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

Authors: Qi Wu; Yuyan Guo
Journal: Journal of Control Science and Engineering
 (2018)

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## Abstract

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.


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## Body

## 1. 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 [12–14] 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 Section2, the control-oriented model (COM) of HFV considered in this paper is simply introduced. In Sections 3 and 4, the back-stepping controller based on neural networks and adaptive fault estimation is designed and system stability is analysed. The simulation results of the designed controller are shown in Section 5. Finally the summary is given in Section 6.


## 2. Longitudinal Dynamics of HFV with Actuator Fault

The COM in [18] is employed for study:(1)V˙=Tcos⁡α-Dm-gsin⁡γh˙=Vsin⁡γγ˙=L+Tsin⁡αmV-gcos⁡γVα˙=q-γ˙q˙=MyyIyy.The detail of the dynamics can be found in [18].Consider the following actuator fault model:(2)δe=u-bifu≥b0if-b<u<bu+bifu≤-b,where u is the designed control input and b<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.


## 3. Adaptive Back-Stepping Controller

The strict-feedback form altitude subsystem based on the hypersonic flight dynamics is considered(3)x˙1=f1+g1x2x˙2=x3x˙3=f3+g3δe,where x1=γ, x2=γ+α, and x3=q. fi, gi are nonlinear functions of the HFV model.Assumption 2.
In altitude subsystem (3), g3 is a bounded function.Flight path angle (FPA) tracking errorx~1 is defined as(4)x~1=x1-x1d,where x1d is the command signal of FPA which is designed via altitude reference signal.Step 1.
Choose virtual control ofx2 as(5)x2d=1g1-f1-ς1x~1+x˙1d,where ς1 is the control gain.
To avoid the continuous derivative of virtual control, the following first-order differentiator is designed:(6)ϖ˙2c=-l20ϖ2c-x2dsgnϖ2c-x2d+ϖ2dϖ˙2d=-l21sgnϖ2d-ϖ˙2c,where l20 and l21 are positive designed parameters.Step 2.
Define the tracking errorx~2:(7)x~2=x2-x2d.
Choose virtual control ofx3 as(8)x3d=-ς2x~2+ϖ˙2c-g1x~1.where ς2 is the control gain.
The following first-order differentiator is designed:(9)ϖ˙3c=-l30ϖ3c-x3dsgnϖ3c-x3d+ϖ3dϖ˙3d=-l31sgnϖ3d-ϖ˙3c,where l30 and l31 are positive designed parameters.Step 3.
Define the tracking errorx~3:(10)x~3=x3-x3d.
Due to the uncertainty caused by imprecise model, the nonlinear functionf3 may be unknown. Therefore its NN-based estimation value is employed in the control law:(11)u0=1g3-ω^3TΘ3-ς3x~3-x~2+ϖ˙3c,where ς3 is the control gain and ω^3 is the estimation of the optimal NN weights ω3∗, which is obtained via adaptive law:(12)ω^˙3=Γ3x~3Θ3x¯3-Γ3δ3ω^3,where Γ3 and δ3 are positive parameters. Θ3 is obtained via radial basis function. Define ω~3=ω3∗-ω^3.In order to eliminate the influence of unknown constantb, signal b^ is introduced in controller design to compensate the actuator:(13)u=u0+b^sgnu0.The adaptive law is designed as(14)b^˙=-Γbg3x~3sgnu0-Γbσbb^,where Γb and σb are positive designed parameters. Define the estimation error b~=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(15)V~=V-Vr, where Vr is the reference signal. Then the following PID controller is designed:(16)Φ=kpvV~+kiv∫V~tdt+kdvV~˙.


## 4. Stability Analysis

Theorem 4.
Consider the HFV altitude system (3) with control laws (5), (8), (11), and (13) and adaptive laws (12) and (14); all of the error signals are uniformly ultimately bounded.Proof.
The Lyapunov function candidate is chosen as(17)VL=∑i=13Vi,where(18)V1=12x~12V2=12x~22V3=12x~32+12ω~3TΓ3-1ω~3+12Γb-1b~2.
The derivative ofVL is obtained as(19)V˙L=∑i=13x~ix˙i-x˙id-ω~3TΓ3-1ω^˙3-Γb-1b~b^˙.
According to the conclusion in [19], ϖ˙2c, ϖ˙3c in differentiators (6), (9) could estimate x˙2d and x˙3d to arbitrary accuracy. Therefore there exists(20)x˙id=ϖ˙ic+χi,i=2,3,where χi≤χim, χim is a positive constant. Substitute the control laws and adaptive laws into (19); there exists(21)V˙L=-∑i=13ςix~i2+δ3ω~3Tω3∗-ω~3Tω~3+ε3x~3+bx~3g3sgnu0-sgnu+σbbb~-b~2-∑i=23x~iχi,where ε3 is the bounded inevitable NN reconstruction error satisfying ε3≤ε3m, where ε3m is a positive constant. Define η=bg3sgnu0-sgnu; consider Assumption 1; η is a bounded signal satisfying η≤ηm, where ηm is a positive constant.
The following inequalities hold:(22)δ3ω~3Tω3∗-ω~3Tω~3≤12δ3ω3∗2-ω~3Tω~3σbbb~-b~2≤12σbb2-b~2-∑i=23x~iχi≤∑i=2312x~i2+12χim2ε3x~3≤12ε3m2+12x~32ηx~3≤12ηm2+12x~32.
The derivative ofVL satisfies(23)V˙L≤-ς1x~12-ς2-12x~22-ς3-32x~32-12δ3ω~3Tω~3-12σbb~2+12δ3ω3∗2+12ε3m2+12ηm2+12σbb2+∑i=2312χim2≤-σVL+P,where σ=min2ς1,2ς2-1,2ς2-3,δ3Γ3,σbΓb and P=(1/2)δ3ω3∗2+(1/2)ε3m2+(1/2)ηm2+(1/2)σbb2+∑i=23(1/2)χim2. Then all of the signals in VL are uniformly ultimately bounded. This completes the proof.


## 5. Simulation

Let altitude increase152.4m from the initial value; notice that the signal will pass the following filter to make sure the reference signal is smooth enough for tracking:(24)ΔrΔ=ω1ω22s+ω1s2+2ξω2s+ω22,where ω1=0.8, ω2=0.5, and ξ=0.1. The FPA command signal is obtained as(25)x1d=arcsin-khh~-ki∫h~dt+h˙rV, where kh=0.5, ki=0.05, h~=h-hr is the altitude tracking error, and hr is the altitude reference signal.The parameters in the controller are set asς1=2, ς2=2, ς3=3, kpv=5, kdv=0.01, and kiv=0.01. The parameters of first-order differentiator and adaptive laws are chosen as l20=120, l30=100, l21=0.9, l31=0.05, Γ3=0.5, δ3=0.002, Γb=0.113, and σb=0.005. Take the initial states as h0=26212m, v0=2743m/s, γ0=0deg, α0=0deg, and q0=0deg/s. The fault parameter is set as b=0.05. The results are as follows.Figure1 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.Figure 1
Altitude.Figure 2
FPA, pitching angle, and pitching rate.Figure 3
Response ofb^.Figure 4
NN weights norm.Figure 5
Control input.


## 6. 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.



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