Research Article | Open Access

Guan Wang, Li Li, Weihua Li, Huajun Zhou, Changbo Ma, Hongwei Xia, "Disturbance Observer-Based Adaptive Control of Hypersonic Vehicles with Constrained Actuators", *International Journal of Aerospace Engineering*, vol. 2020, Article ID 8856676, 12 pages, 2020. https://doi.org/10.1155/2020/8856676

# Disturbance Observer-Based Adaptive Control of Hypersonic Vehicles with Constrained Actuators

**Academic Editor:**Wu Zhonghua

#### Abstract

This study investigates an adaptive controller for the flexible air-breathing hypersonic vehicles (AHVs) subject to external disturbances and actuator constraints. The combination of nonlinear disturbance observer and adaptive mechanism is exploited to design an adaptive controller for each subsystem. For the velocity subsystem, an auxiliary system is employed to handle the scramjet input saturation issue. For the altitude subsystem, the magnitude/rate constraints and the dynamics of aerodynamic control surfaces are addressed by the control allocation module. Simulations show the effectiveness of the proposed control.

#### 1. Introduction

Air-breathing hypersonic vehicles (AHVs) have attracted considerable attention in both military and civil fields. Though AHVs have been developed for more than half a century, various studies have shown that some critical technologies, such as thermal protection, high-fidelity modeling, and robust control, still need to be further investigated.

Based on the control-oriented model developed by Parker et al. [1], numerous theoretical studies on the control of AHVs have sprung up [2]. Early research interests focus on state feedback linearization or nonlinear dynamic inverse-based control of AHVs [3]. In recent years, the adaptive control has always been a research hotspot in the field of AHV control, and one can refer to the robust adaptive control in [4, 5], the sliding mode adaptive control in [6], the trajectory restricted adaptive control in [7], the fault-tolerant adaptive control in [8, 9], and the neural adaptive control in [10, 11].

Nevertheless, a practical problem in AHV control is the physically limited actuators [12, 13]. Recently, some outstanding works have come out focusing on the input saturation problem of AHVs [14, 15]. [16] used the disturbance observer-based (DOB) feedback linearization control to overcome constraints on actuators. A combination of back-stepping design and nonlinear disturbance observer (NDO) is utilized to handle actuator constraints [17]. By introducing the sliding-mode differentiator techniques and the auxiliary systems, [11] developed the distributed finite time fault-tolerant containment control to handle the input saturation. On the other side, control allocation (CA) also plays a good role in dealing with the actuator constraints [18]. For instance, [19] handled the saturation using a robust model predictive control algorithm. [20] developed a control allocation scheme for fault-tolerant control based on a linear parameter varying (LPV) system. But the employed LPV model was complex and time-demanding for the CA problem.

Motivated by the above investigations, this paper proposes a disturbance observer-based adaptive control AHVs with constrained actuators. In the velocity loop, the auxiliary system developed in [12] is employed to handle the saturation issue of the scramjet input. In the altitude loop, a control allocation module is exploited for the magnitude/rate constraints and the dynamics of aerodynamic control surfaces. The main contributions of this paper can be briefly outlined as follows: (1)The saturation problem of the scramjet, as well as the dynamics and magnitude/rate constraints of the control surface, is taken into consideration(2)NDO is developed in each step to further improve the disturbance rejection property. The observation errors are suppressed by adaptive control laws

The rest of this paper is organized as follows: Section 2 formulates the considered problem; Section 3 presents the design process; Sections 4 and 5 give a stability analysis and a simulation study, respectively; Section 6 concludes this study.

#### 2. Problem Statement and Preliminaries

##### 2.1. Control-Oriented AHV Model

In this paper, we consider the longitudinal control-oriented AHV model developed by Parker et al. [1]:

In (1)–(5), velocity , altitude , flight path angle (FPA) , angle of attack (AoA) , and pitch rate (PR) are system states; is the acceleration due to gravity; mass and moment of inertia are decided by the design and fuel level of AHVs; the disturbances , , and are external disturbances with bounded derivatives; is the th generalized elastic coordinate; and are the natural frequency and the damping ratio for flexible mode , respectively; and are coupling coefficients between rigid-body and flexible dynamics; thrust , lift , drag , pitching moment , and th generalized force have the following curve-fitted approximations:

In (7)–(10), scramjet fuel equivalency ratio (FER) , canard deflection angle (CDA) , and elevator deflection angle (EDA) are three system inputs, respectively; is the dynamic pressure; is the reference area; is the thrust-to-moment coupling coefficient; is the mean aerodynamic chord; denotes the fitting coefficients of force and moment.

##### 2.2. Problem Formulation

From a practical viewpoint, FER should comply with the saturation property: where is the FER command; is the constant lower threshold to keep sufficient heat dissipation through active cooling; the upper bound is required to prevent the scramjet from thermal choking [12].

In engineering, the magnitude and rate constraints of the actuators (canard/elevator) have a great influence on control performances. where denote EDA or CDA ; and are lower and upper bounds of the magnitude; and are the corresponding rate bounds. It should be pointed out that the upper and lower amplitude limits of a control surface can be asymmetric in reality, that is, the more common case or is also taken into consideration.

At the end of this subsection, several lemmas crucial for the following design are given.

Lemma 1 (see [21]). *, where , , and .*

Lemma 2 (see [22]). *If a function satisfies with , , it has
*

Lemma 3 (see [23]). *Consider a scalar nonlinear system
where and are system state and control input, respectively; and are known scalar functions with respect to . represents the disturbance and , where is an unknown positive constant.**NDO is constructed as
where is the estimation of ; with ; .**Defining the estimation error , is governed by
The estimation error system (18) is bounded-input-bounded output stability, that is, is bounded with bounded .*

#### 3. Controller Design

##### 3.1. Velocity Tracking Control Design

This section presents an adaptive tracking controller for velocity subsystem (1) ensuring . The velocity dynamics is first expressed in the following form: where

First, similar to [23], we introduce a modified tracking error to handle the saturated as where is governed by with . Then, is calculated as

Taking into account the bounded and the finite deflecting velocity of physical control surfaces, is assumed to be bounded. Hence, the NDO for is constructed.

Defining estimation error as , it has

For the fact that is bounded, is also bounded from Lemma 3, i.e., with .

We design the control law where is estimated by NDO and is the estimate of , generated by the adaptive law: where , , and .

##### 3.2. Altitude Tracking Control Design

In the altitude subsystem, control allocation is employed as an attractive technology to handle both canard and elevator servomechanism dynamics and constraints. For convenience, and represent the desired control command to be designed, and denote the control command after the process of control allocation, and and are the real control inputs applied to AHVs with the dynamics and [24]. We also define for the following design.

###### 3.2.1. Design for -Dynamics

Recalling the kinematical equation (2), the realization of altitude reference tracking can be transferred to the FPA tracking problem. In this paper, to achieve , the control system steers the AHVs to track the FPA reference: where is the smooth altitude reference generated by the traditional second-order command filter; .

Define a time-related function as , which decides the final AoA. The FPA dynamics can be calculated as where

Considering the real small , the approximate residual is assumed to be bounded. Meanwhile, the unknown disturbance introduced by the above control allocation is bounded considering the amplitude and rate constrained actuators. Thus, the boundedness of the disturbance term is guaranteed.

Furthermore, similar to [4], the virtual AoA control is designed as

Then, can be rewritten as

Similarly, the following NDO for is constructed.

Defining estimation error as , and

Further noting the bounded , , and , the boundedness of is guaranteed. From Lemma 3, is bounded for the fact that is bounded, i.e., with .

We design the control law to regulate the system as where is governed by the adaptive law with , , and .

###### 3.2.2. Design for -Dynamics

For the altitude subsystem, the AoA tracking error dynamics have the relationship

Define with , and can be rewritten as

Then, the PR dynamics is formulated as where

Following a similar analysis to previous steps, the boundedness of is also guaranteed. Thus, the PR tracking error dynamics is formulated as

Then, NDO for is constructed as

Defining estimation error as , it has

is bounded according to the corresponding analysis in [25, 26]. Considering the bounded , , and -differentiator, is bounded. Similarly, is bounded for the fact that is bounded, i.e., with .

We design the control law to regulate the system as where is governed by the adaptive law with , , and .

###### 3.2.3. Control Allocation Design for CDA-EDA

In this subsection, a control allocation algorithm on CDA and EDA is integrated to the proposed control considering the hard magnitude and rate constraints of canard and elevator deflections.

Considering the components in (10), we briefly define where denotes the commands generated by control allocation; presents the desired values.

Meanwhile, assume the future control sequence with prediction intervals at the current interval *k* as

As the flight control system is a sampled-data system, the magnitude and rate can establish the following simple first-order approximation: where is the control interval and is the CDA/EDA command generated in the last control interval; the rate constraint can be absorbed into the magnitude constraint:

And new constraints can be defined as

Therefore, the optimization problem of the above control allocation can be expressed in the following form: with where is the tuning weight matrices for the th step.

*Remark 1. *The cost function in (52) is set for two main practical purposes. One goal reflected in is to pursue approximating precision, and the other goal reflected in is to maximize energy savings.

#### 4. Stability Analysis

In this section, stability and performance properties of the closed-loop systems are summarised.

Consider the Lyapunov function , where

Utilizing (23), (27) together with Lemma 1 and applying inequalities: , satisfies

Similarly, and can be calculated from (32), (35), (36) and (41), (44), (45), respectively.

By adding inequalities (56)–(58), we obtain where , , and are constants expressed as

Obviously, provided that . Therefore, all signals involved in the Lyapunov function and utilized in the design are uniformly ultimately bounded.

Applying Lemma 2 to (59), it is concluded that the boundedness of the related variables and signals in the proposed control can be ensured. Meanwhile, the tracking error ex with , retains the uniformly ultimate boundedness property: with

#### 5. Simulation Results

This section implements a simulation study to verify the proposed controller. In the simulating model, the nominal parameter values of the AHV model are selected according to [1]. For comparison, a slightly modified version of the adaptive fault-tolerant control presented in [9] is applied to the same scenario (denoted as “compared control” for convenience).

The simulated AHV is initially cruising at ft/s, ft, rad, deg, , and and plans to implement a maneuver in the next 600 s. and is selected according to [12]; deg, deg, deg/s, and deg/s; the simulating control interval is set as ms.

Velocity and altitude references are generated by passing step command signals: through a second-order filter [4] with the damping ratio of 0.9 and the natural frequency of 0.03 rad/s.

The control parameters are set as follows. In the velocity loop, , , , , and . In the altitude loop, , , , , ; , , , , , and .

The main simulating results are given in Figures 1–9. Figures 1 and 2 show the output tracking results during the ascending maneuver and the descending maneuver, from which we see that the proposed control performs better, though both controls achieve the desired conditions. Time histories of flight attitudes including FPA, AoA, and PR are given in Figure 3. It is seen from Figure 1 that the desired velocity reference cannot be tracked because of the FER saturation property, also see Figure 5. In this situation, the employed auxiliary system generates a modified reference, which is less aggressive and can be well tracked within the ability of scramjet. Figure 4 shows the flexible states in the comparison simulation. Figures 5–7 indicate the control input signals of FER, CDA, and EDA. It is seen from Figures 6 and 7 that both the magnitude constraints and rate constraints of CDA/EDA are well constrained within prescribed bound under the proposed control, which performs better than the compared control. Figure 8 shows the results of lumped disturbance estimation from NDOs. , , and estimated by the adaptive laws are shown in Figure 9.

**(a)**

**(b)**

**(a)**

**(b)**

**(a)**

**(b)**

**(c)**

**(a)**

**(b)**

**(a)**

**(b)**

**(a)**

**(b)**

#### 6. Conclusions

In this paper, a disturbance observer-based adaptive control for a hypersonic vehicle has been addressed to obtain the capability of handling external disturbances and actuator constraints. Fast disturbance estimate has been addressed via NDO and adaptive control compensation in each subsystem. An auxiliary system has been employed subject to the scramjet input saturation. Meanwhile, there is an effective mechanism of dealing with the magnitude/rate constraints and dynamics of aerodynamic control surfaces by introducing a control allocation module. Simulations have been provided to verify the effectiveness of the proposed control.

#### Data Availability

The data used to support the findings of this study are available from the corresponding author upon request.

#### Conflicts of Interest

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

#### Acknowledgments

This work was supported by the National Natural Science Foundation of China (grant number 61304108).

#### References

- J. T. Parker, A. Serrani, S. Yurkovich, M. A. Bolender, and D. B. Doman, “Control-oriented modeling of an air-breathing hypersonic vehicle,”
*Journal of Guidance, Control, and Dynamics*, vol. 30, no. 3, pp. 856–869, 2007. View at: Publisher Site | Google Scholar - B. Xu and Z. Shi, “An overview on flight dynamics and control approaches for hypersonic vehicles,”
*Science China Information Sciences*, vol. 58, no. 7, pp. 1–19, 2015. View at: Publisher Site | Google Scholar - H. Xu, M. D. Mirmirani, and P. A. Ioannou, “Adaptive sliding mode control design for a hypersonic flight vehicle,”
*Journal of Guidance, Control, and Dynamics*, vol. 27, no. 5, pp. 829–838, 2004. View at: Publisher Site | Google Scholar - L. Fiorentini, A. Serrani, M. A. Bolender, and D. B. Doman, “Nonlinear robust adaptive control of flexible air-breathing hypersonic vehicles,”
*Journal of Guidance, Control, and Dynamics*, vol. 32, no. 2, pp. 402–417, 2009. View at: Publisher Site | Google Scholar - H. An, Q. Wu, C. Wang, and X. Cao, “Scramjet operation guaranteed longitudinal control of air-breathing hypersonic vehicles,”
*IEEE/ASME Transactions on Mechatronics*, vol. 1, pp. 1–11, 2020. View at: Publisher Site | Google Scholar - X. Hu, L. Wu, C. Hu, and H. Gao, “Adaptive sliding mode tracking control for a flexible air-breathing hypersonic vehicle,”
*Journal of the Franklin Institute*, vol. 349, no. 2, pp. 559–577, 2012. View at: Publisher Site | Google Scholar - L. Fiorentini and A. Serrani, “Adaptive restricted trajectory tracking for a non-minimum phase hypersonic vehicle model,”
*Automatica*, vol. 48, no. 7, pp. 1248–1261, 2012. View at: Publisher Site | Google Scholar - Z. Yu, Y. Zhang, B. Jiang et al., “Decentralized fractional-order backstepping fault-tolerant control of multi-UAVs against actuator faults and wind effects,”
*Aerospace Science and Technology*, vol. 77, pp. 189–205, 2017. View at: Google Scholar - H. An, B. Fidan, J. Liu, C. Wang, and L. Wu, “Adaptive fault-tolerant control of air-breathing hypersonic vehicles robust to input nonlinearities,”
*International Journal of Control*, vol. 92, no. 5, pp. 1044–1060, 2017. View at: Publisher Site | Google Scholar - X. Bu, X. Wu, D. Wei, and J. Huang, “Neural approximation-based robust adaptive control of flexible air-breathing hypersonic vehicles with parametric uncertainties and control input constraints,”
*Information Sciences*, vol. 346-347, pp. 29–43, 2016. View at: Publisher Site | Google Scholar - Z. Yu, Z. Liu, Y. Zhang, Y. Qu, and C. Su, “Distributed finite-time fault-tolerant containment control for multiple unmanned aerial vehicles,”
*IEEE Transactions on Neural Networks and Learning Systems*, vol. 31, no. 6, pp. 2077–2091, 2020. View at: Publisher Site | Google Scholar - A. Zinnecker, A. Serrani, M. Bolender, and D. Doman, “Combined reference governor and anti-windup design for constrained hypersonic vehicles models,” in
*AIAA Guidance, Navigation, and Control Conference*, pp. 6283–6283, Chicago, IL, USA, August 2009. View at: Publisher Site | Google Scholar - D. Wiese, A. Annaswamy, J. Muse, and M. Bolender, “Adaptive control of a generic hypersonic vehicle,” in
*AIAA Guidance, Navigation, and Control Conference*, pp. 4514–4514, Boston, MA, USA, August 2013. View at: Publisher Site | Google Scholar - H. An, C. Wang, and B. Fidan, “Sliding mode disturbance observer-enhanced adaptive control for the air-breathing hypersonic flight vehicle,”
*Acta Astronautica*, vol. 139, pp. 111–121, 2017. View at: Publisher Site | Google Scholar - X. Bu, X. Wu, M. Tian, J. Huang, R. Zhang, and Z. Ma, “High-order tracking differentiator based adaptive neural control of a flexible air-breathing hypersonic vehicle subject to actuators constraints,”
*ISA Transactions*, vol. 58, pp. 237–247, 2015. View at: Publisher Site | Google Scholar - H. An, J. Liu, C. Wang, and L. Wu, “Disturbance observer-based antiwindup control for air-breathing hypersonic vehicles,”
*IEEE Transactions on Industrial Electronics*, vol. 63, no. 5, pp. 3038–3049, 2016. View at: Publisher Site | Google Scholar - X. Bu, D. Wei, and G. He, “A robust constrained control approach for flexible air‐breathing hypersonic vehicles,”
*International Journal of Robust and Nonlinear Control*, vol. 30, no. 7, pp. 2752–2776, 2020. View at: Publisher Site | Google Scholar - T. A. Johansen and T. I. Fossen, “Control allocation—a survey,”
*Automatica*, vol. 49, no. 5, pp. 1087–1103, 2013. View at: Publisher Site | Google Scholar - W. Qin, B. He, G. Liu, and P. Zhao, “Robust model predictive tracking control of hypersonic vehicles in the presence of actuator constraints and input delays,”
*Journal of the Franklin Institute*, vol. 353, no. 17, pp. 4351–4367, 2016. View at: Publisher Site | Google Scholar - L. Chen, C. Edwards, H. Alwi, and M. Sato, “Flight evaluation of a sliding mode online control allocation scheme for fault tolerant control,”
*Automatica*, vol. 114, article 108829, 2020. View at: Publisher Site | Google Scholar - Z.-P. Jiang and L. Praly, “Design of robust adaptive controllers for nonlinear systems with dynamic uncertainties,”
*Automatica*, vol. 34, no. 7, pp. 825–840, 1998. View at: Publisher Site | Google Scholar - M. Krstic, P. Kokotovic, and I. Kanellakopoulos,
*Nonlinear and Adaptive Control Design*, John Wiley & Sons, Inc., 1995. - H. An, J. Liu, C. Wang, and L. Wu, “Approximate back-stepping fault-tolerant control of the flexible air-breathing hypersonic vehicle,”
*IEEE/ASME Transactions on Mechatronics*, vol. 21, no. 3, pp. 1680–1691, 2015. View at: Publisher Site | Google Scholar - J. Dickeson, A. Rodriguez, S. Sridharan et al., “Control relevant modeling, analysis, and design for scramjet-powered hypersonic vehicles,” in
*16th AIAA/DLR/DGLR International Space Planes and Hypersonic Systems and Technologies Conference*, p. 7287, Bremen, Germany, October 2009. View at: Publisher Site | Google Scholar - B. Xu, D. Wang, Y. Zhang, and Z. Shi, “DOB-based neural control of flexible hypersonic flight vehicle considering wind effects,”
*IEEE/ASME Transactions on Mechatronics*, vol. 64, no. 11, pp. 8676–8685, 2017. View at: Publisher Site | Google Scholar - M. A. Bolender and D. B. Doman, “Nonlinear longitudinal dynamical model of an air-breathing hypersonic vehicle,”
*Journal of Spacecraft and Rockets*, vol. 44, no. 2, pp. 374–387, 2007. View at: Publisher Site | Google Scholar

#### Copyright

Copyright © 2020 Guan Wang 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.