Research Article  Open Access
Ouxun Li, Ju Jiang, Li Deng, Shutong Huang, "Sliding Mode Control Based on HighOrder Linear Extended State Observer for Near Space Vehicle", International Journal of Aerospace Engineering, vol. 2021, Article ID 6657281, 19 pages, 2021. https://doi.org/10.1155/2021/6657281
Sliding Mode Control Based on HighOrder Linear Extended State Observer for Near Space Vehicle
Abstract
Aiming at the uncertainty and external disturbance sensitivity of the near space vehicles (NSV), a novel sliding mode controller based on the highorder linear extended state observer (LESO) is designed in this paper. In the proposed sliding mode controller, the double power reaching law is adopted to enhance the state convergence rate, and the highorder LESO is designed to improve the antidisturbance ability. Moreover, the appropriate observer bandwidth and extended order are selected to further reduce or even eliminate the disturbance by analyzing their influences on the observer performance. Finally, the simulation demonstrations are given for the NSV control system with uncertain parameters and external disturbances. The theoretical analyses and simulation results consistently indicate that the proposed highorder LESO with carefully selected extended order and observer bandwidth has better performance than the traditional ones for the nonlinear NSV system with parametric uncertainty and external disturbance.
1. Introduction
Near space vehicle (NSV) has many excellent characteristics, such as fast speed, strong survival and penetration ability, high flight altitude, wide application range, and strong precision strike ability. Despite the great challenges of NSV control considering the nonlinearity, strong coupling and uncertainty of NSV, in addition with their susceptibility to external disturbances [1], the control technology based on modern control theory has been widely studied. Morden control methods such as the robust control, predictive control, sliding mode control, and intelligent control have been applied to hypersonic vehicle control and achieved certain results. On the other hand, the single control method has its own advantages and disadvantages in the NSV control application, which might not meet the multiple requirements of flight control. Therefore, the combination of different control methods to give full play to their respective advantages has become one of the research focus.
In practical applications, uncertainties such as modeling errors and external disturbances are always existent, which make the system output unstable or asymptotically track the desired target. Therefore, different robust control strategies have been proposed. Thereinto, the sliding mode control has been widely used in the flight control system due to its simple structure, fast response, and its insensitivity to external disturbances [2, 3]. In [4], a finitetime attitude control is developed to ensure that the required thrust vector is met exactly at the prescribed time. In [5], a novel recursive singularity free fast terminal sliding mode strategy for finitetime tracking control of nonholonomic systems is proposed. Specially, the sliding mode control with disturbance observation compensation is quite suitable for uncertain nonlinear systems with external disturbances and parameter perturbations [6, 7]. In [8], a novel nonsingular fast terminal sliding mode control method based on disturbance observer is proposed for the stabilization of the uncertain timevarying and nonlinear thirdorder systems. In [9], the uncertainties are effectively solved by the combination of the sliding mode control and the interval type2 TakagiSugenoKang (TSK) fuzzy control. The double power reaching law sliding mode control is adopted in [10] to track the large range command of the climbing stage of the NSV for better performance. In the aspect of antiinterference, the extended state observer can estimate the total disturbance in real time according to the input and output system information and can be eliminated in the feedback control, so that the closedloop dynamic system has better control performance. Moreover, there is a set of mature empirical formula for the parameter tuning of the observer, which is more convenient for engineering application. In [11], a robust adaptive finitetime fast terminal sliding mode controller is proposed to achieve the desired formation in the presence of model uncertainties and external disturbances. In [12], a novel sliding mode control approach is proposed for the control of a class of underactuated systems which are featured as in cascaded form with external disturbances. Authors in [13] proposed a combination of finitetime robusttracking theory and composite nonlinear feedback approach for the finitetime and highperformance synchronization of the chaotic systems in the presence of the external disturbances, parametric uncertainties, Lipschitz nonlinearities, and time delays. In [14], an adaptive controller is proposed by employing an eventtriggered control and an extendedstateobserver, where a simple strategy to tune the observer parameters is provided. Authors in [15] compensate the disturbance by combining the sliding mode control with the autodisturbance rejection control. A novel antisaturation controller is designed in [16] for the air breathing hypersonic vehicle, using the observer to compensate the mismatched disturbance. A state observer with adaptive extension and a continuous sliding mode controller with disturbance observer are proposed and applied to the hypersonic vehicle system in [17, 18], respectively. The nonlinear disturbance observer is applied to the robust flight control of the air breathing hypersonic vehicle in [19], which can significantly improve the robustness and antidisturbance ability of the system. Authors in [20, 21] have pointed out that the highgain error feedback can guarantee the fast convergence of the observation error and the sufficient estimation accuracy. The performance analyses of the LESO and its higher order form for the secondorder system in [22] indicate that the highorder form has faster response speed and better lowfrequency characteristics.
From the above recent works, it can be seen that the sliding mode control has good robustness in the NSV control system. The disturbance observation compensation can reduce or even eliminate the disturbance and improve the antidisturbance ability. However, considering the highorder, uncertainty, strong coupling of the NSV control system and other factors in the practical application, the traditional LESO could not meet the performance requirements of NSV control system. Given that, a novel sliding mode controller based on the highorder LESO is proposed in this paper. Based on the traditional LESO method, the highorder LESO is used to estimate and compensate the external disturbance. The sliding mode control method with disturbance compensation is used to improve the stability and antiinterference ability of the system. In this paper, we have proven that compared with the traditional LESO, the highorder LESO has better lowfrequency characteristics and antidisturbance ability for the NSV control system. The main contributions of our work can be concluded as follows. (1)The feedback linearization equivalent model of NSV is established. According to the equivalent model, a sliding mode controller with highorder disturbance compensator is designed to improve the system stability and antiinterference ability. The convergence region of highorder disturbance observer under the condition of lumped disturbance is analyzed, and the system stability is also proved by Lyapunov method(2)The sliding mode controller with highorder LESO compensation is proposed, and the design method of the highorder LESO is also provided(3)The transfer functions from the lumped disturbance to the disturbance estimate and the disturbance estimate error in highorder form are deduced, respectively. Moreover, the influence of the observer bandwidth and the extended order on the observer performance are analyzed specifically in the time domain and frequency domain, respectively(4)The nonlinear NSV model with uncertain parameters and external disturbance has been proven, and the simulation results also show that the highorder LESO has better robustness and antidisturbance ability by selecting appropriate order and observation bandwidth
The remainder of this paper is organized as follows. The theoretical bases are presented in Section 2, including the NSV modeling, the sliding mode controller, and the traditional observer design method. The proposed highorder LESO is detailed in Section 3. The aircraft case simulations and discussions are provided in Section 4, followed by the conclusion in Section 5.
2. Preliminaries
2.1. NSV Model
According to [9] and [23–26], the motion model of NSV longitudinal channel is defined as follows: where , , , , and indicate the velocity, track angle, pitch rate, angle of attack, and altitude, respectively; , , and are severally the throttle setting, engine natural frequency, and damping coefficient, followed by , , , , , , , and as the lift, drag, thrust, mass, gravitational constant, distance from the center of mass to the center of gravity, pitching moment, and the moment of inertia, respectively. Thereinto, the lift, drag, and thrust can be expressed as [9] where , , , and are coefficients for the lift, drag, thrust, and pitch moment, respectively; the control input signals include the throttle setting and elevator deflection angle .
Note that the NSV model in Equation (1) is nonlinear. However, according to the nonlinear theory, the inputoutput linearization can be adopted first; then, and are derived three times and four times, respectively. Thus, the linearized model can be obtained as follows [9]: where the first, second, and third derivatives of are obtained according to Equation (1) as where , ,, and.
Note that in Equation (3) and Equation (5), the flight state vector is available for measurement. Accordingly, the linearized model can be described as follows:
When the external disturbances are considered, the linearized model can be uniformly described as [27, 28].
where , and and in Equation (6) are substituted by and as the control inputs. This linearized model considering external disturbances of Equation (8) is defined under the hypotheses as follows [28]:
Assumption 1. The external disturbance and its firstorder differential are bounded as , , where and are known normal numbers.
Assumption 2. The matrix is nonsingular. In the whole branch envelope, it is reasonable to assume that the matrix is nonsingular because the flight path angle of the aircraft satisfies .
Note that Assumption 1 is only suitable for the traditional state expansion observer. The highorder form of observer will be discussed in the following sections.
2.2. Design of the Sliding Mode Controller
The design process of the sliding mode controller can be described as the following two steps. (1)Select the integral sliding surface aswhere and are the tracking errors for the velocity channel and the altitude channel, respectively. and are the positive constants. The derivatives of Equation (9) can be obtained as
Let , and . By introducing the model Equation (6) into Equation (10), we can get (2)Define the sliding mode approaching rate aswhere , ,,,,,, and. When the system state is far away from the sliding mode, and play a dominant role, while and play a dominant role when the system state is close to the sliding mode. According to Equations (8), (11), and (12), the continuoustime controller can be obtained as follows: where
The sliding mode controller can make the system state converge into the neighborhood of the equilibrium zero point quickly, if there are uncertain external disturbances in the system [29]. To ensure the convergence of system state, the observer can be used to compensate the bounded lumped disturbance caused by the system modeling error and bounded external disturbances.
2.3. Design of Traditional Extended State Disturbance Observer
Considering the following thirdorder system with external disturbances and uncertainties, the state space form is described as where is the sum of system uncertainties and external disturbances. Equation (15) can be rewritten as follows: where is the derivative of the system lumped disturbance. Then, the traditional extended state observer can be designed as follows [25, 26]: where , , , and are observer states, , , , and are positive real numbers, and is the observation bandwidth.
According to the above analyses, the state space model of the velocity channel can be expressed as where
The disturbance observer can be described as where , ,, is selected by the bandwidthbased configuration method. More generally, in traditional LESO for any order system, the adjustable parameter only has the observation bandwidth .
Similarly, the altitude channel can be described as where
and the adjustable parameter only has the observation bandwidth .
Combined Equation (20) with Equation (21), the aircraft disturbance observer can be described as where
3. Proposed HighOrder Linear Extended State Observer
3.1. Design of HighOrder Linear Extended State Observer
For the sake of generality, the following th order system with external disturbances and uncertainties are considered. The state space form can be described as where its extended firstorder state observer (traditional LESO) is given as
The following hypothesis proposed in [22] is given as
Assumption 3. The derivatives of the lumped disturbance of the system exist and are bounded as .
Accordingly, the form of state equation and extended th order state observer can be presented as
where is the lumped disturbance , , , , are the first, second, , and th order derivatives of the lumped disturbance, respectively. is the estimate of lumped disturbance , and , , , are severally the estimates of the first, second, , and th order derivatives of the lumped disturbance. The parameter is selected by the bandwidthbased configuration method, and is the observation bandwidth, and indicates the traditional LESO.
3.2. Convergence Analysis
In this section, the convergence analysis is presented. First, the definition of the expansion of th order state observer for the th order system is given below
The differential form of can be described as where is the observation bandwidth.
For clear illustration, the following hypothesis and theorem proposed in [22] are provided first.
Assumption 4. There are certain constants of , , and the positive definite continuous differentiable functions and : , which make where is the Euclid norm.
Theorem 5. If Assumption 3 and Assumption 4 are satisfied, then for the system Equation (28), the following conclusion is true.
According to Assumption 3 and Assumption 4, it can be concluded that
Considering the relationship between and , there is
Then, according to GronwallBellman inequality, we can obtain
Finally, the inequality of can be concluded from the relationship of and as
where .
Accordingly, when the perturbation parameters are small enough, the estimated state of the system Equation (28) is sufficiently close to the state of the system Equation (27). The estimation error converges to and decreases with the increase of the expansion order.
3.3. Stability Analysis
Theorem 6. For the nonlinear dynamic model of NSV control system show in Equation (1), the continuous time controller of Equation (13) is chosen as the disturbance compensation controller, considering the uncertainty and external disturbance. The system is asymptotically stable when the following conditions are satisfied. The Lyapunov function is defined as where , , , is the lumped disturbance, is the estimate of lumped disturbance, , ,,, The inequality (40) can be described as the following two forms: From inequality (42), we can obtain when From inequality (43), we can obtain when In conclusion, the states and converge to the following regions in the finite time.
4. Results and Discussion
In this simulation parts, we first analyze the influence of the extended order and observation bandwidth on the observer performance; then, we further verify the performance of proposed observer in the NSV control system.
4.1. Parameter Analysis
According to Equation (27) and Equation (28), we can deduce the transfer function from lumped disturbances to disturbance estimate and disturbance estimate error as follows: where is the lumped disturbance estimate error, is the system order, is the expansion order, is the observation bandwidth, and .
Furthermore, the performance analysis of the expansion order and the observation bandwidth is also provided in the time domain and the frequency domain over the thirdorder velocity channel and the fourthorder altitude channel, respectively. According to Equation (49), the highorder LESO has the following characteristics.
As can be seen from Figure 1, when the observation bandwidth is a constant, the higher the expansion order, the faster the response speed, but the overshoot will increase. When the expansion order is a constant, the bigger the observation bandwidth, the faster the response speed, and the overshoot remains unchanged. It can be seen from the overall effect that the higher the system order, the greater the overshoot. Therefore, it is particularly important to choose the appropriate observation bandwidth and expansion order.
(a)
(b)
Figure 2 shows the lowfrequency characteristics of highorder LESO lumped disturbance estimates, where the frequency range is . Moreover, the reference frequencies are selected as 0.2, 0.4, and 0.8, respectively. The amplitude frequency characteristics and phase frequency characteristics are shown in Tables 1 and 2, respectively. Compared with the traditional LESO, when the expansion order is , the amplitude errors increase slightly, but the phase errors decrease greatly. When the expansion order is , the amplitude errors and phase errors are smaller than and . When the expansion order is a constant, the larger the observation bandwidth, the smaller of magnitude errors and phase errors. It has been shown in Tables 1 and 2 that the estimation error is also related to the system order. Both of the magnitude errors and the phase errors will increase as the system order becomes higher. Generally, in the lowfrequency band, the estimation errors of highorder LESO are smaller than those of the traditional LESO.
(a)
(b)


According to Equation (50), it can be seen from Figures 3 and 4 that, at the same observer bandwidth, increasing the order of the extended state can significantly enhance the disturbance suppression in the lowfrequency band, which is conducive to better tracking the system state and disturbance. However, in the intermediatefrequency band, the highorder LESO has larger peak amplitudes, which would lead to bigger overshoots or oscillations of the step response, which is not helpful to the system stability. Furthermore, the observation bandwidth only affects the rapidity of highorder LESO, and the overshoot is not affected by the bandwidth, but only related to the expansion order.
(a)
(b)
(a)
(b)
In general, both increasing the observation bandwidth and the expansion order can improve the lowfrequency performance of observer. However, the higher the expansion order, the bigger the peak amplitude of intermediate frequency, which is prone to lead a larger overshoot and even oscillation; the lager the observation bandwidth, the worse the intermediatefrequency and highfrequency characteristics. Furthermore, the higher the order of the system, the greater the estimation errors. Therefore, for highorder system, we use highorder LESO by choosing appropriate observation bandwidth and expansion order, which has better lowfrequency characteristics than traditional LESO.
4.2. Numerical Simulations
In order to verify the effectiveness of the proposed method, the mathematical model of the NSV is built on the MATLAB/Simulink platform, and the designed controller is also simulated and verified with MATLAB.
In this paper, based on the flight conditions of NSV cruise state, the uncertain parameters are adopted as additional variables. The main parameters are set as follows: , , , , , and . The parametric uncertainty is defined as
The expansion orders of , , and in addition with observation bandwidth of , , and are adopted in the simulations, respectively. Thereinto, indicates the traditional LESO, while and are the recommended LESO in this paper. Under the premise of system stability, the parameters of controller and observer are basically unchanged. Moreover, due to the coupling between the velocity and altitude channels, the observer output needs to be decoupled. The step signal for velocity channel and altitude channel are given as and , respectively. The external disturbance and are introduced in the 30th second.
The simulation results are shown in Figures 5–8, where the curves with parameter of indicate the algorithm in [27], the others with parameter of and represent our proposed algorithm, and the compared nonlinear extended state observer (NESO) is proposed in [30]. Figure 5 shows the time response of compared NSV observers over the velocity channel and altitude channel. It can be seen in Figure 5 that both the controllers with LESO and NESO with appropriate parameters can make the system stable. As to the overall effect, the steadystate accuracy of the proposed highorder LESO is obviously better than traditional LESO. When the expansion order parameter is selected as , the steadystate accuracy of proposed observer is similar to that of NESO. Furthermore, when the expansion order parameter is selected as , the steadystate accuracy of proposed observer is better than NESO. Figure 6 shows the time responses of the observer errors over the velocity channel and altitude channel, where the disturbance estimation errors of the proposed highorder LESO are obviously smaller than traditional LESO and NESO. Figure 7 shows the time responses of the actuator on throttle setting and elevator deflection. It can be concluded from Figure 7 that with the increase of the order and observer bandwidth, the control signal will generate peak amplitude and highfrequency oscillations, which are deleterious to the actuator. However, the curve of NESO is smoother than LESO. The higher the system order and the lager the observer bandwidth, the more serious the situation is. Figure 8 shows the time responses of the attack angle and the track angle. The smaller the observer error is, the smaller the change of attitude angle is, and the better the aircraft stability is. The proposed highorder LESO can improve the system antidisturbance ability by selecting an appropriate order and observer bandwidth. Therefore, for the NSV nonlinear model with uncertain parameters and external disturbance, the simulation results show that the proposed highorder LESO with extended order of 3 and observer bandwidth of 10 is recommendable as the disturbance compensator, which is better than the traditional LESO and the NESO.
(a)
(b)
(a)
(b)
(a)
(b)
(a)
(b)
5. Conclusions
In this paper, a sliding mode controller with highorder LESO compensation is designed according to the NSV characteristics, which provides a new way to solve uncertain systems with external disturbances. The proposed highorder LESO is suitable for any order system. The transfer function from the lumped disturbance to the disturbance estimates and disturbance estimates error of any order system with any extended order is derived. It can be found that a carefully selected extended order and observer bandwidth can achieve remarkable observer performance gains, more stable system response, and higher steadystate accuracy. However, there are still some other problems need to be solved in the future work, such as the peak phenomenon of intermediate frequency, the highfrequency antidisturbance ability, the different order coupling, and the design of finitetime sliding mode controller. The future work will be focused on how to design the switching control law between the LESO and the NESO, in order to solve the stability problem of multimode switching for near space variable wing vehicle.
Data Availability
The experimental 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 are no conflicts of interest regarding the publication of this paper.
Acknowledgments
The authors would like to express their thanks for the support from the Natural Science Foundation of China (No. 61673209, No. 61966010, and No. 81960324).
References
 C. Zhang, D. Hu, L. Ye, W. Li, and D. Liu, “Review of the development of hypersonic vehicle technology abroad in 2017,” Tactical Missile Technology, vol. 39, no. 1, pp. 47–50, 2018. View at: Google Scholar
 H. Sun, S. Li, and C. Sun, “Finite time integral sliding mode control of hypersonic vehicles,” Nonlinear Dynamics, vol. 73, no. 12, pp. 229–244, 2013. View at: Publisher Site  Google Scholar
 R. Zhang, L. Dong, and C. Sun, “Adaptive nonsingular terminal sliding mode control design for near space hypersonic vehicles,” IEEE/CAA Journal of Automatica Sinica, vol. 1, no. 2, pp. 155–161, 2014. View at: Publisher Site  Google Scholar
 J. Zhang, J. D. Biggs, D. Ye, and Z. Sun, “Finitetime attitude setpoint tracking for thrustvectoring spacecraft rendezvous,” Aerospace Science and Technology, vol. 96, article 105588, 2020. View at: Publisher Site  Google Scholar
 S. Mobayen, M. J. Yazdanpanah, and V. J. Majd, “A finitetime tracker for nonholonomic systems using recursive singularityfree FTSM,” in Proceedings of the 2011 American Control Conference, pp. 1720–1725, San Francisco, CA, USA, 2011. View at: Publisher Site  Google Scholar
 J. Sun, J. Yi, Z. Pu, and X. Tan, “Fixedtime sliding mode disturbance observerbased nonsmooth backstepping control for hypersonic vehicles,” IEEE Transactions on Systems, Man, and Cybernetics: Systems, vol. 50, pp. 4377–4386, 2018. View at: Publisher Site  Google Scholar
 X. Yin, B. Wang, L. Liu, and Y. Wang, “Disturbance observerbased gain adaptation highorder sliding mode control of hypersonic vehicles,” Aerospace Science and Technology, vol. 89, pp. 19–30, 2019. View at: Publisher Site  Google Scholar
 S. Mobayen and F. Tchier, “Nonsingular fast terminal slidingmode stabilizer for a class of uncertain nonlinear systems based on disturbance observer,” Scientia Iranica, vol. 24, no. 3, pp. 1410–1418, 2017. View at: Publisher Site  Google Scholar
 X. Jiao, B. Fidan, J. Jiang, and M. Kamel, “Adaptive mode switching of hypersonic morphing aircraft based on type2 TSK fuzzy sliding mode control,” Science China Information Sciences, vol. 58, no. 7, pp. 1–15, 2015. View at: Publisher Site  Google Scholar
 C. Gu, J. Jiang, and Y. Wu, “Tracking control study of ascent phase for near space vehicle,” Journal of Harbin Engineering University, vol. 37, no. 11, pp. 1526–1531, 2016. View at: Google Scholar
 J. Zhang, D. Ye, J. D. Biggs, Z. Sun, J. D. Biggs, and Z. Sun, “Finitetime relative orbitattitude tracking control for multispacecraft with collision avoidance and changing network topologies,” Advances in Space Research, vol. 63, no. 3, pp. 1161–1175, 2019. View at: Publisher Site  Google Scholar
 S. Mobayen, “Design of LMIbased sliding mode controller with an exponential policy for a class of underactuated systems,” Complexity, vol. 21, no. 5, 2014. View at: Publisher Site  Google Scholar
 S. Mobayen and J. Ma, “Robust finitetime composite nonlinear feedback control for synchronization of uncertain chaotic systems with nonlinearity and timedelay,” Chaos, Solitons & Fractals, vol. 114, pp. 46–54, 2018. View at: Publisher Site  Google Scholar
 J. Zhang, J. D. Biggs, D. Ye, and Z. Sun, “Extendedstateobserverbased eventtriggered orbitattitude tracking for lowthrust spacecraft,” IEEE Transactions on Aerospace and Electronic Systems, vol. 56, pp. 2872–2883, 2020. View at: Publisher Site  Google Scholar
 M. Cheng, Y. Wu, and F. Ma, “Sliding mode autodisturbance rejection control of hypersonic vehicle under disturbance,” Journal of System Simulation, vol. 29, no. 10, pp. 2391–2406, 2017. View at: Google Scholar
 H. An, J. Liu, C. Wang, and L. Wu, “Disturbance observer based antiwindup control for airbreathing hypersonic vehicles,” IEEE Transactions on Industrial Electronics, vol. 63, no. 5, pp. 3038–3049, 2016. View at: Publisher Site  Google Scholar
 Z. Pu, R. Yuan, J. Yi, and X. Tan, “A class of adaptive extended state observers for nonlinear disturbed systems,” IEEE Transactions on Industrial Electronics, vol. 62, no. 9, pp. 5858–5869, 2015. View at: Publisher Site  Google Scholar
 C. Mu, Q. Zong, B. Tian, and W. Xu, “Continues sliding mode controller with disturbance observer for hypersonic vehicles,” IEEE/CAA Journal of Automatica Sinica, vol. 2, no. 1, pp. 45–55, 2015. View at: Google Scholar
 J. Yang, S. Li, C. Sun, and L. Guo, “Nonlinear disturbance observer based robust flight control for airbreathing hypersonic vehicles,” IEEE Transactions on Aerospace and Electronic Systems, vol. 49, no. 2, pp. 1263–1275, 2013. View at: Publisher Site  Google Scholar
 A. A. Ball and H. K. Khalil, “Highgain observers in the presence of measurement noise: a nonlinear gain approach,” in 2008 47th IEEE Conference on Decision and Control, pp. 2288–2293, Cancun, Mexico, 2008. View at: Publisher Site  Google Scholar
 J. P. V. S. Cunha, R. R. Costa, F. Lizarralde, and L. Hsu, “Peaking free variable structure control of uncertain linear systems based on a highgain observer,” Automatica, vol. 45, no. 5, pp. 1156–1164, 2009. View at: Publisher Site  Google Scholar
 X. Shao and H. Wang, “Performance analysis on linear extended state observer and its extention case with higher extended order,” Control and Decision, vol. 30, no. 5, pp. 815–822, 2015. View at: Google Scholar
 B. Fidan, M. Mirmirani, and P. A. Ioannou, “Flight dynamics and control of airbreathing hypersonic vehicles: review and new directions,” in 12th AIAA International Space Planes and Hypersonic Systems and Technologies, Norfolk, Virginia, 2003. View at: Publisher Site  Google Scholar
 M. A. Bolender and D. B. Doman, “Nonlinear longitudinal dynamical model of an airbreathing hypersonic vehicle,” Journal of Spacecraft and Rockets, vol. 44, no. 2, pp. 374–387, 2007. View at: Publisher Site  Google Scholar
 L. Fiorentini, A. Serrani, M. A. Bolender, and D. B. Doman, “Nonlinear robust adaptive control of flexible airbreathing hypersonic vehicles,” Journal of Guidance Control and Dynamics, vol. 32, no. 2, pp. 402–417, 2009. View at: Publisher Site  Google Scholar
 J. D. Shaughnessy and S. Z. Pinckney, “Hypersonic vehicle simulation model: wingedcone configuration,” in NASA Technical Memorandum, USA, 1991, NASA TM102610. View at: Google Scholar
 Y. Wu, J. Wang, X. Liu, and Y. Liu, “Disturbanceobserverbased sliding mode control for hypersonic flight vehicle,” Control Theory and Applications, vol. 32, no. 6, pp. 717–724, 2015. View at: Google Scholar
 J. Wang, J. Wu, and X. Dong, “Sliding mode control for hypersonic flight vehicle with sliding mode disturbance observer,” Acta Aeronautica et Astronautica Sinica, vol. 35, pp. 1–8, 2014. View at: Google Scholar
 H. Zhang, J. Fan, F. Meng, and J. Huang, “A new sliding control dual power reaching law,” Control and Decision, vol. 28, no. 2, pp. 289–293, 2013. View at: Google Scholar
 J. Chen, D. Nannan, and H. Yu, “Decoupling attitude control of a hypersonic glide vehicle based on a nonlinear extended state observer,” International Journal of Aerospace Engineering, vol. 2020, Article ID 4905698, 11 pages, 2020. View at: Publisher Site  Google Scholar
Copyright
Copyright © 2021 Ouxun Li 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.