system [18] is depicted in Figure 1; where, Thus, the two transfer functions of the above system can be obtained as: Hence, if ( ) IMC G s satisfy the following two conditions: ( ) ( ) Equations ( 19) and ( 20) turn into: Therefore, building on conditions (23) and (24), the system can track the ref- erence signal and at the same time, would theoretically not be influenced by disturbances.

Control Scheme
In this part, the attitude control laws are derived.In controlling a fixed-wing UAV, usually, deflecting aileron generates roll movement, deflecting elevator generates pitch movement, and deflecting rudder generates yaw movement.The overall control scheme is shown as: Applying Equations (( 7), (9), and ( 10)) yields the control laws of roll, pitch, and yaw angular rates, shown as: Roll rate: Pitch rate: Yaw rate: ry ry ru ry ry ry ry r r The Euler angle control laws are included in Equation ( 11) and Figure 2.

Numerical Validation
Simulation was carried out in the MATLAB environment by writing .mfile to demonstrate the feasibility and superiority of the method developed by making Figure 2. Illustration of the overall control scheme.Advances in Aerospace Science and Technology comparison with CPID and NDI methods.In the simulation, the UAV was under external disturbances.

Value of Variables and Parameters
Parameter values of algorithms are shown as (Table 1): Reference signals:

Numerical Validation
In this part, the control simulation of the small UAV was performed with the UAV under external low frequency disturbances.Simulation results are as follows: From the figures the following conclusions can be drawn: Figures 3-5 reveal that, without using any model information of the UAV, the developed data-driven method is obviously superior to the CPID method (also a data-driven method) and the NDI method (requires detailed model information) in control performance.The fundamental reason is that the method develop has a better control performance on body rate control, namely, realizing a better performance in dealing with uncertainties, which is revealed in  The essential difference between the MFAC method and the CPID method is the acquisition of dynamic model, see Equations ( 9) and (10).In the MFAC method, because of the identification of PG, a dynamic model can be obtained, which is foundation for optimal control law design, making deflection angles of control surfaces at each time point are optimum.This is also why the varying frequency of the control surfaces of the method developed is so high in Figures 6-8.The optimum control laws take into account of all uncertainties, thus even they may disturb the system, these control laws will make the output of UAV system track and approximate reference signal (see .While for the Table 1.Parameters of the control scheme.

Conclusion
The attitude control of fixed wing UAVs is studied based on the characteristics of inner loop and outer loop control systems, and realized by using MFAC to design the control law of inner loop angular velocity system and IMC to design that of Euler angle system.Firstly, the MFAC based controller has achieved inner loop angular velocity control using only I/O data without any model information.The IMC based controller has realized outer loop Euler angle control using a few tuning parameters (only one for each channel) and easy tuning process.Secondly, compared with conventional model-free CPID method and detailed model-based NDI method, it is discovered that the method developed is capable of dealing with strong coupling and highly nonlinear system such as that of fixed-wing UAVs; and it shows superiority in dealing with disturbances.
and ( )y k R ∈ represent inputand output of the system, M. L. Chen, Y. Wang DOI: 10.4236/aast.2019.41001 3 Advances in Aerospace Science and Technology respectively.y n and u n are two integers.
with respect to drag.

Figure 1 .
Figure 1.The general structure of internal model control.

Figure 9 .
Figure 9. Angular rate of data-driven method.
the above equations, m and g represent mass and gravitational constant, represent moments of inertia; φ , θ , and ψ represent roll, pitch, and yaw angle, respectively; p, q, and r represent roll, pitch, and yaw angular velocity, respectively; a represents air speed; ρ represents air density; α and β are at- I V