Neural Back-Stepping Control of Hypersonic Flight Vehicle with Actuator Fault

This paper addresses the fault-tolerant control of hypersonic flight vehicle. To estimate the unknown function in flight dynamics, neural networks are employed in controller design. Moreover, in order to compensate the actuator fault, an adaptive signal is introduced in the controller design to estimate the unknown fault parameters. Simulation results demonstrate that the proposed approach could obtain satisfying performance.


Introduction
Fault diagnosis and fault-tolerant control have been a hot issue since its significance on actual systems with actuator faults.By identifying or estimating unknown faults, the influence of faults could be considered and eliminated in controller design.Representative study on this issue could be found in [1,2], while in this paper, fault diagnosis have been studied and applied on hypersonic flight vehicle (HFV) whose control methods have attracted a lot of attention.Hypersonic flight vehicle has a much higher speed than traditional aircraft, however due to its special flight environment as well as structure, the controller design of HFV faces more challenges [3].
Currently, most studies on HFV control did not take the actuator fault into consideration.However, unknown fault such as actuator dead-zone is quite common in nonlinear systems and may cause serious consequences [4].In order to remove the influence caused by actuator fault, several approaches have been studied such as fuzzy control [5], robust control [6,7], and adaptive compensation [8,9].The model of the HFV also contains uncertain nonlinearity.As a result, studies on HFV have concentrated on the estimation of the uncertainty.Among all the methods, neural network has been widely used due to its good performance in approximating unknown nonlinear functions [10].
The model of HFV is highly nonlinear; meanwhile the altitude subsystem could be considered as a strict-feedback system [11].Therefore, back-stepping method [12][13][14] in HFV control has been widely studied.To eliminate continuous derivatives of each virtual control in back-stepping design, multiple methods such as dynamic surface control [15] and differentiator [14] are combined with back-stepping which makes the approach practicable.It is noticed that in [16] the multiple actuator fault is considered and the robust design is presented to guarantee the system stability.Moreover, in [17], the input dead-zone is considered where the Nussbaum design is included to make the adaptive design available.In this paper, we try to estimate the parameters of system fault in the hypersonic flight dynamics and then the "dynamic inversion" of the actuator fault can be included in the control signal.The whole design is using a back-stepping control law with neural networks and adaptive method.
The paper is organized in 6 parts.In Section 2, the control-oriented model (COM) of HFV considered in this paper is simply introduced.In Sections 3 and 4, the backstepping controller based on neural networks and adaptive fault estimation is designed and system stability is analysed.

Longitudinal Dynamics of HFV with Actuator Fault
The COM in [18] is employed for study: The detail of the dynamics can be found in [18].Consider the following actuator fault model: where  is the designed control input and  < 0 is an unknown fault parameter to be estimated.

Remark 1.
In HFV systems, the fault will cause error between designed and actual control input and it will result in tracking error or even flight instability.

Adaptive Back-Stepping Controller
The strict-feedback form altitude subsystem based on the hypersonic flight dynamics is considered where  1 = ,  2 =  + , and  3 = .  ,   are nonlinear functions of the HFV model.
Flight path angle (FPA) tracking error x1 is defined as where  1 is the command signal of FPA which is designed via altitude reference signal.
Step 1. Choose virtual control of  2 as where  1 is the control gain.
To avoid the continuous derivative of virtual control, the following first-order differentiator is designed: where  20 and  21 are positive designed parameters.
Step 2. Define the tracking error x2 : Choose virtual control of  3 as where  2 is the control gain.
The following first-order differentiator is designed: where  30 and  31 are positive designed parameters.
Step 3. Define the tracking error x3 : Due to the uncertainty caused by imprecise model, the nonlinear function  3 may be unknown.Therefore its NNbased estimation value is employed in the control law: where  3 is the control gain and ω3 is the estimation of the optimal NN weights  * 3 , which is obtained via adaptive law: where Γ 3 and  3 are positive parameters.Θ 3 is obtained via radial basis function.Define ω3 =  * 3 − ω3 .
In order to eliminate the influence of unknown constant , signal b is introduced in controller design to compensate the actuator: The adaptive law is designed as where Γ  and   are positive designed parameters.Define the estimation error b =  − b.
Remark 3. In previous work on HFV dead-zone fault control [17], the paper regarded the dead-zone as a part of the compound disturbance, where robust technique is employed.
In our paper, we proposed an adaptive law to estimate the unknown fault parameter and added a compensating signal in the controller so that the influence of dead-zone fault could be directly eliminated, which is shown by Figures 3 and 5 in our manuscript.
Define the velocity tracking error as where   is the reference signal.Then the following PID controller is designed:
Proof.The Lyapunov function candidate is chosen as where The derivative of   is obtained as According to the conclusion in [19], π 2 , π 3 in differentiators ( 6), ( 9) could estimate ẋ 2 and ẋ 3 to arbitrary accuracy.Therefore there exists where   ≤ |  |,   is a positive constant.Substitute the control laws and adaptive laws into (19); there exists The following inequalities hold: The derivative of   satisfies where Then all of the signals in   are uniformly ultimately bounded.This completes the proof.

Simulation
Let altitude increase 152.4 m from the initial value; notice that the signal will pass the following filter to make sure the reference signal is smooth enough for tracking: where  1 = 0.8,  2 = 0.5, and  = 0.1.The FPA command signal is obtained as where  ℎ = 0.5,   = 0.05, h = ℎ − ℎ  is the altitude tracking error, and ℎ  is the altitude reference signal.The parameters in the controller are set as  1 = 2,  2 = 2,  3 = 3,  V = 5,  V = 0.01, and  V = 0.01.The parameters of first-order differentiator and adaptive laws are chosen as  20 = 120,  30 = 100,  21 = 0.9,  31 = 0.05, Γ 3 = 0.5,  3 = 0.002,  Figure 1 indicates that the designed controller could obtain good performance on trajectory tracking.Figure 2 shows that the all state variables are bounded.The response of designed adaptive signal b and NN weights norm are shown in Figures 3 and 4. The designed control input and actual control input are shown in Figure 5, respectively.The results verify that the adaptive compensation design could eliminate the influence of actuator fault.

Conclusion
This paper proposes an adaptive back-stepping control law with NN learning for HFV control.The influence of actuator fault is eliminated by constructing an adaptive compensation signal.Meanwhile, the unknown nonlinearity is estimated by neural networks.The simulation results clearly present the consequence of the above design and verify that the approach could reach the desired tracking performance when actuator fault and model uncertainty exist.