Prescribed Performance Tracking Control for the Hypersonic Vehicle with Actuator Faults

This paper provides a solution for the trajectory tracking control of a hypersonic ﬂ ight vehicle (HFV), which is encountered performance constraints, actuator faults, external disturbances, and system uncertainties. For the altitude and velocity control subsystems, the backstepping-based dynamic surface control (DSC) strategy is constructed to guarantee the prede ﬁ ned constraint of tracking errors. The introduction of ﬁ rst-order low-pass ﬁ lters e ﬀ ectively remedies the problem of “ complexity explosion ” existing in high-order backstepping design. Simultaneously, radial basis function neural networks (RBFNNs) are adopted for approximating the unavailable dynamics, in which the minimum learning parameter (MLP) algorithm brilliantly alleviates the excessive occupation of the computational resource. Specially, in consideration of the unknown actuator failures, the adaptive signals are designed to enhance the reliability of the closed-loop system. Finally, according to rigorous theoretical analysis and simulation experiment, the stability of the proposed controller is veri ﬁ ed, and its superiority is exhibited intuitively.


Introduction
The recent few years have witnessed the burgeoning interest in hypersonic flight vehicles (HFVs) from researchers owing to its unique advantages like rapid maneuver and high efficiency. In light of these inherent characteristics, HFVs have been extensively applied in both military and civilian fields. However, there are still many obstructions in constructing controllers for HFVs, involving the harsh flight environment, the uncertainty of aerodynamics, and the strong nonlinearity and coupling. In order to surmount these challenges, numerous control frameworks have been examined, just to name a few, sliding mode control (SMC) [1][2][3][4], backstepping control [5][6][7][8], dynamic surface control (DSC) [9,10], adaptive control [11,12], and so forth.
As a classical nonlinear control approach, SMC is celebrated for its antidisturbance capability, easy implementation, and excellent robustness [13][14][15][16]. In [1], SMC-based architecture was established to achieve the trajectory tracking control for HFV, while it comes with the undesired phenomenon of chattering. This defect will undoubtedly generate a heavy burden on actuators and shorten their service lives. Fortunately, [2] provided a continuous antichattering sliding mode controller for the flexible HFVs. Besides the SMC methods, the backstepping-based procedure design possesses a high application value in high-order nonlinear systems. The disadvantage of traditional backstepping methods lies in the issue of "complexity explosion," which always leads to waste of computational resources and excessive hardware requirements. Therefore, the DSC-based solution was developed in [9] so that the above problem could be relaxed by resorting to the introduction of low-pass filters. In addition, complicated aerodynamics is another threat for trajectory tracking of HFVs. Inaccurate dynamics makes modeldependent control schemes uncapable for practical engineering. Inspired by this condition, a neural adaptive sliding mode control algorithm was proposed in [17], where the radial basis function neural networks (RBFNNs) were utilized to approximate the available dynamics with an exacting precision.
At the same time, while accounting for the uncertainties of HFV models, great demands were placed on the convergence rate and tracking accuracy of control systems [18]. From this view of point, most of the aforementioned controllers possess satisfactory steady-state characteristics but lack specific constraints on transient performances. Under this urgent requirement, the prescribed performance control (PPC) technology is introduced to restrict certain states within a predefined region globally [19][20][21][22][23][24], such that the system transient indicators can be flexibly specified by the designers. Therefore, Shao et al. imposed time performance constraints in event-triggered robust control for quadrotors, which apparently optimized the stable rate of the closed-loop system [19]. As for the PPC strategy in vehicle control field, Bu constructed a RBFNN-based robust algorithm for air-breathing hypersonic vehicles (AHVs) with unknown dead-zone input nonlinearity, where the trajectory tracking errors are effectively suppressed within a predetermined range [21].
However, the controllers mentioned above are all designed on basis of the assumption that actuators of HFVs work normally. Obviously, it is of impractical conceit for HFV servicing in ideal environments. To prevent the resulting performance degradation and instable phenomenon, the actuator faults must be taken into account while formulating the trajectory tracking control strategies with sufficient applicability and reliability [25][26][27]. In [26], a quasicontinuous high-order sliding mode framework was presented, in which the neural network observer is introduced to ensure the convergence of estimation errors. Further in [28], an adaptive fault-tolerant control scheme was derived in the presence of actuator saturation to improve system reliability and robustness. By the utilization of PPC, a faulttolerant protocol was designed to achieve the accurate trajectory tracking for HFVs.
Driven by the aforementioned observations, this paper aims at the prescribed performance fault-tolerant control for HFV suffering from system uncertainties and actuator failures. The backstepping-based dynamic surface design is conducted to guarantee the finite-time convergence of altitude and velocity tracking errors. Unknown multiplicative and additive faults of actuators are handled by resorting to the construction of adaptive architectures. On the other hand, unmodeled dynamics are approximated via RBFNNs; especially, the minimum learning parameter (MLP) algorithm requires less computational resource. Finally, the stability and superiority of the proposed controller will be corroborated by theoretical analysis and simulation experiments. It will be elaborated that the transient and steadystate performance constraints are always satisfied. Contributions of this article are presented as follows: (1) Different from the traditional approximation of RBFNN [12,17], MLP algorithm is utilized to cope with the uncertain dynamics in this paper. In this way, it is no longer necessary to estimate the entire weight matrix, but the upper bound of its norm is taken as the object of online updating. Therefore, the computational complexity is considerably degraded, and the hardware requirement is reduced significantly (2) In [23,24], PPC control methods were presented for MEMS Gyroscopes and Networked Uncertain Quadrotors, respectively, which cannot be directly applied for hypersonic vehicles. Considering this point, the PPC-based dynamic surface controller is synthesized for hypersonic vehicles in the presence of uncertainties, actuator faults, and disturbances. The results of this paper could be treated as an application of PPC for hypersonic vehicles (3) Instead of restricting the time performance [19,20], the PPC-based amplitude constraints [23,24] of HFVs' trajectory tracking errors are addressed in this paper. In virtue of the hyperbolic tangent function, the open-loop tracking error dynamics with certain designer-specified index constraints is transformed into an equivalent "state-constrained" system, in which both transient and steady-state responses obtain the satisfactory performance The remainder of this paper is organized as follows. Section 2 gives a longitudinal model of HFVs and the relative preliminaries. The altitude and velocity controllers are designed in Section 3 and Section 4, respectively. In Section 5, the validity of the controllers is verified through a simulation experiment. Finally, Section 6 shows the conclusion of this work.

Problem Formulation
2.1. Longitudinal Model of HFV. The trajectory tracking control problem for a HFV is considered in this paper. According to [29], the longitudinal model of the HFV can be expressed as: The corresponding dynamics are given as: where c and c e denote constants; q denotes the dynamic pressure satisfying q = ð1/2ÞρV 2 ; V, h, γ, α, and q denote velocity, altitude, flight path angle, angle of attack, and pitch rate of HFV, respectively [30]; T, D, L, and Myy denote the thrust of the engine, drag force, lift force, and pitching moment, respectively [31]; and S, m, ρ, I yy , μ, and r e denote the reference area, mass, density of air, moment of inertia, the gravitational constant, and radius of the Earth, respectively [32]. As a matter of fact, various unexpected failures occur to the actuators frequently in practical engineering, which will cause severe performance degradation and even collapse for whole close-loop system. Therefore, it is necessary to take the actuator faults into consideration while designing the trajectory tracking controllers. With the introduction of multiplicative and additive faults, the actual input signal is described as: where ρ i denotes the unknown effectiveness factor of actuator and satisfies 0 < ι i ≤ ρ i ≤ 1, ι i ≤ 1 represents the minimum value of ρ i , d i ðtÞ denotes the unknown external disturbance, and Δ i is the additive fault with i = 1, 2.

Fundamental of RBFNNs.
RBFNN is an effective tool to obtain the approximation of unknown system dynamics.
The following lemma sketches the basic principle of RBFNNs.
Lemma 1 [33]. An arbitrary continuous smooth function f ðxÞ can be described as the following form by defining a basis function HðxÞ.
Here, W * = ½W a1 , W a2 , ⋯, W ap T is the ideal weight matrix, and x = ½x a1 , x a2 , ⋯, x am represents the input vector. r and O are the additional approximation error and its upper bound; HðxÞ = ½H a1 ðxÞ, H a2 ðxÞ, ⋯, H am ðxÞ T is selected as the Gaussian basis function, which can be expressed as: with c i ∈ R m and b ∈ R p denoting the center vector and the width of Gaussian basis function, respectively.

Other Preliminaries
Lemma 2 [33]. For arbitrary scalar x ∈ R and μ > 0, the inequation about hyperbolic tangent function is shown as: in which the constant κ = 0:2785.
Assumption 3. The additive actuator fault and external disturbances are both unknown but bounded, which are supposed to satisfy jd i ðtÞ + Δ i j ≤ λ i ði = 1, 2Þ with λ i being positive constants.

Altitude Controller Design
Aiming towards the altitude and velocity tracking missions, two subsystems are formulated on basis of the longitudinal model of HFV. In this section, a DSC-based adaptive faulttolerant controller is constructed for the altitude tracking subsystem. Due to the high-order characteristic of the dynamics, traditional backstepping design is inevitably accompanied by the phenomenon of "complexity explosion," which is arising from the frequent differential to virtual commands. Therefore, low-pass filters are introduced to solve this problem, while system uncertainties are handled via RBFNN. Finally, it is validated via Lyapunov-based analysis that altitude tracking errors exponentially converge to a tiny region containing the origin. In altitude tracking control subsystem, three states are selected as γ, ϑ p , and q. To facilitate the subsequent design, a state vector is defined as: where ϑ p = α + γ. Thus, the altitude dynamics can be described as [30]: with 3 International Journal of Aerospace Engineering Assumption 4. The uncertain term g i ðVÞ possesses an upper bound g i , which satisfies 0 < jg i ðVÞj ≤ g i , i = 1, 3.
At this moment, the control objective has been transformed into forcing x i ði = 0, 1, 2, 3Þ to track the desired trajectories, where x 0 = h. For this purpose, the tracking error variables are introduced as To ensure the prescribed performance, the boundary of variable x ei is defined as: The error transformation is designed as: in which s i is the transformed tracking error. It indicates that the transformed error increases monotonically with the original variable. Note additionally that there exist limits as: Necessarily, the time derivative of s i is derived as: in which Step 1. According to Equation (29), the derivative of s 0 satisfies the following equation: Based on the similar analysis in [34], the asymptotic stability of the altitude tracking error x e0 is ensured if the virtual command is designed as: with k h > 0, k hh > 0. For the purpose of stabilizing x e1 , the following equations are firstly presented by utilizing Equations (24) and (26)- (29): A first-order low-pass filter is introduced as: where the variable x 2c represents the filtered signal and ε 2 > 0 is a constant.
To cope with the unavailable dynamic m 1 f 1 , Lemma 1 is applied here. Then, one has: where h 1 = kH a1 k and w 1 ≥ kW * T a1 k. By defining that R 1 = O 1 , the virtual control input and the relevant adaptive laws are designed as: whereŵ 1 andR 1 are the estimation values of w 1 and R 1 , respectively, and k 1 and k 2 are the positive constants.

International Journal of Aerospace Engineering
Introduce the virtual command x 3c , which is taken as: where k 3 and k 4 are the positive constants.
Step 3. To ensure the convergence of x e3 , Equations (24) and (26)- (29) are combined, and the following equations are obtained: By taking the unavailable dynamic m 3 f 3 into consideration and according to Lemma 1, one can obtain: where h 3 = kH a3 k and w 3 ≥ kW * T a3 k.
Theorem 5. For the HFV system (24), if the virtual commands are designed as Equations (31), (35), and (40), control signals are designed as Equations (45)-(47); it can be concluded that the tracking errors satisfy exponential convergence, and the prescribed performance constraints are guaranteed.
Proof. Lyapunov function candidate is selected as: where φ i are defined as: The differential of y i , i = 2, 3, can be expressed as: It is worthy to point that there must exist constants ο i > 0, i = 2, 3, satisfying the inequality [32]: in which o 1 are unknown positive constants. Therefore, the time derivative of φ i can be developed as:

Velocity Controller Design
The velocity tracking control subsystem for HFVs is established as: The definition of velocity tracking error V e and its time derivative are given as: The predesigned boundary of variable V e is denoted as: The error transformation is redesigned as: where s 4 is the transformed velocity error. Obviously, it indicates that the transformed error increases monotonically with the original error.
Theorem 6. Noting the velocity dynamic (61) of HFV and considering the fault tolerant controller Equations (69)-(71), exponential convergence and prescribed performance constraints will be ensured for the tracking error.       9 International Journal of Aerospace Engineering Therefore, it can be concluded that s 4 is asymptotically stable, and the velocity tracking error V e completes convergence within the prespecified range. Theorem 6 is validated sufficiently.

Simulation
To illustrate the specific performances of the proposed controller, a numerical simulation is executed based on the longitudinal dynamics (1)-(17) of HFV. By giving the initial states and reference trajectories, the tracking control objective will be achieved with predefined performance constraints. The initial values of states are provided in Table 1, and the control parameters are shown in Table 2.
Meanwhile, the desired trajectories, environmental disturbances, and actuator effectiveness parameters are defined as: Consequently, the main results are exhibited in Figures 1-10. Firstly, the curves of tracking errors are shown in Figures 1 and 2. It follows from these two pictures that the altitude tracking error and velocity tracking error are stabilized into a tiny neighborhood around origin within 30 s and 50 s, respectively. Obviously, not only the transient performance constraint is guaranteed but also the steady-state errors are maintained within ranges of 5 m and 5 m/s, which means undoubtedly a very high accuracy for hypersonic flight. What is subsequently given in Figures 3 and 4 is the curves of control inputs. In particular, even if there is an unknown actuator failure occurring at 100 s, the control signal is capable of returning to a constant within an extremely short time. Figures 5 and 6 show the curves of adaptive esti-matesR 3 andR 4 , which always remain bounded under the designed algorithm. The estimations about weight matrixes of RBFNNs are provided in Figures 7 and 8, and the adaptive learning parameters for actuator failure are settled in Figures 9 and 10. As is shown in figures, all the estimations finally converge to constants. Thus, the proposed controller is capable of tracking the desired trajectory in case of actuator faults.
For better illustration of the merits PPC, simulation results without performance constraints are presented in Figures 11 and 12. Observing the tracking errors of altitude and velocity, one can conclude that the performance constraints will be violated before the system's stabilization. From this aspect, it deduces that the PPC control strategies are necessary for hypersonic vehicles' tracking control.

Conclusion
This study concentrates on the robust prescribed performance tracking control for HFV in the presence of system uncertainties and unknown actuator faults. On the basis of longitudinal model of HFV, an antidisturbance neural adaptive control strategy is constructed, in which MLP algorithm is employed to approximate the unavailable dynamics with a reduced computational burden. In addition, the multiplicative and additive failures of actuator are taken into account simultaneously. Finally, the stability and effectiveness of proposed controller are elaborated by rigorous theoretical analysis. The results of simulation example further indicate that the altitude and velocity tracking errors satisfy the prescribed performance constraints.

Data Availability
The data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest
The authors declare that they have no conflicts of interest.