Disturbance Observer-Based Adaptive Control of Hypersonic Vehicles with Constrained Actuators

This study investigates an adaptive controller for the flexible air-breathing hypersonic vehicles (AHVs) subject to external disturbances and actuator constraints. The combination of nonlinear disturbance observer and adaptive mechanism is exploited to design an adaptive controller for each subsystem. For the velocity subsystem, an auxiliary system is employed to handle the scramjet input saturation issue. For the altitude subsystem, the magnitude/rate constraints and the dynamics of aerodynamic control surfaces are addressed by the control allocation module. Simulations show the effectiveness of the proposed control.


Introduction
Air-breathing hypersonic vehicles (AHVs) have attracted considerable attention in both military and civil fields. Though AHVs have been developed for more than half a century, various studies have shown that some critical technologies, such as thermal protection, high-fidelity modeling, and robust control, still need to be further investigated.
Based on the control-oriented model developed by Parker et al. [1], numerous theoretical studies on the control of AHVs have sprung up [2]. Early research interests focus on state feedback linearization or nonlinear dynamic inversebased control of AHVs [3]. In recent years, the adaptive control has always been a research hotspot in the field of AHV control, and one can refer to the robust adaptive control in [4,5], the sliding mode adaptive control in [6], the trajectory restricted adaptive control in [7], the fault-tolerant adaptive control in [8,9], and the neural adaptive control in [10,11].
Nevertheless, a practical problem in AHV control is the physically limited actuators [12,13]. Recently, some outstanding works have come out focusing on the input saturation problem of AHVs [14,15]. [16] used the disturbance observer-based (DOB) feedback linearization control to overcome constraints on actuators. A combination of backstepping design and nonlinear disturbance observer (NDO) is utilized to handle actuator constraints [17]. By introducing the sliding-mode differentiator techniques and the auxiliary systems, [11] developed the distributed finite time faulttolerant containment control to handle the input saturation. On the other side, control allocation (CA) also plays a good role in dealing with the actuator constraints [18]. For instance, [19] handled the saturation using a robust model predictive control algorithm. [20] developed a control allocation scheme for fault-tolerant control based on a linear parameter varying (LPV) system. But the employed LPV model was complex and time-demanding for the CA problem.
Motivated by the above investigations, this paper proposes a disturbance observer-based adaptive control AHVs with constrained actuators. In the velocity loop, the auxiliary system developed in [12] is employed to handle the saturation issue of the scramjet input. In the altitude loop, a control allocation module is exploited for the magnitude/rate constraints and the dynamics of aerodynamic control surfaces. The main contributions of this paper can be briefly outlined as follows: (1) The saturation problem of the scramjet, as well as the dynamics and magnitude/rate constraints of the control surface, is taken into consideration (2) NDO is developed in each step to further improve the disturbance rejection property. The observation errors are suppressed by adaptive control laws The rest of this paper is organized as follows: Section 2 formulates the considered problem; Section 3 presents the design process; Sections 4 and 5 give a stability analysis and a simulation study, respectively; Section 6 concludes this study.

Problem Statement and Preliminaries
2.1. Control-Oriented AHV Model. In this paper, we consider the longitudinal control-oriented AHV model developed by Parker et al. [1]: In (1)-(5), velocity V, altitude h, flight path angle (FPA) γ , angle of attack (AoA) α, and pitch rate (PR) Q are system states; g is the acceleration due to gravity; mass m and moment of inertia I yy are decided by the design and fuel level of AHVs; the disturbances d V , d γ , and d Q are external disturbances with bounded derivatives; η i is the ith generalized elastic coordinate; ω i and ξ i are the natural frequency and the damping ratio for flexible mode η i , respectively; ψ i and ψ i ′ are coupling coefficients between rigid-body and flexible dynamics; thrust T, lift L, drag D, pitching moment M, and ith generalized force N i have the following curve-fitted approximations: In (7)-(10), scramjet fuel equivalency ratio (FER) Φ, canard deflection angle (CDA) δ c , and elevator deflection angle (EDA) δ e are three system inputs, respectively; q is the dynamic pressure; S is the reference area; z T is the thrust-to-moment coupling coefficient; c is the mean aerodynamic chord; C * * denotes the fitting coefficients of force and moment.
2.2. Problem Formulation. From a practical viewpoint, FER should comply with the saturation property: where Φ c is the FER command; Φ cooling is the constant lower threshold to keep sufficient heat dissipation through active cooling; the upper bound Φ choking is required to prevent the scramjet from thermal choking [12]. In engineering, the magnitude and rate constraints of the actuators (canard/elevator) have a great influence on control performances.
where δ denote EDA δ c or CDA δ e ; M L < 0 and M U > 0 are lower and upper bounds of the magnitude; R L < 0 and R U > 0 are the corresponding rate bounds. It should be pointed out that the upper and lower amplitude limits of a control surface can be asymmetric in reality, that is, the more common case jM L j ≠ M U or jR L j ≠ R U is also taken into consideration.
At the end of this subsection, several lemmas crucial for the following design are given.

Controller Design
3.1. Velocity Tracking Control Design. This section presents an adaptive tracking controller for velocity subsystem (1) The velocity dynamics is first expressed in the following form: where First, similar to [23], we introduce a modified tracking error to handle the saturated Φ as where χ V is governed by with k V > 0. Then, _ e V is calculated as Taking into account the bounded _ d V and the finite deflecting velocity of physical control surfaces, _ D V is assumed to be bounded. Hence, the NDO for D V is constructed.
Defining estimation error asD For the fact that _ We design the control law whereD V is estimated by NDO and b Θ V is the estimate of Θ V , generated by the adaptive law: where σ V > 0, k V ′ > 0, and ε V > 0.

Altitude Tracking Control Design.
In the altitude subsystem, control allocation is employed as an attractive technology to handle both canard and elevator servomechanism dynamics and constraints. For convenience, δ 0 c and δ 0 e represent the desired control command to be designed, δ com c and δ com e denote the control command after the process of control allocation, and δ c and δ e are the real control inputs applied to AHVs with the dynamics δ c /δ com c = 20/ðs + 20Þ and δ e /δ com e = 20/ðs + 20Þ [24]. We also define ½Δδ c , Δδ e T = ½δ c − δ 0 c , δ e − δ 0 e T for the following design.

Design for ðh, γÞ-Dynamics.
Recalling the kinematical equation (2), the realization of altitude reference tracking can be transferred to the FPA tracking problem. In this paper, to achieve h → h ref , the control system steers the AHVs to track the FPA reference: where h ref is the smooth altitude reference generated by the traditional second-order command filter; k h > 0. Define a time-related function as α f , which decides the final AoA. The FPA dynamics can be calculated as Considering the real small α, the approximate residual d γ,α f is assumed to be bounded. Meanwhile, the unknown disturbance introduced by the above control allocation d γ,δ is bounded considering the amplitude and rate constrained actuators. Thus, the boundedness of the disturbance term D γ is guaranteed.
Furthermore, similar to [4], the virtual AoA control is designed as Then, _ e γ can be rewritten as Similarly, the following NDO for D γ is constructed.
We design the control law to regulate the _ e γ system as where b Θ γ is governed by the adaptive law with σ γ > 0, ε γ > 0, and k γ ′ > 0.

Design for ðα, QÞ-Dynamics.
For the altitude subsystem, the AoA tracking error dynamics have the relationship Define Then, the PR dynamics is formulated as where Following a similar analysis to previous steps, the boundedness of D Q is also guaranteed. Thus, the PR tracking error dynamics is formulated as International Journal of Aerospace Engineering Then, NDO for D Q is constructed as Defining estimation error asD Q = D Q −D Q , it has is bounded according to the corresponding analysis in [25,26]. Considering the bounded _ d Q , _ d Q,δ , and We design the control law to regulate the _ e Q system as where b Θ Q is governed by the adaptive law with σ Q > 0, ε Q > 0, and k Q ′ > 0.

Control Allocation
Design for CDA-EDA. In this subsection, a control allocation algorithm on CDA and EDA is integrated to the proposed control considering the hard magnitude and rate constraints of canard and elevator deflections.
Considering the components M in (10), we briefly define where u com M denotes the commands generated by control allocation; u des M presents the desired values. Meanwhile, assume the future control sequence with z prediction intervals at the current interval k as As the flight control system is a sampled-data system, the magnitude and rate can establish the following simple firstorder approximation: where t s is the control interval and δðt − t s Þ is the CDA/EDA command generated in the last control interval; the rate constraint can be absorbed into the magnitude constraint: And new constraints can be defined as Therefore, the optimization problem of the above control allocation can be expressed in the following form: where w * j is the tuning weight matrices for the jth step.
Remark 1. The cost function in (52) is set for two main practical purposes. One goal reflected in J Δ u ðΞ k Þ is to pursue approximating precision, and the other goal reflected in J Δ δ ðΞ k Þ is to maximize energy savings.

Stability Analysis
In this section, stability and performance properties of the closed-loop systems are summarised.
By adding inequalities (56)-(58), we obtain where k Q > k α , k γ > −R Γ , and a L , b L ∈ ℝ + are constants expressed as Obviously, _ L < 0 provided that L > b L /a L . Therefore, all signals involved in the Lyapunov function and utilized in the design are uniformly ultimately bounded.
Applying Lemma 2 to (59), it is concluded that the boundedness of the related variables and signals in the proposed control can be ensured. Meanwhile, the tracking error ex with x = V, h retains the uniformly ultimate boundedness property: with

Simulation Results
This section implements a simulation study to verify the proposed controller. In the simulating model, the nominal parameter values of the AHV model are selected according to [1]. For comparison, a slightly modified version of the adaptive fault-tolerant control presented in [9] is applied to the same scenario (denoted as "compared control" for convenience). The simulated AHV is initially cruising at Vð0Þ = 7846 ft/s, hð0Þ = 85000 ft, γð0Þ = 0 rad, αð0Þ = 2:6 deg, η 1 ð0Þ = 0:92 ft ⋅ ffiffiffiffiffiffiffiffiffi ffi slugs p /ft, and η 2 ð0Þ = 0:61 ft ⋅ ffiffiffiffiffiffiffiffiffi ffi slugs p /ft and plans to implement a maneuver in the next 600 s. Φ cooling = 0:01 and Φ choking is selected according to [12]; M L = −20 deg, M U = 20 deg, R L = −60 deg/s, and R U = 60 deg/s; the simulating control interval is set as t s = 10 ms.
The main simulating results are given in Figures 1-9. Figures 1 and 2 show the output tracking results during the ascending maneuver and the descending maneuver, from which we see that the proposed control performs better, 6 International Journal of Aerospace Engineering 7 International Journal of Aerospace Engineering though both controls achieve the desired conditions. Time histories of flight attitudes including FPA, AoA, and PR are given in Figure 3. It is seen from Figure 1 that the desired velocity reference cannot be tracked because of the FER saturation property, also see Figure 5. In this situation, the employed auxiliary system generates a modified reference, which is less aggressive and can be well tracked within the ability of scramjet. Figure 4 shows the flexible states in the comparison simulation. Figures 5-7 indicate the control input signals of FER, CDA, and EDA. It is seen from Figures 6 and 7 that both the magnitude constraints and rate constraints of CDA/EDA are well constrained within prescribed bound under the proposed control, which performs better than the compared control. Figure 8 shows the results   International Journal of Aerospace Engineering

Conclusions
In this paper, a disturbance observer-based adaptive control for a hypersonic vehicle has been addressed to obtain the capability of handling external disturbances and actuator constraints. Fast disturbance estimate has been addressed via NDO and adaptive control compensation in each subsystem. An auxiliary system has been employed subject to the scramjet input saturation. Meanwhile, there is an effective mechanism of dealing with the magnitude/rate constraints and dynamics of aerodynamic control surfaces by introducing a control allocation module. Simulations have been provided to verify the effectiveness of the proposed control.

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 there is no conflict of interest regarding the publication of this paper.