Adaptive sliding mode tracking control for a flexible air-breathing hypersonic vehicle

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Abstract

This paper is concerned with the adaptive sliding mode control (ASMC) design problem for a flexible air-breathing hypersonic vehicle (FAHV). This problem is challenging because of the inherent couplings between the propulsion system, the airframe dynamics and the presence of strong flexibility effects. Due to the enormous complexity of the vehicle dynamics, only the longitudinal model is adopted for control design in the present paper. A linearized model is established around a trim point for a nonlinear, dynamically coupled simulation model of the FAHV, then a reference model is designed and a tracking error model is proposed with the aim of the ASMC problem. There exist the parameter uncertainties and external disturbance in the model, which are not necessary to satisfy the so-called matched condition. A robust sliding surface is designed, and then an adaptive sliding mode controller is designed based on the tracking error model. The proposed controller can drive the error dynamics onto the predefined sliding surface in a finite time, and guarantees the property of asymptotical stability without the information of upper bound of uncertainties as well as perturbations. Finally, simulations are given to show the effectiveness of the proposed control methods.

Introduction

Air-breathing hypersonic vehicles (AHVs) are crucial because they may represent a more cost efficient way to make access to space routine, or even make the space travel routine and intercontinental travel as easy as intercity travel [1], [2]. Although ordinary rocket-based propulsion systems can reach orbital speeds, they are much more expensive to operate in that they must carry oxygen. Being different from ordinary flight vehicles, AHVs adopt scramjets [3], so the AHVs can carry more payload than rocket-powered ones. AHVs use the technology of airframe integrated with scramjet engine configuration [4], which makes the interactions between the elastic airframe, the propulsion system, and the structural dynamics very strong [5]. In addition, the requirements of flight stability and high speed response, the existence of various random interference factors and large uncertainties make it more difficult [6], [7].

Flight control design for AHVs is highly challenging, due to the sensitivity changes in fight condition and the difficulty in measuring and estimating the aerodynamic characteristics of the vehicle. Thus, the problem of flight control design is one of the key techniques to the application of AHVs. Because of the dynamics' enormous complexity, only the longitudinal dynamics models of AHVs have been used for control design. In [8], [9], a comprehensive analytical model of hypersonic vehicles was first developed. This model does not has the flexible models, but is still highly nonlinear, multivariable and has strong couplings between the propulsive and aerodynamic effects [10]. Based on this model, several results are available in the literature, see, e.g. [11], [12], [13], [15], [16]. In addition, due to the slender geometries and light structures of this generic vehicles, significant flexible effects cannot be neglected in the control design [19]. A flexible air-breathing hypersonic flight vehicle (FAHV) model, which includes the flexible dynamics, was developed in [17], [18]. For this kind of models, control design and simulation have been studied in resent years. The equations of this models become exceedingly complex when flexibility effects are considered, so these models can be used only for simulations or validation purposes [6], [7]. In [19], a control-oriented model was derived for the FAHV models using curve fits calculated directly from the forces and moments included in the truth model, then an approximate feedback liberalization example of control design was given to derive a nonlinear controller. In [20], the authors presented two output feedback control design methods for the FAHV models, and adaptive control techniques were also considered in [21]. In [22], dynamic output feedback techniques was used to provide reference robust velocity and altitude tracking control in the presence of model uncertainties and varying flight conditions, and in [23], [24], linear controllers with input constraints using on-line optimization and anti-windup techniques were also proposed. More recently, a nonlinear robust adaptive control design method was presented in [25], and in [26], the authors considered the modeling of aerothermodynamics effects and gave a Lyapunov-based tracking controller.

In practice, it is difficult to measure or estimate the atmospheric properties and aerodynamic characteristics in the flight envelop of FAHV [27]. Therefore, modeling inaccuracies are always in existence, also, various disturbances are presented. This can result some strong adverse effects on the performance of FAHV control systems. Therefore, robust control has been the main technique used for FAHV flight control [28]. On the other hand, sliding mode control (SMC) is well known for its robustness to deal with parametric uncertainties and external disturbances for dynamic systems [39], [35], [14]. In real system, the upper bound of uncertainties and disturbances are difficult to obtain, and this will reduce the system's robustness. Adaptive control law could lead to a stable closed-loop system and the deviation from the sliding surface is bounded [29]. The adaptive sliding mode control design method for the stability problem have been discussed in [30], [31], [32], [41], but the result on tracking problem is very limited. Also, few results are given on the unmatched disturbances. As mentioned above, in [16], an adaptive sliding controller was designed for the model developed in [8], [9], but unfortunately, this method cannot be used on the model developed in [17], [18], since the model is non-minimum phase.

Motivated by the above discussions, in this paper, we propose a robust adaptive SMC design method for the longitudinal model of FAHV with system parameter uncertainties and nonlinear perturbations. Firstly, for the longitudinal motion of the FAHV at a special trim condition, a linearized model is formulated for the control design problem. Then, based on the given reference model, a robust sliding surface is developed and the robustness of the designed sliding surface is also discussed. An adaptive law is proposed such that the tracking dynamic is globally stable without the information of disturbances' upper bound. Finally, an illustrative example is provided to show the effectiveness and advantage of the proposed control design methods.

The rest of this paper is organized as follow. Section 2 presents the model of FAHV and the control objectives of this article. Sections 3 and Section 4 give the main results on the robust adaptive SMC design for the longitudinal model of the FAHV. Simulation results are given in Section 5 and we conclude this paper in Section 6.

Notation: The notations used throughout the paper are fairly standard. Throughout this paper, the superscript “T” stands for matrix transposition; and Rn denotes the n-dimensional Euclidean space and Rn×m denotes the set of all n×m real matrices; I and 0 denote the identity matrix and zero matrix with compatible dimensions. · refers to the Euclidean vector norm or spectral matrix norm. Matrices are assumed to be compatible for algebraic operations if their dimensions are not explicitly stated.

Section snippets

Model description and control objectives

The hypersonic vehicle model considered in this paper is developed by Bolender and Doman [17], [18]. Flexibility effects are included in the equations of motion for the longitudinal dynamics of FAHV, by modeling the vehicle as a single flexible structure, whereas the scramjet engine model is adopted from Chavez and Schmidt [9]. A longitudinal sketch of the vehicle is given in Fig. 1. The nonlinear equations is described as follows:h˙=Vsin(θα),V˙=1m(TcosαD)gsin(θα),α=1mV(TsinαL)+Q+1Vcos(θα

Robust stability analysis of the sliding mode

To stabilize the tracking error's dynamic (8), the SMC technique is utilized here. First, a sliding mode surface function is designed asσ(e(t),t)=Se(t)=BTP1e(t),where PRn×n is a positive-definite matrix which needs to be designed. Similar to the methods in [38], a transformation matrix is defined as follow:T=(B˜TPB˜)1B˜T(BTP1B)1BTP1=T1T2,where B˜ is any basis of the null space of BT. For a given matrix B, B˜ is non-unique, but any choice satisfying the condition is acceptable. T1R(nm)×n

Adaptive SMC synthesis

In the above section, a sufficient condition to ensure the asymptotic stability of the sliding mode dynamics is discussed. After designing the sliding surface, the next phase of the traditional SMC is to design an appropriate SMC law such that the error dynamics will be driven onto the sliding surface, and remain on it. When utilizing the conventional SMC technique, it is necessary to have the information of the upper bound of ψ(x(t),t) in order to design a control law with switching part

Simulation results

In this section, an example is provided to illustrate the effectiveness of the robust adaptive sliding mode control proposed in the previous sections. The hypersonic vehicle model parameters are borrowed from [19]. The equilibrium point of the nonlinear vehicle dynamics described by the system in Eq. (1) is listed in Table 1. By using the parameters, the matrices A and B in Eq. (2) can be written as A=0077027702000000.2433×1030.1349×10220.9531.92000001.769×1071.027×1060.0696101000000001

Conclusion

In this paper, a robust adaptive SMC strategy has been presented for the tracking control problem of the longitudinal dynamics of FAHV model. The linearized model and a reference model have been established, and an error dynamic model has been obtained. Then, an adaptive SMC law has been proposed, which can guarantee the global stability of the closed-loop system without the information of the upper bound of the uncertainties and the external disturbance. Simulation results have validated the

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    This work was partially supported by National Natural Science Foundation of China (61174126, 90916005, 61025014 & 60736026), Aviation Science Fund of China (2009ZA77001), and the Natural Science Foundation of Heilongjiang Province of China (F201002).

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