Abstract

This paper investigates a tracking problem for flexible air-breathing hypersonic vehicles (FAHVs) with composite disturbance. The composite disturbance produced by flexible effects, parameter uncertainties, and external interferences is modeled as a kind of unknown derivative-bounded disturbance in this paper. Then a novel composite control strategy is presented for the nonlinear FAHV model with the composite disturbance, which combines a nonlinear disturbance-observer-based compensator (NDOBC) and a dynamic-inversion-based sliding mode controller (DIBSMC). Specifically, the NDOBC is constructed to estimate and compensate for the composite disturbance, and the DIBSMC is designed to track desired trajectories of velocity and flight path angle. Moreover, the uniformly ultimate boundedness of the composite system can be guaranteed by using Lyapunov theory. Finally, simulation results on a full nonlinear model of FAHVs demonstrate that the proposed nonlinear disturbance-observer-based sliding mode controller is more effective than the traditional DIBSMC. Specifically, it is shown that the chattering of traditional DIBSMC in presence of composite disturbances can be attenuated with the NDOBC.

1. Introduction

Flight control of air-breathing hypersonic vehicles (AHVs) is very important and difficult. AHVs use scramjet engines integrated with the airframe, so the vehicle dynamics display strong interactions between the elastic airframe, the propulsion system, and the structural dynamics (see [1–3] and references therein). In addition, significant flexible effects cannot be neglected in the control design due to the slender geometries and light structures of AHVs (see [4–6] and references therein). It has been shown that these strong couplings, undesired flexible effects, large parametric uncertainties, and various external disturbances may cause degradation of the performance of flight control systems. Thus, the desired control scheme should be robust enough to overcome these factors.

In the past few years, many effective robust control laws have been presented (see, e.g., [7–13]) for a nonlinear AHV model developed at NASA Langley Research Center. For example, an observer-based passive fault-tolerant control scheme was successfully applied for this AHV model with both parameter uncertainty and actuator faults in [13]. But only rigid modes were considered in the nonlinear AHV model. Thus a new FAHV model was developed in [6], which includes not only the interactions between the propulsion system and the airframe dynamics but also the strong flexible effects. After that, many nonlinear controllers have been applied effectively to this nonlinear model developed by Parker et al. and Fiorentini et al. (see, e.g., [14–18]). An effective adaptive sliding mode controller was designed for the linearized FAHV model in [15]. But the capability of the linear model to represent the dynamics and the coupling effects is limited. So nonlinear control methods were concerned for the nonlinear FAHV model in [16, 17]. However, it was based on control-design models without any consideration of rigid-flexible coupling dynamics. Due to these couplings, rigid-body state and control input will be affected by flexible modes. In fact, it has been shown that the undesired flexible effects may cause degradation to the performance of flight control systems (see [6, 18]). Therefore, the flexible effects were studied and transformed into elastic-mode-related terms that appeared in forces and moment in [18]. However, it may lead to a large amount of online calculations because many coefficients have to be estimated using adaptive methods. To avoid these problems, a novel nonlinear disturbance-observer-based sliding mode control strategy was proposed to achieve control objective and attenuate the composite disturbance produced by flexible effects, parameter uncertainties, and external interferences in this paper.

Recently, the disturbance-observer-based control (DOBC) method has attracted considerable attention as a robust control scheme. Many effective DOBC schemes have been developed for spacecraft, missiles, and hypersonic vehicles (see, e.g., [19–23]). Chen firstly designed a dynamic inversion controller based on disturbance observer for missile systems in [20]. An antidisturbance PD control scheme was successfully applied for the attitude control of flexible spacecrafts in [21]. Chen et al. proposed a robust control law using disturbance observer and neural networks for the linearized AHV model in [22]. Li et al. presented a composite controller based on disturbance observer for the nonlinear AHV model in [23]. However, the study for the nonlinear FAHV model with composite disturbance using DOBC strategy is still insufficient. In this paper, a novel composite control strategy, which combines a nonlinear disturbance-observer-based compensator (NDOBC) and a dynamic-inversion-based sliding mode controller (DIBSMC), is presented for the nonlinear FAHV model with composite disturbance.

In conclusion, the main contributions in this paper can be summarized as follows. (1) The composite disturbance produced by flexible effects, parameter uncertainties, and external interferences is formulated as a kind of unknown disturbance, which is considered in the control-oriented model and estimated by a nonlinear disturbance observer. (2) Combining a NDOBC with a DIBSMC, a nonlinear disturbance-observer-based sliding mode controller (NDOBSMC) is proposed to make velocity and flight-path angle track desired signals and reject the composite disturbance. (3) The stability of composite closed-loop system which includes two tracking-error equations and a disturbance estimation error equation is analyzed.

The remainder of this paper is organized as follows. In Section 2, the nonlinear FAHV model with composite disturbance is introduced. In Section 3, the NDOBC and the NDOBSMC are designed, and stability of the composite system is analyzed. In Section 4, the effectiveness of the proposed NDOBSMC is confirmed through numerical simulations. Conclusions are provided in Section 5.

2. Problem Formulation

A nonlinear model for the longitudinal dynamics of flexible air-breathing hypersonic vehicles, as given by Bolender and Doman in [6], is described by the following nonlinear equations:where , , , and , in which and are the mode shapes and and are mass densities. It is noted that a pitching disturbance moment, , is considered in this paper, which may originate from parameter uncertain, sensor noise, actuator error, and external disturbance.

The approximations of the forces and moment given by Parker et al. in [17] are described as follows:

Coefficients that appear in the forces and moment (i.e., , , , , , and ) are subject to uncertainty in this paper. Denote , , , , , and as nominal values of the coefficients. The uncertainty is modeled as an additive variable to the nominal values used for control design; then the coefficients can be expressed as , , , , , and . The forces and moment are given by , , , , and , where the detailed expressions of , , , , , , , , , , and are given in the appendix.

In this paper, the control design problem is to select control inputs, and , that force the outputs, and , to track commanded values, and , in the presence of the composite disturbance produced by the couplings, uncertainties, and external interferences.

3. Composite Controller Design

This section presents a novel composite controller with a hierarchical architecture for the nonlinear FAHV model. Firstly, a nonlinear control-oriented model (COM) with composite disturbance is provided. Secondly, a nonlinear disturbance observer (NDO) based compensator (NDOBC) is constructed to estimate and compensate for the composite disturbance. Then a NDO-based sliding mode controller (NDOBSMC) is designed to track the desired trajectories in the presence of composite disturbance. Finally, stability of the composite system is analyzed.

3.1. COM with Composite Disturbance

In this part, a novel COM with composite disturbance is obtained from the FAHV nonlinear model (1) by removing the altitude dynamic and flexible modes and setting the weak elevator couplings to zero. The new COM is given by the following set of equations:where , , and . Besides, the second-order dynamic equation for fuel equivalence ratio, , is added to obtain the COM with full vector relative degree. In addition, the composite disturbance, , is considered in this COM, which contains and a pitching disturbance moment, , produced by the couplings between flexible and rigid modes. Denote in order to simplify the following design of disturbance observer.

Remark 1. It is different from [17] that the composite disturbance, , produced by parameter uncertainties, external disturbances, and couplings between rigid and flexible modes is considered in the proposed COM.

3.2. NDOBC Design

The nonlinear COM described by (3) is a special case of the generic muti-input/multioutput (MIMO) nonlinear system:where state , control input , output , and composite disturbance . Nonlinear functions , , and are sufficiently smooth with respect to , and the concrete forms of , , and are given byin which

For the MIMO nonlinear system (4), a NDO is designed to estimate the unknown composite disturbance according to [20], which is given bywhere and are the estimate of the unknown disturbance and the internal state of the nonlinear observer, respectively. Besides, is a nonlinear function to be designed and the nonlinear observer gain is defined byDefine the estimation error by . The error dynamic is described aswhere is the first derivative of composite disturbance . It is assumed that in this paper, where is a known positive constant.

Substituting and into (9), the differential equation (9) can be rewritten aswhere . An appropriate fourth element of observer gain, , needs to be designed such that converge to zero.

The following result provides a design scheme for the disturbance observer (7) with guaranteed uniformly ultimate boundedness of system (9) or (10).

Lemma 2 (for details see [24]). Let be a domain that contains the origin and let be a continuously differentiable function such thatfor all and for all , where and are class functions and is a continuous positive definite function. Then, there is (dependent on and ) such that the solution of the systemsatisfiesMoreover, if and belongs class , then (13) holds for any initial state , with one restriction on how large is. Note that inequality (13) shows that is uniformly ultimately bounded (UUB) with the ultimate bound .

Theorem 3. For the estimation error system (10), if there exists a constant satisfyingwhere the fourth element of observer gain, , is chosen as and is a bounded nonlinear function with respect to satisfying , then the solution of system (10) is UUB.

Proof. Denote the Lyapunov functionComputing the derivative of along the trajectories of (10), it can be obtained that Thus it can be concluded from Lemma 2 that the solution of estimation error system (10) is uniformly ultimately bounded, if . Obviously, there exits a positive constant such that . And is a negative definite matrix when hold. According to Lemma 2, the ultimate bound of is ; that is, .

Remark 4. The NDO (7) reduces to a linear disturbance observer (LDO) if . Obviously, the performance of NDO is better than LDO when LDO takes the same constant as NDO.

3.3. NDOBSMC Design

The nonlinear COM described by (3) can be linearized completely using the input/output linearization technique of full state feedback, so the linearized model is developed by repeated differentiation of and as follows:where , , , , and . The detailed expressions of , , , and are given in the appendix.

The right-hand sides of (17) involve second derivatives of and . The expression of the second derivatives for and consists of three parts: the first part is control relevant, the second part is disturbance relevant, and the third part is neither control relevant nor disturbance relevant. Considerwhere and .

When is defined, the output dynamics of and can be written as follows:where control inputs, and , and composite disturbance, , appear explicitly. Disturbances, and , are produced by parameter uncertainties. The concrete forms of , , , , and are given byin which

Two decoupled sliding mode surfaces and are defined by and , where is tracking error of velocity, is tracking error of flight path angle, and and are strictly positive constants defining the bandwidth of the tracking error dynamics. The sliding mode surfaces and represent linear differential equations where their solutions and converge to zero exponentially with the time constants and , respectively.

Differentiating and , we havewhere and .

With (22), recall that estimation value of the disturbance is developed by the NDO (7) and the NDOBSMC can be designed aswhere and are sliding mode gains of the NDOBSMC (23), is the disturbance estimation value obtained by the NDO (7), and is assumed to be invertible.

Remark 5. When the output of disturbance observer , that is, there is no disturbance observer, the NDOBSMC (23) obviously reduces to a DIBSMC, which is written as

As discussed in [25], the control laws (23) and (24) may result in control chattering because of the discontinuity across the sliding surfaces. As a practical matter, chattering is undesirable because it involves very high control action and may excite high frequency dynamics neglected in the modeling. The discontinuity in the control law can be dealt with by defining two thin boundary layers of widths and around the sliding mode surfaces, that is, replacing and with continuous saturation functions and , whereSo the NDOBSMC (23) and the DIBSMC (24) are modified as follows:

Considering (10) and substituting (26) into (22), the augmented closed-loop system can be obtained as follows:

Next, uniform ultimate boundedness (UUB) of the composite system (28) will be considered.

Theorem 6. For the composite system (28), if the sliding mode gains, and , satisfyand the fourth element of observer gain, , is chosen as with the constant satisfyingthen the solutions of composite system (28) are UUB, where positive constants , , , , and are tuning parameters to obtain the appropriate values of control gains, and positive constants , , , , and are the upper bounds of , , , , and ; that is, , , , , and .