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The inversion design approach is a very useful tool for the complex multiple-input-multiple-output nonlinear systems to implement the decoupling control goal, such as the airplane model and spacecraft model. In this work, the flight control law is proposed using the neural-based inversion design method associated with the nonlinear compensation for a general longitudinal model of the airplane. First, the nonlinear mathematic model is converted to the equivalent linear model based on the feedback linearization theory. Then, the flight control law integrated with this inversion model is developed to stabilize the nonlinear system and relieve the coupling effect. Afterwards, the inversion control combined with the neural network and nonlinear portion is presented to improve the transient performance and attenuate the uncertain effects on both external disturbances and model errors. Finally, the simulation results demonstrate the effectiveness of this controller.

For a general longitudinal model of the airplane, the flight control law tends to be designed in terms of the linearized model corresponding to the given trim points. On this basis, the proportional-integral-derivative (PID) controller is used to achieve the desired flight performance under the assumption that the short-period dynamics are faster than the phugoid mode [

In some studies, the inversion control design is realized by adopting feedback signals to offset inherent coupling dynamics, thus guaranteeing the satisfactory decoupling control ability. In particular, an investigation example was illustrated using the dynamic-inversion methodology for the linear model of a generic X-38 type reentry vehicle [

In this paper, the flight control law is proposed using the neural-based inversion design method and nonlinear compensation for a general longitudinal model of the airplane. In particular, the dynamic-inversion control can relieve the strong coupling effects regarding the model dynamics, whereas the neural-based compensation is helpful in improving the robust performance to suppress the uncertain disturbances. There are three aspects of this problem that have to be addressed. First, the inversion design method is introduced to convert the nonlinear mathematic model to the equivalent model accurately. After that, the inversion control law is designed to stabilize the system and relieve the coupling effects. Furthermore, the compensation using the neural network and nonlinear portion is introduced to improve the transient performance and system robustness. Lastly, an airplane example is provided to verify the feasibility of the proposed controller.

The longitudinal motion of the airplane involves only vertical motion parameters and aerodynamic actions, so the airplane dynamics can be described based on the velocity coordinate. While the elevator deflection (

In (

Also, the gravity constant (

Based on (

For any

For the nonlinear model of the airplane in (

First, selecting

Considering the presence of

Equation (

In (

Similarly, the differentiation of

Substituting (

With the integration of (

If the matrix

As long as the output of (

Let the inversion control law be [

In (

The approximate linearization approach is considered that the airplane movement is associated with small deviations from the steady flight state. And all high-order dynamics are regarded to be small such that their actions are negligible in contrast to the first-order model dynamics. When the first-order terms are kept in (

Correspondingly, the inversion control law based on this approximate model is expressed by

In (

Improving the transient performance is very important for the aircraft model to follow the expected command rapidly without deviating from the design point. Alternatively, the system robustness will guarantee flight stability with the existence of the large model uncertainties and external disturbances. As a result, the transient performance and system robustness can be an issue for the aircraft model to realize the challenging tasks.

To this end, this work combines the above dynamic-inversion control with the compensation of the neural network and nonlinear potion in order to ensure system robustness and self-adaption and to improve the transient performance. This is because the inversion control is sensitive to modeling errors due to the need of the detailed knowledge of the nonlinear airplane model. In this case, the application of the neural network can alleviate this sensitivity, and the nonlinear portion can ameliorate the transient performance associated with the inversion controller [

First, the inversion design idea based on the feedback linearization principle transforms the nonlinear model in (

Afterwards, the inversion error is defined by

Based on (

Furthermore, the pseudo-control vector consisting of the proportional controller, command derivative, and adaptive signal is selected [

After substituting (

By selecting the suitable control parameters, (

To this end, the adaptive compensation includes the nonlinear portion and output of the neural network, and it is provided as

In addition, the update laws of the weights

Let

Also, the errors between

After taking the derivative with respect to (

Equation (

Therefore, when

These functions,

In general, the flight control law using

In particular, the structure diagram of this robust adaptive control system is shown in Figure

Structure diagram of robust adaptive control for the airplane.

Figure

In this study, the airplane properties are used in [^{3}, ^{2}, and

First, the control goal is that the speed

Response curves with regard to command signals.

Figure

Furthermore, the change curves corresponding to the angle of attack and control inputs are demonstrated in Figure

Change curves of angle of attack and control inputs.

From Figure

Compensation outputs.

Figure

Furthermore, we assume that the model parameter matrix

Response curves using adaptive control law in the uncertain condition.

Change curves of angle of attack and control inputs in the uncertain condition.

Figures

This paper proposes a control law using the neural-based inversion design approach with the nonlinear compensation for a general longitudinal model of the airplane. First, the nonlinear model of the airplane is established, and the balance equation is gotten for the given altitude and velocity. Next, the inversion control law is designed based on the feedback linearization principle. Furthermore, the control law in combination with the neural network and nonlinear portion is proposed. For this controller, the inversion control can realize the decoupling operation concerning the nonlinear model dynamics, whereas the adaptive outputs of the neural network and nonlinear portion can improve system robustness, transient performance, and adaptability. Finally, the simulation is conducted to show that the proposed control methods are feasible for a general longitudinal model of the airplane.

The authors declare that there are no conflicts of interest regarding the publication of this paper.

This work is supported by the Fundamental Research Funds for the Central Universities (no. NJ20160052).