Mathematical Problems in Engineering

Volume 2015 (2015), Article ID 506906, 26 pages

http://dx.doi.org/10.1155/2015/506906

## Continuous Recursive Sliding Mode Control for Hypersonic Flight Vehicle with Extended Disturbance Observer

School of Automation Science and Electrical Engineering, Beihang University, Beijing 100191, China

Received 9 August 2015; Accepted 20 October 2015

Academic Editor: Ricardo Aguilar-López

Copyright © 2015 Yunjie Wu and Jianmin Wang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

A continuous recursive sliding mode controller (CRSMC) with extended disturbance observer (EDO) is proposed for the longitudinal dynamics of a generic hypersonic flight vehicle (HFV) in the presence of multiple uncertainties under control constraints. Firstly, sliding mode tracking controller based on a set of novel recursive sliding mode manifolds is presented, in which the chattering problem is reduced. The CRSMC possesses the merits of both nonsingular terminal sliding mode controller (NTSMC) and high-order sliding mode controller (HOSMC). Then antiwindup controller is designed according to the input constraints, which adds a dynamic compensation factor in the CRSMC. For the external disturbance of system, an improved disturbance observer based on extended disturbance observer (EDO) is designed. The external disturbance is estimated by the disturbance observer and the estimated value is regarded as compensation in CRSMC for disturbance. The stability of the proposed scheme is analyzed by Lyapunov function theory. Finally, numerical simulation is conducted for cruise flight dynamics of HFV, where altitude is 110000 ft, velocity is 15060 ft/s, and Mach is 15. Simulation results show the validity of the proposed approach.

#### 1. Introduction

Hypersonic flight vehicle (HFV) is a cost-effective and reliable space aircraft. It can realize prompt global responses for its high speed. So HFV has attracted more and more attention from the civil and military field. However, HFV is highly coupling and nonlinear aircraft, and it is extremely sensitive to changes in physical and aerodynamic parameters due to its peculiar structure and rigorous flight environment. Hence, it is a significantly challenging task to model and control HFV. Since the 1960s, a large number of scholars had done research in this field. Up to now, abundant research results have been achieved.

The aeropropulsive and aeroelastic properties of HFV were analyzed in detail by Chavez and Schmidt [1], which laid the foundation for establishing HFV model. Bolender and Doman built a nonlinear longitudinal dynamics model for the air-breathing HFV in [2, 3], which captured the complex interactions between the aerodynamics, propulsion system, and structural dynamics. Then, robust flight control systems based on Monte Carlo evaluation were synthesized in [4], and stochastic robust nonlinear dynamics inversion (NDI) control law was presented for longitudinal motion of HFV in [5]. For the sake of applying linear control method, control-oriented linearized model was derived in [6]. Obaid proposed minimax linear quadratic regulator (LQR) optimal controller in [7, 8] and minimax optimal linear quadratic Gaussian controller in [9] based on feedback linearization model of HFV by virtue of minimax linear quadratic theory, where the optimal control minimizes the maximum value of the quadratic cost function and gives an optimal solution for the selected cost function. The optimal controller can achieve good control performance; however, the construction of Riccati equation parameter matrixes in LQR and LQG is a difficult work. Qi et al. [10] designed an adaptive backstepping fault-tolerant control scheme for HFV by means of backstepping theory. Butt et al. [11] designed adaptive controller for HFV in terms of dynamic surface method. Xu et al. [12] proposed adaptive sliding mode control for longitudinal dynamics of input-output linearization model of HFV, in which the uncertain parameters were estimated by adaptive laws based on Lyapunov synthesis approach. Among all the above methods, sliding mode control is the most attractive method, because of its robustness to uncertainty, and it is simple and designed easily.

Terminal sliding mode control is a kind of finite time convergence control, which appeals to many researchers due to its controllable convergence time. Until now, there are numerous research results about terminal sliding mode control, such as continuous terminal sliding mode control design [13] and fast terminal sliding mode control [14]. Nevertheless, terminal sliding mode control usually has singularity problem, which reduces its performance. According to this, Feng et al. [15] proposed nonsingular terminal sliding mode control and applied it to the rigid manipulators successfully.

It is worth noting that HFV is a high-order nonlinear system. Hence, the low-order and linear sliding mode control is conservative and cannot acquire better performance. References [16, 17] have presented high-order sliding mode control for HFV, but their first sliding surfaces remain in a linear form. And the chattering problem of sliding mode control affects the engineering application. In addition, the operation of mechanical actuators for HFV is restricted, which is easy to reach saturation in the near space where the execution efficiency of actuators drops. According to the input saturation, a large number of research results can be consulted, such as [18–20]. While the articles about antiwindup control for HFV are few, only [21] takes into account actuator restraints under complex flight conditions. Aiming at all the above, a continuous recursive sliding mode control (CRSMC) under control constraints is proposed for HFV in this paper, in which a compensation factor is added in CRSMC for actuator restraints.

Extended state observer (ESO) was proposed by Han in [22]. In ESO, the disturbance is regarded as a system state and then state observer is constructed. Through tuning the observer gain, which is the only adjustable parameter, all states of the system can be estimated. At the same time, the disturbance is estimated as the extended state. ESO is a simple and effective observer, which has been applied in many projects successfully, such as speed control of induction motor drive [23], attitude tracking of rigid spacecraft [24], robotic uncalibrated hand-eye coordination control [25], and boiler-turbine-generator control systems [26]. In addition, Gao developed ESO by proposing scaling and bandwidth-parameterization theory [27], which simplifies and facilitates the parameter tuning process. Ginoya et al. [28] proposed an extended disturbance observer (EDO) on the basis of disturbance observer (DO), which was projected by Chen in [29–31]. The EDO is appropriate for the system having disturbances in all channels. In accordance with the theory of ESO and EDO, an improved disturbance observer (IEDO) is designed for the external disturbance of HFV in this paper. The external disturbance is estimated by IEDO. Then, the estimated value is regarded as compensation for the above CRSMC. Therefore, the proposed scheme of this paper is CRSMC under control constraints plus IEDO. The major contributions of this paper are as follows:(I)A novel recursive sliding mode control is applied in the flight control of HFV, in which the recursive sliding manifolds are nonlinear and nonsingular. In addition, this sliding mode controller is continuous due to its nonlinear reaching law. Compared with the linear sliding mode control for HFV in [12, 16, 17], the proposed scheme has faster response and is less conservative.(II)A compensation controller for control constraints is proposed for HFV, for which the articles are few. The compensation controller works after the CRSMC as a compensation factor, which not only plays a role in antiwindup controller but also has no effect on the performance of the CRSMC. The simulations in following part demonstrate its effectiveness.(III)A novel disturbance observer is designed for the external disturbance for HFV. The disturbance observer is improved extended disturbance observer (EDO) in [28], in which there are more parameters that need to be regulated because of each channel with multiple gain parameters. The improved EDO has only one parameter that needs to be regulated, which is more convenient for engineering application.

The remainder of this paper is organized as follows. Section 2 formulates the HFV longitudinal model and control problem. Section 3 reveals the design process of CRSMC under control constraints and IEDO for external disturbance in detail. The stability of the proposed scheme is analyzed in Section 4 by Lyapunov function method. Numerical simulations are conducted in Section 5. Section 6 provides conclusions of the paper.

#### 2. Model and Problem Formulation

##### 2.1. HFV Model

The control plant of this paper is a model of longitudinal dynamics of a generic HFV, which is developed by NASA Langley research center. The model consists of some differential equations that describe velocity, altitude, flight path angle, angle of attack, and pitch rate [12, 18], which are expressed as follows:where , , , , are velocity, altitude, flight path angle, angle of attack, and pitch rate of HFV, respectively; is moment of inertia of the aircraft and is mass; , , , and are, respectively, lift, drag, thrust, and pitching moment acting on the aircraft; is radial distance from Earth’s center. The detailed expressions of , , , and , and are, respectively, as follows:where , , , and denote, respectively, density of air, reference area, mean aerodynamic chord, and radius of the Earth; is the elevator deflection angle; , , , and , , are, respectively, relevant aerodynamic coefficient parameters aswhere means the external disturbance reflected on the elevator.

The engine dynamics are modeled as a second-order system aswhere and are throttle setting and throttle setting command, respectively; is damping ratio and is natural frequency of engine; is the external disturbance on behalf of torques and generalized elastic forces.

Because of the peculiar structure of HFV and complex flight conditions, some certain parameters uncertainties are taken into consideration; namely,where represents the nominal value of parameter and denotes the parameter uncertainties.

*Assumption 1. *The external disturbance is assumed to be upper bounded; that is, ; here, are known positive constants for .

*Assumption 2. *The parameter uncertainties in (5) are assumed to be bounded. And they satisfy the following conditions:where , , , , , and are all positive real constants.

*Remark 3. *According to nonlinear model (1), velocity and altitude are regarded as output variables, while the input variables are chosen as engine throttle setting command and elevator deflection angle . The control task is designing an appropriate controller such that the output variables track the relevant command in finite time in the presence of disturbance, respectively.

##### 2.2. Input-Output Linearization

Because the model of (1) is highly nonlinear and strong coupling, the linearized model is needed for the sake of designing control law. Here, the input-output linearization method is adopted for linearizing. In accordance with Remark 3, the linearized target is that the input variables appear apparently in the expressions of output variables . In terms of Nonlinear System Theory and employing mathematical tools of Lie derivative, the input variables can appear in the motion equations by differentiating three times and four times, respectively [12]; namely,where is the system state vector and , , , and are the system equations’ first-order and second-order partial differential to state variables, respectively, whose detailed expressions are exhibited in the Appendix.

The expressions of and can be divided into two parts aswhereGiven , thenwhere

Then, (10) is rewritten aswhere , .

*Assumption 4. *Matrix is assumed to be invertible.

*Remark 5. *Matrix is nonsingular during the entire flight envelope except on a vertical flight path [5] for the input-output combination. Hence, Assumption 4 is reasonable.

#### 3. Controller and Observer Design

In this section, a continuous recursive sliding mode tracking controller is firstly designed and then a compensation controller is designed for the input saturation. In the third subsection, an improved disturbance observer based on EDO is presented for estimating the external disturbance.

##### 3.1. Tracking Controller Design

Before introducing the tracking controller, two lemmas are presented which will assist in analyzing and proving the theorem.

Lemma 6 (see [13]). *Consider the following first-order nonlinear differential equation:where , , and . Then, the system state will converge to zero in finite timewhere is the initial condition; that is, .*

Lemma 7 (see [32]). *Consider the following first-order nonlinear differential equation:where , , , , , and . Then, the system state will also converge to zero in finite timewhere is the initial condition.*

*Proof. *Due to and , that is, , (15) can be changed asIn accordance with Lemma 6, the solution of (17) will converge to zero in finite timewhere and , . This completes the proof.

Theorem 8. *Consider nonlinear model (1) and linearized model (12) with Assumption 4; the system output variables and can track the reference commands and , respectively; meanwhile, they keep stable and robust in the presence of parameter uncertainties, if the controller is selected aswhere , , , , , , for , , and (; ) are the recursive sliding mode manifolds which are defined aswhere and are tracking errors and is strictly positive constant.*

*Proof. *Taking into account the above recursive sliding mode manifolds, the time differential of (20a), (20b), and (20c) is thatIn line with , (23) changes aswhereSimilarly, the time differential of (21a), (21b), and (21c) is thatIn terms of , (28) is changed aswhereLike that, (24) and (29) are written together asSubstituting (12) into (31) in the absence of external disturbance yieldsChoose a Lyapunov function candidate aswhere .

Taking first-order time derivative of Lyapunov function in (33) yieldsSubstituting (19) into (34), we obtain thatHence, the sliding mode manifolds and will converge to zero in finite time; that is, the finite convergence time is for . This means thatSubstituting (36) into (20c) and (21c), we obtainBy virtue of Lemma 7, the secondary sliding mode manifolds and will converge to zero in finite timeSimilar to the above process, the last level sliding mode surfaces and in (20b) and (21b) will also converge to zero in finite timeSubstituting and into (20a) and (21a), respectively, it is obtained thatIt is seen that velocity tracking error converges to zero. However, (41) is asymptotic convergence. Define an arbitrary small neighborhood for ; that is, , . Once the altitude tracking error converges to the neighborhood, it is considered that the altitude has tracked the command . So the approximate convergence time isSummarizing the above contents, the total convergence time for velocity channel isLikewise, the total approximate convergence time for altitude channel (in fact, the altitude channel is asymptotic convergence) isIn other words, it is obtained thatIn a word, the system output variables and can track the velocity command and altitude command and afterwards keep stable under the proposed controller in (19), respectively. Furthermore, in accordance with the character of the sliding mode control, the controller is robust in the presence of parameter uncertainties. The proof is completed.

*Remark 9. *It is worth noting the terms and in the location of the denominator for controller (19). They may cause singularity problem once the convergence speed of sliding mode surface equals zero. Hence, the following constraint is given:where is a small positive constant, .

*Remark 10. *There are three layers of sliding mode manifolds for each channel. In each channel, these sliding surfaces are recursive. The last layer sliding surface first arrives; in this moment, the system states start to move toward the secondary sliding surface. After a period of time, the second layer sliding surface arrives. Then, each sliding surface arrives successively. Finally, the system tracking error converges to zero in limited time. The flowchart of these three layers of sliding mode manifolds is shown in Figure 1.