Adaptive Finite-Time Control for a Flexible Hypersonic Vehicle with Actuator Fault

The problemof robust fault-tolerant tracking control is investigated. Simulation on the longitudinalmodel of a flexible air-breathing hypersonic vehicle (FAHV) with actuator faults and uncertainties is conducted. In order to guarantee that the velocity and altitude track their desired commands in finite time with the partial loss of actuator effectiveness, an adaptive fault-tolerant control strategy is presented based on practical finite-time sliding mode method. The adaptive update laws are used to estimate the upper bound of uncertainties and the minimum value of actuator efficiency factor. Finally, simulation results show that the proposed control strategy is effective in rejecting uncertainties even in the presence of actuator faults.


Introduction
Air-breathing hypersonic vehicles (AHVs) are intended to be a reliable and cost-effective technology for access to space.Because the slender geometries and light structures cause significant flexible effects and strong coupling between propulsive and aerodynamic forces resulting from the integration of the scramjet engine, AHVs are confronting many complex problems and challenges, involving many different research areas, such as aerodynamics, thermal protection, and communication, and many problems of these fields have been reported [1][2][3].Meanwhile, flight control design for AHVs is a hot topic and a challenging task [4,5].
During the last decades, a kind of flexible hypersonic vehicle model including flexible dynamics has been developed in [6,7].Based on this model, there have been several papers discussing the challenges associated with the control of air-breathing hypersonic vehicle (AHV) [8,9] and many control methods have been employed in the flight control system.In [10], a linear quadratic regulator (LQR) was presented for a linearized FAHV model.In [11][12][13], sequential loop closure controller was designed for the FAHV based on adaptive dynamic inversion together with backstepping structure.In [14,15], approximate feedback linearization based on dynamic inversion method was adopted to design controller for the FAHV.In [16,17], a nonlinear tracking controller was constructed by using a minimax LQR control approach, which provides robust stability and excellent tracking performance with parameter uncertainties.
The approaches mentioned above do not specifically consider possible actuator faults, which deteriorate the control performance, affect stability, and security of the AHVs, and sometimes even lead to catastrophic accidents.Consequently, it is essential that the actuator faults must be taken into account in the controller design.In the current papers, some fault-tolerant control schemes for AHVs have attracted more and more research attention and gained fruitful results, which can be reported in [18][19][20][21].In [18][19][20], the results mainly concentrate on the reentry attitude control of the AHV.Meanwhile, the fault tolerant control strategies for the longitudinal model of the AHVs are studied.In [21], an observer-based fault-tolerant control approach using both robust control and LMI techniques is designed for a linearized longitudinal AHV model in the presence of parameter uncertainties and actuator faults, but this method was effective only in the neighborhood of the operating point.On the other hand, nonlinear fault-tolerant control design methods have been devoted to the longitudinal AHV model.A finite-time integral sliding mode control method was proposed in [22], which could achieve superior velocity and altitude tracking performance with actuator fault.In [23], the longitudinal AHV model with unknown parameters and uncertain actuator faults is formatted into a parametric strictfeedback form, and then an adaptive fault-tolerant control scheme based on a combination of back-stepping control and dynamic surface control techniques is applied to make the velocity and altitude track the desired value.
However, the aforesaid methods only consider the rigid body of AHVs without flexible effects.A fault-tolerant control scheme for the FAHV was presented in [24], according to the model obtained by approximate linearization in given flight conditions.So, this scheme may not obtain good control performances when flight dynamics undergo great parameter perturbations.To the best of our knowledge, although considerable effort has been made on the control design for the AHVs, the important issue of fault-tolerant control of the FAHV dynamical system has not been fully investigated yet, which remains challenging and motivates us to do this study.
As a typical robust control method, sliding mode control (SMC) scheme is regarded as an effective method to cope with external disturbances and parametric uncertainties [25].Recently, the SMC method has been widely applied for the fault tolerant control of aircraft system, spacecraft, and so on.In [26], a fault-tolerant sliding mode controller was presented for an aircraft system, which requires the message of the effectiveness factor, while it may be difficult and expensive to obtain the actuator faults online.In [27], a finite-time convergent SMC scheme is developed to solve the problem of fault-tolerant control for a rigid spacecraft.The drawback of this method is that the message of the lower bound of the effectiveness factor and the upper bound of system uncertainties needs to be known in prior.
The aforementioned references could achieve desired performance through the SMC methodology affected by actuator faults.Although the traditional SMC can guarantee the stability of the system, it adopts a linear switching function.Then the system states and the errors converge to an equilibrium point asymptotically in infinite time.In other words, it means that finite-time convergence is not ensured.Motivated by the above discussions, we propose a novel adaptive sliding mode control scheme for the longitudinal model of the FAHV with uncertainties and actuator faults in this paper.As compared with the existing results, the main contributions are as follows.Firstly, the design method of sliding mode surface based on homogeneous geometry could assure practical finite-time converged tracking of the desired command.Secondly, the upper bounds of aerodynamic uncertainties and the minimum value of actuator efficiency factor are not required in prior.The adaptive law is designed to adjust the control gains dynamically so as to ensure the establishment of sliding mode motion, and the robustness against uncertainties is ensured at the same time.After the uncertainties and actuator faults are compensated using adaptive sliding mode control scheme, the stability of the closed-loop system can be maintained.
The rest of this paper is organized as follows.In Section 2 the FAHV model is introduced and control objective is stated.Section 3 designs the sliding mode surface and the corresponding adaptive finite-time fault tolerant controller was proposed with actuator fault.Simulation results are discussed in Section 4 and the conclusions are provided in Section 5.

Problem Statement
The considered FAHV model is derived from [6,28], and the longitudinal equations of motion of the FAHV are given by where  = [, , ℎ, , ]  is a vector of rigid-body state, which includes the vehicle speed, flight path angle, altitude, angel of attack, and pitch rate, respectively;   ,  , , and   are the generalized flexible coordinate, natural frequencies, and damping coefficients of the th elastic mode.The readers may refer to [7] for a full description of the variables in this model.Because of coupling in aerodynamic forces of the FAHV model (1), some simplifications must be carried out for the purpose of feedback linearization.The simplification of the model is necessary because we want to obtain a linearized model, and the same simplified process can be found in [29].An input-output linearization model is developed by repeated differentiation of the outputs  and ℎ as follows: where   and   are control inputs and the specific expressions of   ,  ℎ ,  11 ,  12 ,  21 , and  22 are presented in [29, equation (17)].
Compared with [29], the main propose of this study is discussing the fault tolerant controller design for the FAHV to follow a given desired output reference signals   = [  , ℎ  ]  in the presence of partial loss of actuator effectiveness.

Adaptive Finite-Time Fault-Tolerant Controller Design
The specific controller design step includes two parts: sliding mode surface design and sliding mode control design, which can be described as follows.

Sliding Mode Surface Design. Define tracking error variable as follows:
Differentiating ( 4) and ( 5) three times, and four times respectively, results in Equations ( 6)-( 7) can be expressed in matrix form: Note that the additional item Δ is introduced to represent the flexible effects and coupled uncertainties described in [29, equation ( 14)].
Introduce new control variable: Then ( 8)-( 9) can be rewritten as Assumption 1.The uncertainties discussed in the research are bounded ‖Δ‖ ≤ , but the value  is unknown in advance.
Assumption 2. The matrix  denoted in ( 8) is nonsingular over the entire flight envelope given in [12], so Assumption 1 is reasonable to be assumed.Now, according to the definition of HOSM [30,31], our objective is to design controller which makes the   ,  ℎ and their derivatives converge to the neighborhood of origin.
Based on the homogeneity theory provided in [32], it is easily shown that   , ė  , ë  and  ℎ , ė ℎ , ë ℎ , ⃛  ℎ will converge to the neighborhood of origin in finite time if it is satisfied that   ,  ℎ converge to the neighborhood of origin in finite time.

Adaptive Sliding Mode Controller
Design.Now, let us consider the situation in which the actuator experiences partial loss of effectiveness fault.Then, differentiating (11) and (12), we obtain where  = diag( 1 ,  2 ) ∈  2 × 2 is a matrix characterizing the health condition of the actuators with 0 ≤   ≤ 1 ( = 1, 2).Note that the case   = 1 means that the th actuator is totally healthy, the case   = 0 implies that the th actuator completely fails, and the case 0 <   < 1 corresponds to the case in which the th actuator partially loses its effectiveness, but it still has effect all the time.In this sense, the matrix  becomes uncertain and even time varying but remains positive definite.In this study, an assumption 0 <   ≤ 1 is given.The control objective is to design the control inputs for   and   such that all of the closed-loop signals are bounded and the velocity  and altitude ℎ track desired command trajectories   and ℎ  in the presence of flexible uncertainties and loss of effective actuator faults.That is to say, the velocity sliding mode surface   and altitude sliding mode surface  ℎ converge to an arbitrary small set containing the origin in finite time  0 , which is ‖  ‖ ≤   and ‖ ℎ ‖ ≤  ℎ for  ≥  0 , where   and  ℎ are arbitrary small positive constant numbers.
Similar to (33), the following inequality can be obtained: Thus, from ( 28)- (34), the derivative of the Lyapunov function (28) becomes where According to Lemma 3.6 in [33], the decrease of can drive the sliding mode surfaces   and  ℎ to converge to a neighborhood of the sliding surface in finite time.Furthermore, selecting 0 <  ≤ 1, inequality (35) can be expressed as If −(1 − ) (+1)/2 +  0 < 0, then Ẇ ≤ − (+1)/2 .Based on the conclusion from [30], the decrease of drives the trajectories of the closed-loop system into  (+1)/2 ≤  0 /(1 − ).Therefore, the trajectories of the closed-loop system is bounded in finite time as lim where 0 <  0 < 1 and  is a small set containing the origin of the closed-loop system.And the time needed to reach (38) is bound as where (0) is the initial value of .After that, the control objective that the   , ė  , ë  and  ℎ , ė ℎ , ë ℎ , ⃛  ℎ converge to the neighborhood of origin is established.
When the control  = [ 1 ,  2 ]  is designed via (9), according to Assumption 2 the actual control variable is calculated as It is evident from (40) that the finite-time convergent performance of the proposed adaptive fault tolerant controller can be obtained without the knowledge of the minimum value of actuator effectiveness factor.Meanwhile, the upper bound of uncertainties does not need to be known in advance.

Simulation
To illustrate the efficiency of controller designed previously, a climbing maneuver with longitudinal acceleration for a 100 ft/s velocity change and a 1000 ft altitude change is considered.Simulation studies have been done on the full nonlinear flexible hypersonic vehicle defined in (1).The reference commands have been generated by filtering step reference commands by a second-order prefilter with natural frequency   = 0.06 rad/s and damping ratio   = 0.95.
The initial trim condition is selected as  = 7710 ft/s and ℎ = 85000 ft.Simulation parameters are provided in Table 1.
The simulation results are provided in Figures 1-4. Figure 1 denotes the response to the 100 ft/s step velocity and 1000 ft step altitude.It has been observed that the velocity and altitude converge to the desired value.The control inputs of   and   could be seen in bottom plots of Figure 1.
Figure 2 shows the performance of the angle of attack  and the pitch rate  at the top, as well as the canard deflection   and the flight path angle  at the bottom.
The velocity and altitude sliding mode surfaces   ,  ℎ are shown in Figure 3, which are oscillation with small magnitudes when actuator fault occurred.The convergent performance verifies the effectiveness of the proposed control strategy.The adaptive parameters υ and θ in control laws of (15)-( 18) could be seen in bottom plots of Figure 3, where the convergence of υ is confirmed.From the simulation results in Figure 3 The value of   in our research is in tolerance.Meanwhile, the stability of flexible states is depicted by Figure 4.And it can be seen that the flexible states  1 ,  2 , and  3 converge to constant values, respectively.
In summary, the simulation results demonstrate that, although there are actuator faults and uncertainties in the system, the good tracking performance and satisfactory system responses can be guaranteed.

Conclusions and Future Work
In this paper, an effective method has been proposed for linearizing the nonlinear model of the FAHV via feedback, which simplifies the complexity of the controller design process.Furthermore, an adaptive fault-tolerant control scheme based on finite-time sliding mode control technique has been brought forward for the FAHV without any information about the upper bound of uncertainties or the minimum value of actuator effectiveness.Simulation results have been presented to evaluate the validity of the proposed control scheme and to show its robustness to uncertainties and the loss of actuator effectiveness.
Further research work includes two aspects.Firstly, only the loss-of-effectiveness fault has been investigated in this paper; other types of actuator faults such as float failure and actuator faults in FAHV with unknown structure are worth being dealt with.Furthermore, the FAHV model considered in this paper is highly nonlinear and strongly coupled, and a more general active FTC scheme as adaptive fault diagnosis observer in [34,35] should be investigated in our future study.

Figure 2 :
Figure 2: Other flight states with actuator faults.

Figure 3 :
Figure 3: Sliding mode surface and adaptive parameters υ and θ with actuator faults.

Figure 4 :
Figure 4: The dynamic response curves of flexile modes with actuator faults.