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Mathematical Problems in Engineering
Volume 2015 (2015), Article ID 293480, 11 pages
http://dx.doi.org/10.1155/2015/293480
Research Article

Robust Adaptive Attitude Control for Airbreathing Hypersonic Vehicle with Attitude Constraints and Propulsive Disturbance

1College of Energy and Power Engineering, Nanjing University of Science and Technology, Nanjing 210094, China
2College of Automation Engineering, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, China

Received 16 July 2015; Accepted 26 November 2015

Academic Editor: Marek Lefik

Copyright © 2015 Jian Fu et al. 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 robust adaptive backstepping attitude control scheme, combined with invariant-set-based sliding mode control and fast-nonlinear disturbance observer, is proposed for the airbreathing hypersonic vehicle with attitude constraints and propulsive disturbance. Based on the positive invariant set and backstepping method, an innovative sliding surface is firstly developed for the attitude constraints. And the propulsive disturbance of airbreathing hypersonic vehicle is described as a differential equation which is motivated by attitude angles in this paper. Then, an adaptive fast-nonlinear disturbance observer for the proposed sliding surface is designed to estimate this kind of disturbance. The convergence of all closed-loop signals is rigorously proved via Lyapunov analysis method under the developed robust attitude control scheme. Finally, simulation results are given to illustrate the effectiveness of the proposed attitude control scheme.

1. Introduction

In recent years, a new kind of aerospace vehicle, which is called airbreathing hypersonic vehicle (AHV), has attracted the considerable attention in both military and civil communities. Comparing with conventional flight vehicles, this new aerospace vehicle can sustain flight for a long time with high flight speed and cover a large envelope [1, 2]. Due to the strong disturbance and fast changed pneumatics parameters, it is still a challenging task to design a robust AHV control system. Therefore, considerable efforts have been made to develop an efficient flight controller for this kind of aerospace vehicles. In [3, 4], adaptive function link network control and fuzzy logical system were proposed for the aerospace vehicle with uncertainties. And baseline nonlinear model predictive controller was presented in [5] for AHV with nonvanishing mismatched disturbances. In [6], an adaptive fault-tolerant control strategy with input saturation is proposed for AHV flight system with actuator faults and external disturbance. However, the robust attitude control method should be further developed for the AHV with attitude constraints and propulsive disturbance.

In order to suppress the unknown external disturbance from changeable flight environment, many papers have focused on the disturbance observer of AHV flight control system [711]. A nonlinear disturbance observer based control (NOBC) method is proposed in [7] for the longitudinal dynamics of a generic airbreathing hypersonic vehicle under mismatched disturbance. In [8], the NDO technique is combined with Takagi-Sugeno (T-S) fuzzy linear model to design a disturbance observer based guaranteed cost fuzzy controller (GCFC) for the AHV flight systems. To achieve satisfactory tracking performance, a composite tracking controller is proposed for AHV using NDO and backstepping technologies [9]. However, there are few disturbance observer based control results for AHV with the propulsive disturbance. This kind of disturbance for AHV is usually motivated from the coupling between the scramjet propulsion system and attitude angles [10, 11]. Thus, this kind of disturbance will lead to a challenging problem for the attitude control of the AHV. In this paper, the propulsive disturbance is described as a differential equation which is motivated by attitude constraints in this paper. And a composite controller is proposed to handle this disturbance using adaptive NDO and backstepping technologies.

Furthermore, sliding mode control (SMC) is considered as an important method due to its high precision control, relative simplicity of control design, and high robust features with respect to system internal perturbations and external disturbances. Thus, the SMC and its application of AHV have been widely studied in [1221]. A composite control treatment based on sliding model method is presented to the nonlinear longitudinal dynamics of AHV vehicles [12]. And dynamic terminal sliding mode technique is proposed for the robust control design of AHV [13, 14]. Considering the input constraints, a second-order terminal sliding mode control method is discussed for AHV flight system in reentry phase [15]. Meanwhile, an adaptive dynamic sliding mode control method is presented for the fault-tolerant control problem of AHV in [16]. However, most of the AHV control systems combining backstepping control with SMC have not considered the attitude constraints. Thus, it is necessary to design an innovative sliding mode control for the AHV using backstepping technique and considering the attitude constraints.

This work is motivated by the robust attitude control of AHV with attitude constraints and propulsive disturbance. The control objective is that the proposed robust control scheme can track a desired trajectory in the presence of the unknown propulsive disturbance and attitude constraints. An innovative sliding surface is firstly developed in this paper. And the invariant sets and saturation function are utilized in this surface for the attitude constraints of AHV. To guarantee the control effects under propulsive disturbance, the switching information from invariant-set-based sliding surface is included in the adaptive disturbance observer to increase the convergence speed. Therefore, the organization of the paper is as follows. Section 2 details the problem formulation, while main results are given in Section 3. Simulation results are presented in Section 4 to show the effectiveness of proposed robust adaptive backstepping attitude control for AHV. And some conclusion results are shown in Section 5.

2. Problem Statement

2.1. Model Description

To study the robust attitude controller, a nonlinear attitude motion model of AHV flight system is described as the following nonlinear affine nonlinear system [2224]:where is the attitude angle vector of slow-loop states, is the body-axis angular rate vector of fast-loop states, is the deflection vector of control surfaces which are the system control inputs, is the system output, and the unknown disturbance is defined aswhere is the unknown propulsive disturbance which is coupling with the attitude angles and is the rest of unknown disturbance. The detailed expression of ,  ,  , and  can be found in [24].

2.2. Assumptions and Control Objective

In order to reduce the coupling effects between the scramjet propulsion system and attitude angles, the attitude constraints are defined aswhere is the attitude angle vector and is the desired output signal. When the attitude angle vector satisfies constraints (3), the propulsive disturbance will be a small value which can be neglected in the control design process. However, if exceeds the boundary of constraints (3), this disturbance will be increased significantly. Then, it is supposed that the disturbances and will satisfy the following assumptions.

Assumption 1. For all , there exists such that ,  .

Assumption 2. The propulsive disturbance can be described as the following differential equation:where , and we can obtain if . For the element , there exists an unknown positive constant such that ,  . The vector is a special designed sliding surface as shown in (22), and there exists while .

Assumption 3. The disturbance can be described as the following differential equation:where , and there exists a known positive constant such that ,  .

Assumption 4. The generalized matrix inverses of and are always existing for the nonlinear attitude motion model of the AHV, and the elements in matrixes , , , and are always continuous for the nonlinear attitude motion model of the AHV.

Definition 1 (see [25]). The set is said to be positively invariant (PI) for nonlinear system (1), if, for all , there exists the solution ,  .

3. Robust Adaptive Backstepping Attitude Control for the AHV with Attitude Constraints and Propulsive Disturbance

3.1. Preliminary

In this section, a few results are given to support the design process of robust adaptive backstepping attitude controller. Considering the standard backstepping control design method, we definewhere is a designed virtual control law. Then, considering (1), the derivatives can be written aswhere are the state errors of slow-loop states and are the state errors of fast-loop states. Invoking (6), the attitude constraints (3) can be rewritten aswhere and .

Considering the attitude constraints of AHV, the unidirectional auxiliary surfaces (UAS) for slow-loop and fast-loop states are utilized to design the invariant-set-based sliding mode controller. The detailed design process for these surfaces is given as follows.

Step 1. Considering the slow-loop system (7), we define the switching surfaces aswhere ,  ,  , ,  ,  , and is denoted by for brevity. The conditions and   are given to guarantee the stability of switching surfaces and . And the condition is used to avoid the overlap of switching surfaces.

Step 2. Invoking the coefficients in (10), the unidirectional auxiliary surface can be designed aswhere where are the designed parameters which satisfy ,  , and  , and the rest of coefficients are given as follows:where and  . Then, the compact form of unidirectional auxiliary surfacewhere ,  ,  ,  ,   and is a constant vector.
As shown in Figure 1, the UAS can form a convex set which satisfies , and the expression of set is written asThe compact form of convex set can be written as , where implies ,   .

Figure 1: The convex set in attitude constraint .

Step 3. Similarly, we can define the following switching surfaces for the fast-loop system (8):where , ,  , , and  ,   .

Step 4. Invoking the coefficients in (16), the unidirectional auxiliary surface can be designed aswhere where ,  , and  are the designed parameters, and the rest of coefficients can be designed as follows:where and   . Then, the compact form of unidirectional auxiliary surfacewhere ,  ,  ,  , and is a constant vector. Then, there exist the following results for the unidirectional auxiliary surfaces and  .

Lemma 2. For the points and  ,  , it follows that and . And if , there exists ; if , there exists .

Proof. See Appendix A.

Lemma 3. For all ,  , the functions and in Lemma 2 are continuous functions.

Proof. See Appendix B.

3.2. Robust Adaptive Backstepping Attitude Control Based on Disturbance Observer and Attitude Constraints

In the attitude control design, we combine the backstepping method, adaptive invariant-set-based sliding mode control, and the disturbance observer technique to design a robust adaptive backstepping attitude controller for the AHV system (1) with attitude constraints and propulsive disturbance, and the detailed design process is appended as follows.

3.2.1. Design Process for the Controller

Step 1. Invoking (14) and (20), we can define and aswhere and . Then, the innovative sliding mode surface can be defined aswhere , , , and is the saturation function. From (22), it is clear that

Remark 4. It is clear that when . Considering in (15), we can obtain that while . Therefore, the attitude constraints will be satisfied when system states are sliding on the sliding surface .

Step 2. In this paper, the nonlinear disturbance observer which is employed to estimate is defined aswhere is a design parameter which should be chosen to render ,  .

Assumption 5. For the estimation error , there exists positive constant such that ,  , where the coefficient is a bounded element in (4).

Step 3. To suppress the estimation error of disturbance, an adaptive item is defined aswhere and is a design parameter.

Step 4. Defining and and considering (7), the virtual control law is designed aswhere ,  ,   are designed approaching laws. Using the output of the designed nonlinear disturbance observer (24) and considering Assumptions 3 and 4, the robust attitude control law is designed aswhere is the generalized matrix inverse of , , and  ,  .

Remark 6. Defining and considering and , we can obtain

3.2.2. Proof for the Stability of Sliding Mode Surface

The discussion for the stability of sliding mode surface is given in this part. Considering , the control law in (26) can be rewritten as follows:

Substituting (29) into (7) yields

Choose the Lyapunov function candidate asFrom Lemmas 2 and 3, it is clear that the Lyapunov function is a positive and continuous function. And there exists while . Invoking (14) and (21), the time derivative of can be written as

Substituting (30) into (32) yields

It is apparent that the first term on the right-hand side of (33) is stable. Furthermore, the second term will be canceled as follows.

Invoking (4), it is noted that the time derivative of propulsive disturbance while . Then, we can obtain from (2), (3), and (4) and . Considering the definition of and substituting (27) into (8) yieldFor considering the stability of all signals for the closed-loop control system, the Lyapunov function candidate is chosen asThe time derivative of isInvoking (8), (21), (24), (27), and Assumption 3, the time derivative iswhere .

Theorem 7. Considering the error systems (7) and (8) with the unknown propulsive disturbance, the disturbance observer is designed as (24) and the robust backstepping attitude control scheme is proposed as (26) and (27). Then, the attitude control error of AHV is convergent when the system states are sliding on the innovative sliding surface in (22).

Proof. Invoking (37) and considering ,  , and ,  , we have Thus, the tracking errors and the disturbance approximation error are convergent while the system states are sliding on the innovative sliding mode surface (22).

3.2.3. Proof for the Reaching Phase for Sliding Mode Surface

The discussion for the reaching phase for sliding surface is given in this part. Choose the Lyapunov function candidate asFrom Lemmas 2 and 3, it is clear that the Lyapunov function is a positive and continuous function, and there exist while . Assume that and considering (23), there exists the following equation:Substituting (7) and (26) into (40) yieldsInvoking (26) and (39), the time derivative of can be written asSubstituting (25), (28), and (40) into (42) yieldsInvoking (2), (4), and (5) yields

Considering the stability of all signals for the closed-loop control system, the Lyapunov function candidate is chosen asThe time derivative of isInvoking (21), (24), (27), (43), and (44), the time derivative of isConsidering Assumption 5, , (22), and (23), we haveSubstituting into (48) yields

Theorem 8. Considering the attitude motion dynamics (1) of the AHV with attitude constraints and unknown propulsive disturbance, the disturbance observer is designed as (24) and the robust backstepping attitude control scheme is proposed as (26) and (27). Then, attitude controller error of AHV is convergent under the attitude constraints (3). Meanwhile, the system states will arrive at the sliding surface , and this surface also can be proved as a positively invariant set.

Proof. According to (49), we haveSubstituting ,  , and  into (50) yieldsThus, the system states will arrive at the sliding surface . From Remark 4, it is noted that attitude constraints (3) will be satisfied when system states are sliding on sliding surface .
Considering the stability of sliding surface as shown in Theorem 8, the attitude control error of AHV is convergent under the attitude constraints (3). From (39), (45), and (51), there exists for all if . And the sliding surface can be rewritten as a closed set as shown in (15) and Remark 4. Then, the sliding surface can be seen as a positively invariant set according to Definition 1.

4. Simulation Results

In this section, simulation results are given to illustrate the effectiveness of the proposed adaptive UAS-SMC schemes for AHV with attitude constraints. Suppose that the AHV vehicle lies in the cruise flight phase with the velocity 3000 m/s and flight altitude 30 km. The initial attitude and attitude angular velocity conditions are chosen as ,  deg,   deg, and  deg/s. The attitude constraints for the state errors are given as In the simulation, we assume that the unknown time-varying disturbance moments imposed on the AHV arewhere ;  ;  ;   and the coefficients , , and can be found in [24].

According to the design steps in Section 3, the adaptive UAS-SMC control design parameters are chosen as ,  ,  , ,  ,  , and  ;   and  ,   . The detailed approaching laws ,   can be found in [25], and the innovative sliding mode surface can be defined as Then, the simulation results are given as in Figures 25.

Figure 2: The attitude responses of AHV system under SMC and UAS-SMC methods.
Figure 3: The state error responses under SMC + NDO and adaptive UAS-SMC + NDO.
Figure 4: The disturbance estimations under NDO [3] and UAS-NDO with the same coefficient .
Figure 5: Adaptive UAS-SMC + NDO attitude control input.

The attitude and state error responses are shown in Figures 25 under SMC and adaptive UAS-SMC methods, respectively. From Figure 2, attack angle in system (1) is unstable when propulsive disturbance is not considered in the design process of SMC controller. To handle this problem, the conservative parameters and nonlinear disturbance observer are given in the SMC control design as shown in Figure 3. However, the undesirable overshoots are often found with conservative parameters since attitude constraints have not been considered in the design process of SMC scheme. And these overshoots are harmful for the AHV system because they might be out of the attitude constraints as shown in Figure 3. From the state error responses in Figure 4, the attitude constraints can be satisfied with adaptive UAS-SMC method. And harmful overshoots will be removed by the designed positively invariant sets. Therefore, we know that the proposed adaptive UAS-SMC control scheme can efficiently track the desired trajectories with attitude constraints and propulsive disturbance. Figure 4 shows the disturbance estimation under NDO [3] and UAS-NDO with the same coefficient . Unlike the previous NDO, the switching matrix of UAS is introduced in the UAS-NDO. This matrix will improve the performance of disturbance observer (24). Then, we can obtain a new fast-nonlinear disturbance observer for the time-varying propulsive disturbance of AHV systems.

5. Conclusion

In this paper, the adaptive UAS-SMC controller with nonlinear disturbance observer has been proposed for the AHV with the unknown propulsive disturbance and the attitude constraints. To handle the propulsive disturbance, a developed fast-nonlinear disturbance observer is proposed to estimate the propulsive disturbance using adaptive method. And an innovative sliding surface is firstly proposed for the attitude constraints with invariant set theory. Rigorous analysis has been given for the convergence of all closed-loop signals under the proposed control schemes. Simulation results show the effectiveness of the robust adaptive UAS-SMC scheme for the AHV. In the following study, the robust attitude control scheme can be further developed for the AHV with time-varying attitude constraints.

Appendices

A. Proof of Lemma 2

Take the point and as examples; we have the following discussions.

From the definition of in (11), the function is switching among the following subspaces: Since the proofs for the other subspaces are similar, we just prove the conclusions in No. , No. in this lemma.

Invoking (11) and (13), the function can be written as the following equation for every point in No. Subspace,Since the point is located in No. Subspace, there existsFrom (10) and (11), it is noted that ,  , and . Then, we can obtain that for every point in No. Subspace.

Invoking (11) and (13), the function can be written as the following equation for every point in No. Subspace:Since the point is located in No. Subspace, there existsFrom (10) and (11), it is noted that and . Then, the coefficient can be expressed aswhere . Substituting (A.6) into (A.4) and considering (A.5) yieldThus, there exists for every point in No. Subspace.

Similarly, we have for every point in No. and Subspace. According to above discussion, there exists for every , ,  . And the function will satisfy the following equation: When , it is clear that the point is located in No. Subspace. Substituting into (A.7) and considering ,  , and  yield Considering , we have ,  . Hence, if , there exists . And we can also obtain the similar conclusion for and .

B. Proof of Lemma 3

Take as an example; we have the following discussions.

Invoking (11) and (13), the function can be written aswhere From the definition of , function is switched on switching surfaces and . Then, the discussion for the continuity of function can be given as follows.

For every point on switching surface , (10) yieldsAssuming and considering the definition of , there exists (B.4) for and  :Substituting (B.3) into (B.4) yieldsAssuming and considering the definition of , there exists (B.6) for and   :Substituting (B.3) into (B.6) yieldsSince , function is continuous on switching surface while . Similarly, function is continuous on switching surface while . Hence, is a continuous function for every point on switching surface . And we can also have the same result for every point on switching surface .

From the above discussion, function is continuous on switching ,  . Then, we can obtain that the function in Lemma 2 is a continuous function for all . Meanwhile, we can obtain the same conclusion for in Lemma 2.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This work was partially supported by National Natural Science Foundation (NNSF) of China under Grants 61374212, 61174102, 61304099, and 11402117; Postdoctoral Fund of Jiangsu Province 1401023; Pre-Research Foundation of General Equipment Department 9140C300305140C30140.

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