In this paper, a new adaptive sliding mode control method is presented for the longitudinal model of a generic hypersonic vehicle subject to uncertainties and external disturbance. Firstly, an oriented-control model with mismatched uncertainties is built for a generic hypersonic vehicle. Secondly, the back-stepping technique is introduced to design a sliding mode controller with an adaptive law to adapt to the disturbance and uncertainty. Thirdly, a set of nonlinear disturbance observers are designed to estimate the lumped disturbance and compensate the sliding mode controller, and the stability of the proposed controller is analyzed by utilizing Lyapunov stability theory. Finally, simulation results show that the effectiveness of the proposed controller is validated by the nonlinear model and the proposed method exhibits promising robustness to mismatched uncertainties.
Generic hypersonic vehicles (GHVs) provide a reliable way to enter space and attract worldwide attentions in recent years. As GHVs are sensitive to physical and aerodynamic parameter changes, a concernful task is to design an efficient control system that makes the flight of GHVs feasible.
Faced with the complexities of GHV dynamics, the design methods of the guidance and control system have attracted considerable interests [
Sliding mode control (SMC) is widely used in dealing with parameter uncertainties and external disturbances for the flight of GHVs [
Meanwhile, the adaptive control approach is applied to adapt to the parameter uncertainties as well as constraint on states and control inputs [
It is well known that the disturbance observer is an efficient and active method to compensate the controller against uncertainties and external disturbances [
Motivated by the abovementioned researches, a new adaptive SMC strategy that consists of the adaptive control method, back-stepping method, and nonlinear disturbance observer method is proposed in this paper. In the proposed controller framework, a new nonlinear disturbance observer (NDO) is employed to estimate the lumped disturbances that are introduced into the sliding surface and virtual control input at each step to compensate the effects of disturbances. It is proved that the closed-loop system is asymptotically stable here. Finally, simulation results show that the proposed method has a good disturbance rejection performance without sacrificing the nominal control performance.
The key innovations are listed below:
A new adaptive SMC method is proposed to meet the flight performance for the GHV with highly nonlinear and mismatched uncertainties. A nonlinear disturbance observer is introduced into the control system to estimate the lumped uncertainties and external disturbance to compensate the sliding mode controller. The compute explosion problem is solved in the back-stepping method by utilizing the adaptive controller.
The longitudinal dynamics of a GHV can be described with a set of differential equations composed by velocity
The engine dynamics is modeled by a second-order system
In this paper, the aerodynamics and physical coefficients are simplified around the nominal cruising flight. The terms of
The velocity is mainly related to throttle setting
The model of a GHV described by (
The lumped disturbances are bounded and the maximum value is as follows:
The variable states are chosen as Velocity subsystem:
Altitude subsystem: the tracking error is described by
And the composited controller, consisting of an adaptive back-stepping method and nonlinear disturbance observer, is designed for a GHV. The NDO is added into the controller for improving performance of the controller.
The proposed controller utilizes the back-stepping method while the virtual control inputs can be obtained at each step. The designed adaptive law can compensate for the modeling uncertainties effectively.
The velocity tracking error can be defined as
A new sliding mode is chosen as
The adaptive parameter is chosen as
When the differentiation of tracking error is taken into the dynamics, the equation can be obtained as
The command of throttle setting can be designed as
The back-stepping method is used to design the controller for altitude subsystem.
(the control input design for the flight path angle). The tracking error in this step can be defined as
For making the tracking error converge to zero, the sliding mode surface can be designed as
The adaptive parameter is chosen as
If the differentiation is taken into the sliding mode, it can be obtained as
It can be obtained as
(the control input design for the pitching angle). The tracking error and sliding mode surface can be defined as follows:
The adaptive parameter is chosen as
The equation can be transformed when the derivation of
It can be obtained as
(the control input design for the pitching rate angle). The tracking error
The adaptive parameter is chosen as
The equation can be transformed when the derivation of
The slide mode controller can be obtained as follows:
Inspired by the works of Zhang et al. [
A nonlinear disturbance observer for (
Similarly, an NDO for (
A nonlinear disturbance observer for (
A nonlinear disturbance observer for (
Similarly, a nonlinear disturbance observer for (
The stability of the closed control system is proved by the Lyapunov stability theory.
Firstly, a Lyapunov function is chosen as
The derivation of the Lyapunov function is obtained as
(stability analysis for the velocity subsystem). The derivation of the Lyapunov function for system (
If the parameters of the controller (
(stability analysis for the angles)
Flight path angle:
If Pitching angle:
If Pitching rate:
If The Lyapunov stability is proved.
The Lyapunov function is chosen as
The derivation of the Lyapunov function can be obtained as
(stability analysis for velocity). The Lyapunov function for system (
The derivation of the Lyapunov function can be obtained as
The
It can be obtained as
(stability analysis for the altitude subsystem).
Altitude:
The derivation of Flight path angle: the Lyapunov function is chosen as
The derivation of
The parameter Pitching angle: the Lyapunov function is chosen as
The derivation of
If the parameters Pitching rate: the Lyapunov function is chosen as
The derivation of
If the
Then it can be obtained as
The convergence of tracking error is proved now.
In this section, the effectiveness and performance of the developed controller are verified by simulations. The longitudinal model is considered under its cruise flight condition. The initial values are chosen as
The controller parameters are chosen as
The external disturbances are chosen to be
In this part, the square wave and step are applied in command generator, respectively.
The square wave is adopted to prove the effectiveness of controller. The uncertainties are added into this system. The simulation results are shown in Figures
Velocity command, response, and controller. The uncertainty terms are added into the system.
Altitude command, response, and controller. The uncertainty terms are added into the system.
It is obtained from Figure
The proposed controller is compared to the back-stepping method [
Altitude command, response, and controller. The uncertainty terms and external disturbances are added into the system.
Lumped disturbances estimated by NDO.
Velocity command, response, and controller. The uncertainty terms and external disturbances are added into the system.
Lumped disturbance estimated by NDO.
Command, response, and controller. The uncertainty terms and external disturbances are added into the system.
Lumped disturbance estimated by nonlinear disturbance observer.
The performance of the controller is proved with the existence of uncertainties and external disturbances. At
In order to verify the effectiveness of the controller against parameter perturbation, the coefficients of deflection are chosen as
Altitude command, response, and controller. The uncertainty terms and external disturbance are added into the model.
Lumped disturbance estimated by nonlinear disturbance observer. The parameters deflection, uncertainty terms, and external disturbance are added into the model.
The system under the proposed controller exhibits good performance against both positive and negative parameter perturbation.
A new adaptive sliding mode control method combined with the nonlinear disturbance observer is proposed to solve the tracking problem for the longitudinal model of a GHV. The compute explosion problem is solved by utilizing the new adaptive control algorithm. In addition, the proposed controller for a GHV model has achieved favorable results in terms of robustness without the cost of sacrificing the nominal control performance. Finally, the performance of the proposed control algorithm has been demonstrated by simulation results.
The authors declare that they have no conflicts of interest.
This work was supported by the National Natural Science Foundation of China (no. 61703339).