An autopilot inner loop that combines backstepping control with adaptive function approximation is developed for airdrop operations. The complex nonlinear uncertainty of the aircraft-cargo model is factorized into a known matrix and an uncertainty function, and a projection-based adaptive approach is proposed to estimate this function. Using projection in the adaptation law bounds the estimated function and guarantees the robustness of the controller against time-varying external disturbances and uncertainties. The convergence properties and robustness of the control method are proved via Lyapunov theory. Simulations are conducted under the condition that one transport aircraft performs a maximum load airdrop task at a height of 82 ft, using single row single platform mode. The results show good performance and robust operation of the controller, and the airdrop mission performance indexes are satisfied, even in the presence of ±15% uncertainty in the aerodynamic coefficients, ±0.01 rad/s pitch rate disturbance, and 20% actuators faults.

Heavyweight airdrop is an essential capability of a large transport aircraft, and it is critical to the success of many military tasks, such as precision delivery of heavyweight equipment and supplies [

Over recent years, various meaningful achievements have been reported in developing advanced aircraft controllers that are compatible with airdrop tasks. Several control methods that use a linearized model at a given operating point have been proposed in the literature, including

The nonlinear system can be decoupled via exact state transformations rather than linear approximations. However, to perform perfect linearization, accurate knowledge of the plant dynamics must be available. This is not the case for airdrop flight controller design, as the complex nonlinear aerodynamic characteristics are very difficult to ascertain and model precisely [

In these cases, nonlinear adaptive control methods are called for. Adaptive backstepping control [

The main motivation for the current work is to propose a simplified controller design for the airdrop mode that can accommodate large changes in aircraft dynamics and reject uncertainties of both constant and time-varying types, as well as matched and unmatched types. The contributions of this paper are (1) a flight controller design that inherits the merits of the backstepping approach, thus solving the unmatched control problem of cargo airdrop; (2) the introduction of adaptation theories to estimate the system uncertainties, which overcomes the conservative drawbacks of [

The structure of this paper is as follows. The aircraft-cargo model with cargo extraction is presented in Section

As stated in the previous section, this paper studies the design of a flight control law for the airdrop operations. The governing equations of motions are recalled from [

The pitch aerodynamic moment is obtained as

It is observed from (

From (

Here,

The overall control system is designed using three feedback loops, as shown in Figure

Autopilot control architecture with three layers of feedback.

The steps in the adaptive backstepping control law are described below.

Consider the first equation in system (

Consider the second equation in system (

The projection-based adaptation law ensures that

The proof of stability of the control law is achieved by augmenting Lyapunov functions for the state tracking errors and parameter estimation errors. We define

Given the system in (

Consider the Lyapunov function candidate:

Such a projection-based adaptation law can bound the estimated function, and this theoretically guarantees the robustness of the controller against time-varying disturbances and uncertainties. However, the stability properties of systems using conventional adaptation laws are provable only under the assumption of constant disturbances and uncertainties [

It follows from (

To verify the proposed controller design, a 24,955 kg transport aircraft performing an airdrop of cargo weighing 8,000 kg is simulated. The cargo is initially locked at the CG of the aircraft. The aircraft is trimmed at the following conditions:

To satisfy the requirements of mission completeness and flight safety, the performance indexes for the heavyweight airdrop are given as follows [

The performance and robustness of the controller are first tested in the presence of constant disturbances, actuators faults, and uncertainties. The following three cases are simulated and compared.

In the first case,

In the second case,

In the third case,

The compact sets can be conservatively set to

Aircraft responses of the dropping process in the presence of constant disturbances and uncertainties (Cases

Next, we test the performance and robustness of the control system for the condition of time-varying disturbances and uncertainties, without retuning the parameters of the controller. Note that no actuators faults are considered in the following test (i.e.,

In this case,

In this case,

In this case,

Figure

Aircraft responses of the dropping process in the presence of time-varying disturbances and uncertainties (Cases

This paper focused on the problem of designing an aircraft controller compatible with heavyweight airdrop operations. To achieve good stability and robust characteristics, a novel flight controller combining backstepping control with adaptive function approximation was developed for pitch attitude and velocity control. This method uses projection-based adaptation strategies to achieve robustness against uncertainties. Lyapunov-based analysis shows that the controller ensures uniformly bounded steady-state tracking errors in the presence of constant actuators faults, time-varying external disturbances, and aerodynamic uncertainties. The performance of the controller was evaluated in a maximum load airdrop mission. Simulation results verified that the controller performance satisfies the airdrop mission performance indexes in the presence of pitch rate disturbances, aerodynamic uncertainties, and actuators faults. The application of this research can be used to achieve higher levels of performance and safety in practical airdrop missions.

The projection operator introduced in [

Consider a convex compact set with a smooth boundary given by

The properties of the projection operator are given by the following lemma.

Let

The projection operator

From the convexity of function

The authors declare that there is no conflict of interests regarding the publication of this paper.

This work is supported by the National Natural Science Foundation of China (Grant no. 60904038) and the Aviation Science Foundation of China (Grant no. 20141396012).