# Fault-Tolerant Control of Hypersonic Vehicle Using Neural Network and Sliding Mode

Authors: Peng Cui; Changsheng Gao; Wuxing Jing; Ruoming An
Journal: International Journal of Aerospace Engineering
 (2022)

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## Abstract

In this paper, the tracking control of the air-breathing hypersonic vehicle with model parameter uncertainties and actuator faults is studied. Firstly, a high-order linearization model is used to build an adaptive terminal sliding mode that eliminates chattering and provides increased robustness for unknown disturbances in the system. Second, a fault-tolerant control method mixing the radial basis function neural network with adaptive sliding mode control is suggested, with the addition of a hyperbolic tangent function to avoid controller input saturation. Finally, the stability of the controller is proved strictly by the Lyapunov theory, and the robustness and effectiveness of the controller are further verified by numerical simulations of the longitudinal model of the hypersonic vehicle.


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## Body

## 1. Introduction

Air-breathing hypersonic vehicles (ABHV) are an appealing technology for military and low-cost space access needs. The aerodynamic properties of ABHV make flight control more susceptible to parameter errors and actuator faults than traditional flying aircraft at high attitude and Mach numbers. Strong fault-tolerant control systems should therefore be created for ABHVs to guarantee flight safety [1]. Various researchers have been researching hypersonic vehicle modelling since the 1990s, which is used to describe the dynamic characteristics of hypersonic vehicles. Marrison and Stengel suggested a high-fidelity longitudinal dynamics model based on the universal hypersonic vehicle developed at NASA Langley Research Center [2]. Strong nonlinear, time-varying, and nonminimum phase characteristics are the main features of a hypersonic vehicle. The longitudinal dynamics of ABHV are linearized to higher order [3]. Many sophisticated control techniques [4], including adaptive control [5], backstepping control [6], predictive control [7], fuzzy control [8], and feedback linearization control [9], can be employed to form fault-tolerant controllers for ABHV.Sliding mode control is one of the most appealing nonlinear control tools among all types of control approaches. The feedback linearization model is used to construct an adaptive sliding mode controller with model uncertainty [10]. Because the linear sliding mode surface can only converge exponentially, the integral sliding mode controller is employed to achieve finite-time state convergence [11]. Based on this finite-time integral sliding mode surface, a simplified smooth second-order sliding mode approach law can realize synchronous convergence to zero [12]. In addition, finite-time convergence of states can be realized by terminal sliding mode, and unknown actuator faults can be estimated by adaptive law [13]. In the meantime, the sliding mode observer (SMO) may be used to assess actuator fault information and model uncertainty, and the estimation error converges in a finite amount of time [14] .In actuality, system uncertainty and actuator faults are difficult to model. For dynamics that cannot be modeled, neural networks have been shown to be able to approximate any continuous function [15]. The typical way is always to build the feedback controller according to the model and then use the neural network to estimate the system disturbance to adjust the control input [16] . It is easy to convert model parameter uncertainties into system disturbances using neural network compensation to increase the robustness of the feedback controller [17]. In some cases [18], neural networks may approximate the model function and gain function for a nonlinear system, hence increasing robustness [19]. Further, the control input may be defined as a new object function [20], and a radial basis function neural network may also be employed to approximate it. As a result, the uncertainty induced by actuator failure may be predicted using a neural network [21, 22].Despite substantial advances in fault-tolerant hypersonic vehicle management, the accompanying literature analysis demonstrates that there is still room for improvement [23, 24]. For example, just the actuator failure was analyzed, ignoring bias and outage failures. Although the failure observer’s assistance was used to estimate failure information, the overall system’s stability was not revealed, making it impossible to assess the process’s condition at any time. We investigate the failure tracking control of a hypersonic vehicle employing terminal sliding mode theory and a neural network approach in this research [25–29]. In this paper, a fault-tolerant control (FTC) method combining radial basis function neural network (RBFNN) and adaptive terminal sliding mode (ATSM) is proposed, which can track the ABHV trajectory of an ABHV in the presence of air density, mass, and moment of inertia uncertainties as well as actuator faults. The main contributions of this paper are as follows:
(1)
This scheme resolves the coupling problem in MIMO systems like ABHV, which has theoretical implications(2)
The RBFNN may approximate the actuator’s fault characteristics and provide the hyperbolic tangent function to prevent saturation of the control input, thus assuring the control system’s practical application value(3)
This fault-tolerant control scheme ensures the ABHV closed-loop control system’s stability and allows for a fast and effective reaction to the reference trajectoryThe arrangement of this paper is as follows:In Section2, the longitudinal dynamics model of ABHV is input and output linearized, as well as actuator fault modelling and control input antisaturation processing. Then, Section 3 introduces the prior knowledge of RBFNN and the relevant lemma needed for controller design. The FTC scheme based on RBFNN is designed for the control model in Section 2. In Section 4, the ABHV model with and without faults is numerically simulated. Finally, this paper is summarised in Section 5.


## 2. Problem Statements

The longitudinal model of the ABHV is described as follows [8]:
(1)V̇=Tcosα−Dm−gsinγ,ḣ=Vsinγ,γ̇=L+TsinαmV−gV−Vrcosγ,α̇=q−γ̇,q̇=MyIyy,where V is the flight velocity, α is the angle of attack, h is the flight altitude, γ is the flight path angle (FPA), and q is the pitch rate. Iyy is the moment of inertia, m is the ABHV’s mass, g=μ/Re+h2 is the gravitational acceleration, and Re is the earth’s radius. L is the lift force, T is the thrust of engine, D is the resistance, and M is the pitching moment. These forces, torques, and coefficients of ABHV are as follows:
(2)L=0.5ρV2SCL,CL=0.6203α,D=0.5ρV2SCD,CD=0.6450α2+0.0043378α+0.003772,T=0.5ρV2SCT,CT=0.02576ϕ,ϕ<1,0.0224+0.00336ϕ,ϕ≥1,M=0.5ρV2Sc¯CM,CM=CMα+CMq+CMδ,CMα=−0.035α2+0.036617α+5.3261×10−6,CMq=c¯2Vq−6.796α2+0.3015α−0.2289,CMδ=0.0292δe−α,where δe is the elevator angular deflection and ϕ is fuel-to-air ratio. In fact, the actuator output is bounded and can be expressed as ϕ∈ϕmin,ϕmax and δe∈δmin,δmax, where ϕmin, ϕmax, δmin, and δmax are known constants. A second-order system describes the dynamics of the engine [11]:
(3)ϕ¨=−2ζωnϕ̇−ωn2ϕ+ωn2ϕc,where ϕc signifies the control input demand, ωn means the engine dynamics’ natural frequency undamped, and ζ is the damping ratio. Because the control input is bound by Equation (4), the following hyperbolic tangent function is used to avoid control input saturation:
(4)ϕL=ϕmin−ϕmax2tanh2ϕc−ϕmin+ϕmaxϕmin−ϕmax,δL=δmin−δmax2tanh2δe−δmin+δmaxδmin−δmax.Actuator faults can generally be described as a linear model:(5)uF=kFuL+dF,where uF is the control signals under faults, uL=ϕLδLT is the saturation input, kF=diagkF1kF2 is the fault gain matrix, and dF=dF1dF2T is the vector of fault deviation.


## 3. Controller Design

Lemma 1 (approximation based on RBFNN).
The following RBFNNs can estimate an unknown smooth nonlinear functionfx on a compact set:
(6)fx=W∗Thx+ε,where x is the input of network, W∗T is the ideal weight matrix, hx is the convolution vector, and ε is the approximation error. hx=h1h2⋯hlT is chosen as the convolution formula:
(7)hjx=expx−cj22bj2,j=1,2,⋯l,where cj and bj are the center and width of the j-th neuron, respectively, and l is the node number of the neural networks.In Figure1, the FTC scheme framework combining RBFNN and ATSM proposed in this paper is shown.Figure 1
Control scheme.In order to track ABHV’s reference velocityVd and reference altitude hd, the state variable is set to x=Vγαqϕϕ̇hT, the input variable to u=δeϕcT, and the output variable to y=VhT. The tracking errors can be defined as eV=V−Vd and eh=h−hd.The high-order linearization model of ABHV is shown as follows:(8)V3h4=fVfh+b11b12b21b22ϕcδe,where fV, fh, b11, b12, b21, and b22 are defined in [11]. The auxiliary variables [13] are introduced as
(9)e1=eV+ėVeh+ėh+e¨h,e2=ė1=ėV+e¨Vėh+e¨h+e⃛h.Considering the influence of parameter uncertainty and external disturbance, the system can be defined as follows:(10)ė1=e2,ė2=fx+gxu+d,where fx, gx, and d are known in [13] as follows:
(11)fx=Vd3hd4−fVxfhx+e2x−ėVxėhx,gx=b11b12b21b22,d=d1d2,u=φcδe.Assumption 2.
There exists a positive valuedu with an unknown upper bound, such that the system disturbance d satisfies:
(12)d∞≤du.Because the state variables and aerodynamic parameters are practically bounded, the disturbanced is a function of them and has a certain range of values.In order to satisfy Lemma1, a terminal sliding manifold is chosen as follows:
(13)s=e2+kae1+kbPe1,qp,Pe1,qp=signe11e11q/psigne12e12q/pT,where s=s1s2T, e1=e11e12T, e2=e21e22T, ka, and kb are positive constants. When q and p are positive odd numbers, and p>q, the sliding mode is Hurwitz.The law of the sliding mode approach is as follows:(14)ṡ=−k1signs1signs2T−k2s,where k1 and k2 are positive constants.Since the system disturbanced cannot be accurately obtained, a robust term d^ is introduced in the form of
(15)d^=d^u2s+14λ12s,where d^u is the estimated value of du and λ1 is a positive constant. The estimated error of du is defined as d~u=du−d^u.Considering system (13), create a Lyapunov function by following these steps:
(16)L1=12sTs+12d~u2.The derivative ofL1 can be obtained:
(17)L̇1=sTṡ+d~ud~̇u=sTfx+gxu+d+kae2+kbṖe1,qp−d~ud^̇u,Ṗe1,qp=qpsigne11e11q/p−1e21signe12e11q/p−1e22T.Applying Assumption2 and the approach law (14), the ATSM system’s input is constructed as follows:
(18)us=g−1−f−d^−kae2−kbṖe1,qp−k1signs1signs2T−k2s.By substituting the derivative from Equation (18) into Equation (17), the following may be deduced:
(19)L̇1≤−k1s1−k2sTs+dus1−sTd^u2s+14λ12s−d~ud^̇u.The adaptive law of parametersd^u is chosen as
(20)d^̇u=λ12sTs−λ2d^u,where λ2 is positive constant.Substituting Equation (20) into Equation (19), the following inequality relation can be obtained:
(21)dus1−d^u2sTs−14λ12sTs−d~uλ12sTs≤du4λ1.Therefore, Equation (19) can be simplified as
(22)L̇1≤−k1s1−k2sTs+du4λ1−12λ2d~u2+12λ2du2−12λ2d^u2≤−k2sTs−12λ2d~u2+du4λ1+12λ2du2≤−mink2,12λ2L1+du4λ1+12λ2du2.Taking into account the actuator faults and saturations, the system (10) can be represented as follows:
(23)ė1=e2,ė2=fx+gxu−σx+d,where σx=u−uF are the errors between the control inputs and the control signals under faults.The RBFNN is used to estimateσx the disturbance term given by the following:
(24)σ=W∗Thx+ε.The estimate error ofσ can be described as
(25)σ~=σ−σ^=W~Th+ε,where W^ is the estimate value of W∗ and W~=W∗−W^ is the estimated error.Assumption 3.
The approximation errorε is supposed to be bound by an unknown positive constant ε0 as follows:
(26)ε∞≤ε0.According to Equation (18), the actual input of the FTC scheme is constructed as follows:
(27)u=g−1−f−d^uTd^us−14λ12s−kae2−kbṖe1,qp−k1signs1signs2T−k2s+W∧Thx.For the system (23), the Lyapunov function L2 corresponding to Equation (16) is created:
(28)L2=L1+12trW~TΓ−1W~.The derivative ofL2 can be found as follows:
(29)L̇1=sTṡ−d~ud^̇u−trW~TΓ−1W^̇.Applying Equations (20), (23), and (27), the time derivative of L3 can be written as
(30)L̇2≤−k1s1−k2sTs−12λ2d~u2+du4λ1+12λ2du2−sTgW~Th−sTgε−trW~TΓ−1W^̇.The adaptive adjustment law forW^ is chosen as follows:
(31)W^̇=−ΓhsTg+λ3W^,where λ3 is a positive constant and Γ is a diagonal positive matrix.Substituting Equation (31) into Equation (30), the time derivative of L3 can be written as
(32)L̇2≤−k1s1−k2sTs−12λ2d~u2+du4λ1+12λ2du2−sTgW~Th−sTgε−trW~ThsTg+λ3W~TW^.When k1>ε0Tg∞, Equation (32) can be rewritten as:
(33)L̇2≤−k1s1−k2sTs−12λ2d~u2+du4λ1+12λ2du2+λ3trW~TW^.According to the unequal relationship,(34)2W~TW~=W~TW∗−W^+W∗−W∧TW=W~TW∗−W~TW~+W∧TW∗−W∧TW^=W∗TW∗−W~TW~−W∧TW^,trW~TW^=W∗2−W~2−W^2≤12W∗2−W~2.Substitute into Equation (32) to obtain
(35)L̇2≤−k2sTs−12λ2d~u2−12λ3W~2+du4λ1+12λ2du2+12λ3W∗2.Whenλ4 is the largest eigenvalue of the matrix Γ−1, trW~TΓ−1W~≤λ4W~2, the following relationships exist:
(36)L̇2≤−kLL2+Δ,where kL=mink2,1/2λ2,λ3/λ4 and Δ=du/4λ+1/2λ2du2+1/2λ3W∗2.The solution to formula (35) is as follows:
(37)L2t≤L20e−kLt+ΔkL≤L20+ΔkL∀t≥0.According to Equations (22) and (37), it can be seen that all signals in the closed-loop control system are bounded. The state tracking errors eV, eh, e1, and e2, as well as the estimation errors of the disturbance’s upper bound du and the neural network weights W∗, are bounded. Based on Lyapunov theory, the stability of this closed-loop system is uniformly ultimately stable and exponentially converge to a bounded neighbourhood of the system original point.


## 4. Simulation Results

In order to verify the effectiveness of the proposed control strategy, according to the dynamics models of Equations (1), (2), and (3), and by referring to the parameters of reference [2], simulation is carried out to make the aircraft track the given altitude command and speed command. The initial values of the states are set as V0=15060ft/s, h=110000ft, γ=0rad, α=0.0312rad, ϕ=0.1802, ϕ̇=0, and q=0rad/s. The step command is Vd=15160ft/s and hc=112000ft. The reference commands of Vr and hr are calculated as follows:
(38)ẋi=xi+1i=1,2,⋯,n−1,ẋn=λnxd−ddt+λnx1,with λV=0.5 and λh=0.3. To verify the robustness of the controller, the model parameters are processed as follows: Δm=−0.03, ΔIyy=−0.03, Δc¯=0.03, Δce0=0.03, Δρ0=0.03, and ΔS0=0.03. The external disturbances [11–13] are given as d1=0.002sin0.1t and d2=0.01sin0.2t.The design parameters of the ATSM controller method in this paper are, respectively, set askα=5, kb=0.01, p=3, q=1, k1=5, k2=10, λ1=1.5, λ2=0.02, and λ3=0.01. The design parameters of the FTC method in this paper are, respectively, set as kα=3, kb=0.1, p=3, q=1, k1=10, k2=10, λ1=1.5, λ2=0.02, λ3=0.01W^0=011×2, Γ=200I11×11, and bj=100.The center value matrixC=C1C2⋯C11 of the Gaussian basis function is chosen as
(39)C5×11=A5×6B5×5,A5×6=140001420014400146001480015000−0.0175−0.0140−0.0105−0.0070−0.00350−0.0873−0.0698−0.0524−0.0349−0.0175000.10000.20000.30000.40000.5000100000102000104000106000108000110000,B5×5=15200154001560015800160000.00350.00700.01050.01400.01750.01750.03490.05240.06980.08730.60000.70000.80000.90001.000112000114000116000118000120000.Example 1 (the failure-free simulation analysis between ATSM and FTC).
In the example, the parameters of Equation (5) are set as kF=0 and dF=0. The range of control input is set as 0≤ϕc≤1 and −30≤δe≤30deg. The simulation results are shown in Figure 2.Figure 2
Response curves of the ABHV’s variables without the actuator faults.
(a)
Velocity(b)
Altitude(c)
Flight path angle, attack angle, and pitch rate(d)
Velocity error and altitude error(e)
Fuel-to-air ratio(f)
Elevator deflection(g)
Adaptive parameters(h)
NN weight adaption(i)
NN estimate valuesThe tracking curves of velocity and altitude can be obtained from Figures2(a) and 2(b), allowing the ATSM controller to track the reference command with a small tracking error. It is visible from Figures 2(a)–2(c) that the states converge to steady-state values in a brief period. Figures 2(d) and 2(e) show the control inputs of fuel-to-air ratio and deflection in the limit range, which are smooth. Figure 2(f) describes the curves of the adaptive parameters in the case of unknown uncertainties. The NN weight adaption and estimate values fluctuate quickly owing to tracking errors, as seen in Figures 2(g) and 2(h), and the variables converge to a constant in a limited amount of time.Example 2 (the failure-tolerant simulation analysis between ATSM and FTC).
In the example, the range of control input is set as in Example1, and the parameters of Equation (5) are set as follows:
(40)kFt=0,t<40s,0.50.3T,t≥40s,dFt=0,t<40s,0.0010.001∗sin0.2t+π6,t≥40s.
The simulation results are shown in Figure3.Figure 3
The response curves of the ABHV’s variables under the actuator faults.
(a)
Velocity(b)
Altitude(c)
Flight path angle, attack angle, and pitch rate(d)
Velocity error and altitude error(e)
Fuel-to-air ratio(f)
Elevator deflection(g)
Adaptive parameters(h)
NN weights adaption(i)
NN estimate valuesThe hypersonic vehicle is marginally impacted by velocity and altitude in time-varying fault conditions, as seen in Figures3(a)–3(c). Figures 3(e) and 3(f) illustrate the curves of the input signal. From Figure 3(d), when simulating for 40 s, the actuator faults occur, the ATSM controller adjusts the control gain rapidly, and the control input is out of the lower bound of the fuel-to-air ratio. The results of simulation show that the amplitude of deviation can be reduced with the method and improve the convergence of the states. From Figures 3(g)–3(i), it can be seen that the FTC scheme has a strong fault-tolerant capacity to adjust the adaptive parameters and the NN estimate values online and handle failure effectively under the occurrence of failure.


## 5. Conclusions

In this paper, the ABHV longitudinal model’s parameter uncertainties, external disturbances, and actuator defects are investigated using radial basis function neural networks (RBFNN) and adaptive sliding mode (ATSM). The input-output linearization approach is used to create an adaptive terminal sliding mode controller based on the simplified nonlinear dynamics model of a hypersonic vehicle, which achieves speed and altitude tracking convergence in a short time. A fault-tolerant control (FTC) system was created to further tackle the actuator defect problem, and RBFNN was employed to simulate the unknown actuator dynamics. The Lyapunov theorem verifies the closed-loop system’s uniformly ultimate stability. The FTC scheme is numerically simulated. The analysis and simulation findings reveal that the fault-tolerant control system described in this study can provide quick tracking response and has high resilience in the presence of actuator faults.



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