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Aiming at the longitudinal motion model of the air-breathing hypersonic vehicles (AHVs) with parameter uncertainties, a new prescribed performance-based active disturbance rejection control (PP-ADRC) method was proposed. First, the AHV model was divided into a velocity subsystem and altitude system. To guarantee the reliability of the control law, the design process was based on the nonaffine form of the AHV model. Unlike the traditional prescribed performance control (PPC), which requires accurate initial tracking errors, by designing a new performance function that does not depend on the initial tracking error and can ensure the small overshoot convergence of the tracking error, the error convergence process can meet the desired dynamic and steady-state performance. Moreover, the designed controller combined with an active disturbance rejection control (ADRC) and extended state observer (ESO) further enhanced the disturbance rejection capability and robustness of the method. To avoid the differential expansion problem and effectively filter out the effects of input noise in the differential signals, a new tracking differentiator was proposed. Finally, the effectiveness of the proposed method was verified by comparative simulations.

Air-breathing hypersonic vehicles (AHVs) are a new type of aircraft, which fly at speeds greater than Mach 5 at near space altitudes. AHVs exhibit fast flying speeds, strong penetration abilities, and long combat distances. It is difficult to detect and intercept AHVs. AHVs have strong survivability, and they have outstanding advantages in strategy, tactics, and cost-effectiveness compared to traditional aerospace vehicles [

Most of the previous research on the modelling and control of AHVs has mainly focused on the AHV’s longitudinal motion plane. On the one hand, the longitudinal motion model is complex enough to require flight control. On the other hand, due to the scramjet engine’s extreme sensitivity to the flight attitude, AHVs should avoid horizontal manoeuvres during actual flight [

Although the methods in the above literature achieved certain control effects, the research focus was on the robustness and steady-state performance of the AHV closed-loop control system, neglecting the dynamic performance of the control system. However, AHV’s super-manoeuvrable, large-envelope, and hypersonic flight demands better dynamic performance of the control system than any other existing aircraft. In most cases, a small control delay will cause significant errors in the hypersonic flight. Therefore, to guarantee the robustness and steady-state accuracy of the control system, more attention should be paid to the dynamic performance and real-time performance of the control system. To consider both the steady-state and dynamic performances, the concept of prescribed performance control was proposed by Charalampos and George [

In view of the deficiencies of the research in the above studies, an elastic hypersonic vehicle was taken as the research object. In this study, a new prescribed performance function was designed based on the hyperbolic cosine function, which avoided the singular control problem caused by the improper initial value setting. Thus, the steady-state performance and dynamic performance of the control system could be guaranteed. Meanwhile, active disturbance rejection control was introduced and an ESO was designed for each unknown nonaffine function in the AHV system [

To better describe the longitudinal motion of the AHV, American scholar Parker used the research conclusions of Bolender and Doman [

Equations (

AHV force diagram.

Equations (

The AHV prescribed performance control method based on the nonaffine model is studied in this paper. To make the tracking error convergence processes meet the desired dynamic and steady-state performance, the following new performance function

The prescribed performance is defined as follows:

The prescribed performance defined by Equation (

For any arbitrary bounded

When the

Prescribed performance defined by Equation (

The transformed error

If

Because

Based on Equation (

Substituting

Therefore, Theorem

The control law below will be designed based on the transformed error

On the one hand, since the thrust

Based on the previous studies [

Similarly, the altitude subsystem of the AHV can be expressed as the following nonaffine pure feedback system:

Most previous studies on AHV control issues designed the control law based on the affine model. However, the AHV motion model is nonaffine. If the nonaffine model of the AHV is forcibly simplified to an affine model, the loss of certain key dynamics is inevitably. The designed control law has the risk of partial or complete failure. The proposed control law in this paper will be designed based on the nonaffine model (Equations (

To design the active disturbance rejection control law, the extended state observer is applied to estimate the uncertainty of the AHV model and the external disturbance. The following system is considered:

Aiming at the system specified by Equation (

The proof process is shown in the appendix.

Some signals in the control law design process are often difficult to obtain by the model construction. Many scholars have proposed using the tracking differentiator to estimate the signal [

If the new TD (Equation (

The certification process is detailed elsewhere [

Compared with traditional tracking differentiators [

The velocity subsystem (Equation (

The velocity tracking error can be defined as follows:

The first derivative of Equation (

According to Equation (

The first derivative of Equation (

To estimate the unknown term

The active disturbance rejection control law

Considering the velocity subsystem (Equation (

Substituting Equation (

The following Lyapunov function was selected:

The first derivative of

Combined with Equation (

Considering that

With

Combining Equations (

The above proves that the transformed error is bounded. According to Theorem

Considering the altitude subsystem (Equation (

Furthermore, the first derivative of

The flight-path angle command is defined as follows:

If

The two characteristics roots of Equation (

For the rest of the altitude subsystem (Equation (

For

Combined with the concept of active disturbance rejection,

The flight-path angle tracking error is defined as follows:

The first derivative of Equation (

The mathematical expression of

To estimate the unknown term

The pitch angle command

For

The pitch angle tracking error is defined as follows:

If the AHV is affected by an external disturbance,

The first derivative of

To avoid the differential expansion problem and effectively filter out the effects of input noise in the differential signals, the new TD proposed in this paper is applied to estimate

To estimate the unknown term

The pitch rate command

For

The pitch rate tracking error is defined as follows:

If the AHV is affected by an external disturbance, combined with the concept of active disturbance rejection, the equation

The first derivative of

The following new TD is applied to estimate

To estimate the unknown term

The control law

Considering the altitude subsystem (Equation (

The following Lyapunov function is selected:

The first derivative of

According to Equations (

Combining Equations (

With

According to Equation (

Taking the longitudinal motion model of the AHV (Equations (

Initial values of the AHV state variables.

Parameter | Value | Unit |
---|---|---|

2500 | m/s | |

27000 | m | |

0 | ° | |

1.5 | ° | |

0 | °/s | |

0.29 | — | |

0.26 | — |

The velocity reference command

When the control algorithm proposed in this paper was used for the simulation, the prescribed performance parameters were selected as follows:

The prescribed performance-based active disturbance rejection control (PP-ADRC) method proposed in this paper with the robust back-stepping control method (RBC) from a previous study [

The simulation results for Scenario

The PP-ADRC was compared with the neural back-stepping control method (NBC) from a previous study [

Simulation results for Scenario

Velocity tracking performance

Velocity tracking error

Altitude tracking performance

Altitude tracking error

Flight-path angle

Estimation of

Elastic state

Elastic state

Fuel equivalence ratio

Elevator angular deflection

The simulation results of the Scenario

Simulation results for Scenario

Velocity tracking performance

Velocity tracking error

Altitude tracking performance

Altitude tracking error

Flight-path angle

Estimation of

Elastic state

Elastic state

Fuel equivalence ratio

Elevator angular deflection

An active disturbance rejection control for an AHV based on the prescribed performance function is proposed in this paper. The proposed method guarantees the stability of the AHV closed-loop control system. The desired dynamic and steady-state performances of the convergence process of the tracking error were ensured

In the controller design process, the adopted active disturbance rejection method and extended state observer further enhanced the capacity to resist the disturbances, which guaranteed the robustness of the method

The simulation results in the paper proved the effectiveness of the proposed method. The comparison with the related publications showed that the dynamic and steady-state performances of the proposed method were superior

Based on Equations (

The following Lyapunov function was selected:

The first derivative of

Values of

The following compact set is defined as follows:

According to Equation (

The data used to support the findings of this study are available from the corresponding author upon request.

The authors declare that they have no conflicts of interest.

This study was supported by the National Natural Science Foundation of China (Grant no. 61573374 and no. 61703421). The funding did not lead to any conflict of interests.