Back-Stepping Control for Flexible Air-Breathing Hypersonic Vehicles Based on Uncertainty and Disturbance Estimator

Lin Cao， Dong Zhang and Ao Zhang

（1. System Engineering Institute of Sichuan Aerospace, Chengdu 610100, China；2. College of Astronautics, Northwestern Polytechnical University, Xi’an 710072, China）

Abstract: A theoretical framework of nonlinear flight control for a flexible air?breathing hyperson?ic vehicle(FAHV) is proposed in this paper. In order to suppress the system uncertainty and ex?ternal disturbance, an uncertainty and disturbance estimator(UDE) based back?stepping control strategy is designed for a dynamic state?feedback controller to provide stable velocity and altitude tracking. Firstly, the longitudinal dynamics of FAHV is simplified into a closure loop form with lumped uncertainty and disturbance. Then the UDE is applied to estimate the lumped uncertainty and disturbance for the purpose of control input compensation. While a nonlinear tracking differen?tiator is introduced to solve the problem of “explosion of term” in the back?stepping control. The stability of the UDE?based control strategy is proved by using Lyapunov stability theorem. Finally,simulation results are presented to demonstrate the capacity of the proposed control strategy.

Key words: flexible air?breathing hypersonic vehicle(FAHV)；uncertainty and disturbance estim?ator(UDE)；back?stepping control

Air?breathing hypersonic vehicles(AHVs)have attracted lots of attention for years due to their reliability and feasibility for access to space in both civilian and military applications. In the past few decades, numerous meaningful work has been done to further their development and ap?plications, by US Air Force and NASA. Al?though X?43A and X?51A trial vehicles have achieved great success in recent years, flight con?trol for AHVs is still a challenging task due to the very high speed which cause the vehicle to be very sensitive to changes in the flight condi?tions[1]. In addition, flight at high altitude and Mach numbers results in difficulties in measur?ing and estimating the aerodynamic properties[2].Therefore, there is no doubt that the practical controllers for AHVs should be designed with strong robustness.

In the past few years, robust and adaptive control approaches have been employed to deal with flight control problems for AHVs with para?meter uncertainty. Refs. [3?4] provided a linear?quadratic robust control scheme, based on stochastic robustness analysis and nonlinear dy?namic inversion, to design the control system of an air?breathing hypersonic vehicle. Ref. [5] pro?posed an adaptive sliding mode controller based on input/output linearization for a generic hyper?sonic vehicle. The controller is evaluated by the robustness with respect to parameter uncertain?ties. In order to enhance the robustness of velo?city and altitude tracking in the presence of ex?ternal disturbances, Ref. [6] offered a robust com?posite control scheme based on continuous finite time control laws for the longitude dynamics of an air?breathing hypersonic vehicle, combined with a nonlinear disturbance observer to estim?ate the lumped disturbances.

Because of strong couplings among structure,engine, and aerodynamics, the flexible effects cannot be ignored in control design of AHVs[7?8],which could result in negative factors to system stability. Recently, an air?breathing hypersonic vehicle model with flexible dynamics was de?veloped by Bolender and Doman[9]. Based on this kind of flexible air?breathing hypersonic vehicle models, several investigations on flight control design have been published in recent years. To avoid dealing with the complex nonlinear dynam?ics, linear control methods have been widely used for flight control designs through linearizing the flexible air?breathing hypersonic vehicle(FAHV)models. In Ref. [10], Groves adopted numerical methods to linearize the nonlinear longitudinal dynamics of an FAHV at a specified trim condi?tion, and presented the techniques of linear quad?ratic regulator(LQR) design with integral aug?mentation. Additionally, the technique of feed?back linearization was also used for FAHV mod?el linearization, and the LQR was design to force the vehicle to track the reference trajectory[11].On the other hand, nonlinear control methods also have been investigated for FAHV flight con?trols, such as minimax LQR control[12], sliding mode control[13?14], higher?order sliding mode con?trol[15], fuzzy control[16], and back?stepping con?trol[17?20]. Especially in back?stepping controls,disturbance observers are commonly used for dis?turbance estimation to compensate the control inputs. In Ref. [18], a nonlinear Luenberger ob?server is constructed to estimate the unknown disturbance. Then a nonlinear composite control strategy is proposed to reject the flexible effects on pitch rate. In Ref. [19], a new nonlinear dis?turbance observer is designed based on hyperbol?ic function to estimate the model uncertainties and varying disturbances. In Ref. [20], the radial basis function neural network(RBFNN) is em?ployed to approach the unknown functions with any desired accuracy. This method can be con?sidered as an indirect observation of system un?certainty and external disturbance. In Ref. [21],the high order sliding mode disturbance observ?er(SMDO) is also applied to estimate the uncer?tainties to compensate the controllers and dis?turbance suppression. Obviously, the disturb?ance observer plays a very important role in non?linear control approaches, which could make the control laws achieve adaptability and disturb?ance rejection.

Recently, the disturbance rejection control based on an uncertainty and disturbance estimat?or is widely investigated in modern control the?ory. The uncertainty and disturbance estimator(UDE) based control scheme is developed from time?delay control (TDC), but presents better control capacities[22?23]. Therefore, in order to provide more plentiful resolutions, this paper pro?poses an uncertainty and disturbance estimator based control strategy for the flight control of FAHVs. First, the longitudinal dynamics of FAHVs is simplified into the forms of closure loop with lumped uncertainty and disturbance.Then the UDE is applied to estimate the lumped uncertainty and disturbance to achieve the tar?get of controller compensation and disturbance rejection. While a nonlinear tracking differentiat?or(NTD) is employed to estimate the derivat?ives of virtual controls, which could solve the problem of “explosion of term” in the traditional back?stepping control. Finally, Lyapunov stabil?ity theory is employed to prove the stability of the control laws.

The rest of this paper is organized as follows.Section 1 introduces the longitudinal dynamic models of a flexible air?breathing hypersonic vehicle. Then the detailed UDE?based control design for FAHVs is demonstrated in Section 2.Several representative simulations are conducted in Section 3. Finally, Section 4 concludes this pa?per briefly.

1 Mathematical Model of FAHV

The first principle model(FPM) of an air?breathing hypersonic vehicle considered in the paper is presented as[24?25]

2 UDE-Based Control Design for FAHV

The control objective considered in this pa?per is to design a nonlinear controller for FAHV to provide a robust tracking of step?velocity and step?altitude commands Vrefand hef. It is easy to denote that the rigid equations of motion can be decomposed into two subsystems, namely, the ve?locity subsystem and the altitude subsystem. Ac?cording to the dynamics of FAHV, we know that the velocity V is mainly related to the fuel equi?valence ratio ? and the altitude h is mainly af?fected by the elevator deflection δe. Therefore, the UDE?based control laws combined with the back?stepping design are designed for the flight con?trol of FAHV, by considering the ? and δeas the control inputs of velocity and altitude subsys?tems respectively. In the control laws, the virtu?al controls and their derivatives are estimated by NTDs[26], which avoids the traditional problems of “explosion of complexity” and “circular con?struction problem”. While UDE is adopted to es?timate the inertial system uncertainties and ex?ternal disturbances for mismatched uncertainty and disturbance rejections. The diagram of the UDE?based control strategy is presented in Fig. 1.

Fig. 1 Diagram of UDE?based control strategy

2.1 Reference model

2.2 UDE-based control law by dynamic inversion

In some way, five all?pass filters for the sig?nals of uncertainty and disturbance need to be designed in the control laws. However, in real physical applications, various stochastic noises al?ways exist as high?frequency signals in the obser?vation of system states. Thus to eliminate the ef?fects of these noises, a low?pass filter(LPF) is employed to filter the signal of uncertainty and disturbance, the bandwidth of which is chosen appropriately to cover the spectrum of uncer?tainty and disturbances, but to be lower than that of the noises.

2.3 Stability analysis

Theorem 1 Consider the closed?loop system comprising of FAHV dynamics (1) with control?lers (10), NTDs (11), and UDEs (15). Then,tracking errors of the system states are semi?globally uniformly bounded.

Proof Define the estimations of virtual con?trol laws by NTDs as

3 Simulation Results

Due to the physical constraints of hyperson?ic flight and the operability of the scramjet en?gine, the rigid?body states are bounded to satisfy the admissible range, which determines the flight envelope, together with the admissible range of the control inputs. The admissible range for vari?ables are the same as the ones of Tab. 1 in Ref. [27].

According to the flight envelope, the vehicle is assumed to climb a maneuver from the initial trim conditions to the final trim conditions. The reference commands are 30.5 m/s step?velocity command and 610 m altitude command. Then the UDE?based control laws for FAHV derived by Eq. (10) have been validated in MATLAB envi?ronment.

The system responses to a 30.5 m/s step?ve?locity and 610 m step?altitude command are presented in Figs. 2?3. Fig. 2 shows that the pro?posed UDE?base control strategy can provide an excellent performance of tracking velocity and altitude reference command, when the perturba?tions of aerodynamic coefficients increase from?=?0.3 (Case I) to ? =+0.3 (Case II). The velocity and altitude tracking errors are very small. And the control inputs ?, δedemonstrated in Fig. 3 are quite smooth and bounded ration?ally.

Fig. 2 Velocity and altitude tracking and tracking error

Fig. 3 Control inputs: ? and δe

Furthermore, the rigid?body statesx=[V,h,γ,α,Q]Tare assumed to be measured for uncer?tainty and disturbance estimation, combining with stochastic measurement noises. These meas?urement noises are considered as Gaussian white noises, and bounded as [–1 m/s, 1 m/s], [–2 m,2 m], [–0.1 °, 0.1 °], [–0.1 °, 0.1 °], and [–0.5 (°)/s,0.5 (°)/s] for V, h, γ, α and Q respectively. For illustrating the advantages of the proposed UDE,a nonlinear disturbance observer (NDO) descr?ibed in Ref. [26] is employed to make a comparison.

The system responses to a 30.5 m/s step?ve?locity and 610 m step?altitude command, com?bined with states measurement noise, are presen?ted in Figs. 4?5. For NDO?based control, meas?urement noise leads to high?frequency chattering in control inputs. However, the UDE?based con?trol provides more smooth control inputs. Thus as shown in Fig. 4, compared to NDO?based con?trol, the system response with UDE?based con?trol presents more steady tracking errors in velo?city and altitude channels.

Fig. 4 Velocity and altitude tracking and tracking error, with measurement noise

Fig. 5 Control inputs: ? and δe, with measurement noise

4 Conclusions

In this paper, a new UDE?based control strategy is proposed for the flight control of FAHVs. For the sake of control design conveni?ence, firstly the longitudinal dynamics of FAHVs is decomposed into two functional subsystems, as the velocity system and the altitude system, with lumped uncertainty and disturbance, then a UDE is introduced for the lumped uncertainty and dis?turbance estimation to compensate the control input. The new NTD from Ref. [26] is adopted to estimate the derivatives of the virtual control sig?nals, which could overcome the intractability of“explosion of term” in the traditional back?step?ping control. Lyapunov stability theory is em?ployed to prove the stability of the closed?loop control system. Simulation results demonstrate that the UDE?based control strategy presents the robust tracking of the reference trajectories. Es?pecially, compared to generic nonlinear disturb?ance observers, the UDE?based control strategy can provide more smooth system responses and control signals in the presence of high?frequency measurement noises.

Actually, it is not easy to select proper con?trol parameters that could satisfy the inequality condition described in Eq. (23) all the time.Therefore, our future work will focus on the is?sues of control parameter determination, which could make the proposed controller adapt to any situations of uncertainty and disturbance.