Research Article  Open Access
ThreeDimensional Integrated Guidance and Control for Near Space Interceptor Based on Robust Adaptive Backstepping Approach
Abstract
This study presents a novel integrated guidance and control method for near space interceptor, considering the coupling among different channels of the missile dynamics, which makes the most of the overall performance of guidance and control system. Initially, threedimensional integrated guidance and control model is employed by combining the interceptortarget relative motion model with the nonlinear dynamics of the interceptor, which establishes a direct relationship between the interceptortarget relative motion and the deflections of aerodynamic fins. Subsequently, regarding the acceleration of the target as bounded uncertainty of the system, an integrated guidance and control algorithm is designed based on robust adaptive backstepping method, with the upper bound of the uncertainties unknown. Moreover, a nonlinear tracking differentiator is introduced to reduce the “compute explosion” caused by backstepping method. It is proved that tracking errors of the state and the upper bound of the uncertainties converge to the neighborhoods of the origin exponentially. Finally, simulations results show that, compared to the conventional guidance and control design, the algorithm proposed in this paper has greater advantages in miss distance, required normal overload, and flight stability, especially when attacking highmaneuvering targets.
1. Introduction
The guidance and the control systems of interceptors are complicated because they are nonlinear and coupled with each other. On the one hand, the guidance system guides the interceptor to the target, during which the variation of trajectory parameters intensifies the uncertainty of the attitude model and brings a burden to the control of attitude. On the other hand, the control system adjusts the attitude in order to track the guidance command, during which the action of the actuators exerts influence on the trajectory and the autopilot lag reduces the performance of the guidance system. Conventionally, guidance and control systems are designed separately, which is proved efficient and convenient merely for the interception of nonmaneuvering target. However, the conventional design approach does not make full use of the coupling between guidance and control and hence has great limitations in improving the overall performance of the interceptor. For the near space interceptor, especially when attacking highmaneuvering target, the coupling between guidance and control is significant while the conventional design method may result in deterioration of the overall performance, sometimes even failure of the interception. Therefore, the study of integrated guidance and control, which makes full use of the coupling between the guidance and the control, is of great significance for better overall performance of the interceptor and higher accuracy for near space interception.
Integrated guidance and control design can be mainly classified into two categories: the partial one and the complete one. Partial integrated guidance and control is that the guidance law is designed accounting for autopilot dynamics, by which the negative influence of autopilot lag is compensated. In [1–3], guidance laws are studied for the near space interception of maneuvering target, in which the autopilot is approximated as firstorder dynamics. Actually, the autopilot is of highorder dynamics. As an improvement, guidance laws in [4–6] are designed accounting for the autopilot as secondorder dynamics. In [7–9], finitetime convergence guidance laws in the presence of autopilot lag are studied, which guarantee that the lineofsight angular rate converges to zero in finite time. For better attack performance, guidance laws with impact angle constraints and autopilot dynamics compensation are designed in [10–12]. Compared to conventional design approach, partial integrated guidance and control can improve the overall performance of the system to some extent, in which the coupling between guidance and control is approximately considered. However, partial integrated guidance and control still keeps the outline of conventional design scheme, that is, the outer guidance loop and inner control loop, which has some limitations in improving the overall performance. On the one hand, the coupling between guidance and control is normally approximated as the firstorder dynamics or the secondorder one, both of which cannot represent the actual coupling. On the other hand, all the couplings are regarded as disturbances to compensate, resulting in the underutilization of conductive couplings.
Complete integrated guidance and control is that the guidance and control is designed as a whole and the control command is produced directly according to the interceptortarget relative motion, which breaks through the outline of conventional design scheme entirely. Complete integrated guidance and control has great advantage in making full use of the coupling between guidance and control and improving the overall performance of the system, which is the trend of the guidance and control design. For convenience, most of the complete integrated guidance and control are studied in twodimensional space [13–19]. Actually, all the near space interceptions are conducted in threedimensional space, so achievements of twodimensional integrated guidance and control cannot be applied in threedimensional interception directly. By viewing the coupling among the pitch, yaw, and roll channels as external disturbances, threechannel independent integrated guidance and control are studied in [20–23], assumption of which is that the movement of yaw and roll channels is in a small range. When attacking highmaneuvering target, however, this assumption cannot be satisfied, which limits the application of threechannel independent integrated guidance and control. To realize threechannel coupling integrated guidance and control for interception of maneuvering target, threedimensional integrated guidance and control model of strict feedback cascade is established in [24–26], based on which integrated guidance and control algorithms are designed using dynamic surface control. Meanwhile, robust functions [24, 25] or extended state observer [26] are introduced to compensate for the disturbances. In [27], threedimensional full states coupling integrated guidance and control model is established and adaptive integrated guidance and control algorithm is designed. But the resulting control algorithm is relatively complicated and hard to be realized in engineering.
Based on the idea of complete integrated guidance and control, a novel integrated guidance and control considering the coupling among three channels is proposed in this paper, aiming at threedimensional near space interception of maneuvering target. Firstly, considering the coupling among the pitch, yaw, and roll channels, threedimensional integrated guidance and control model is built, which establishes the direct relationship between the interceptortarget relative motion and the deflections of aerodynamic fins. Secondly, regarding the acceleration of the target as bounded uncertainty of the system, an integrated guidance and control algorithm is designed based on robust adaptive backstepping method, with no information of the upper bound of the uncertainties. Thirdly, a nonlinear tracking differentiator is introduced to reduce the “compute explosion” caused by backstepping method. It is proved that tracking errors of the state and the unknown parameters converge to the neighborhoods of the origin exponentially. Finally, simulation and comparison are conducted between the proposed integrated guidance and control method and the conventional guidance and control design. Simulation results show that the proposed algorithm in this paper has greater advantages in miss distance, required normal overload, and flight stability, especially when attacking the highmaneuvering target.
2. Integrated Guidance and Control Model
The interceptortarget engagement geometry in threedimensional space is shown in Figure 1, where and are, respectively, the inertial and lineofsight coordinate systems; and are, respectively, the velocity and half velocity coordinate systems; and represent the centroid of the interceptor and target, respectively. Supposing that skidtoturn (STT) control scheme is adopted in the flight of interceptor, the velocity coordinate system coincides with half velocity coordinate system.
According to the above definitions, integrated guidance and control model for near space interceptor in threedimensional space can be described as the following form [24]:where the state vectors are defined as , , and ; the input vector is defined as ; the output vector is defined as ; , are the bounded uncertainties of the system, including the modeling errors and external disturbances; concrete expressions of , , , , , and are as follows:From the integrated guidance and control model of (1), we can see that the direct relationship between the interceptortarget relative motion and the deflections of aerodynamic fins is built, in which the coupling among the pitch, yaw, and roll channels is considered sufficiently. What is more, the state variables of (1) are of cascaded feedback form. That is,
3. Integrated Guidance and Control Algorithm
3.1. Robust Adaptive Backstepping Controller
Before the controller design process, some assumptions are made as follows.
Assumption 1. , are bounded and invertible.
Assumption 2. There exist a set of unknown positive constant and known nonnegative smooth function , such that .
Design objective of integrated guidance and control is selected to be zero lineofsight angular rate. What is more, roll angle of STT interceptor should be kept zero during the whole flight. That is, Based on robust adaptive backstepping approach, integrated guidance and control algorithm can be designed as follows.
Step 1. The tracking error of state is defined as and the derivative of with respect to time is given byAccording to exponential approaching law, the virtual control is constructed as the following form: where is the gain matrix with symmetric positive definite form; is the robust function introduced to compensate for the uncertainty , which is given by [28]where is the gain parameter; is the approximation of the unknown positive constant , which can be designed by the following adaptive algorithm:where , , and are positive design parameters.
Step 2. Let ; then . The tracking error of state is defined as and the derivative of with respect to time is given byThe virtual control is constructed as the following form: where is the gain matrix with symmetric positive definite form; is introduced to compensate for the coupling between and ; is the robust function compensating for the uncertainty , which is given bywhere , , , and are positive design parameters.
Step 3. The tracking error of state is defined as and the derivative of with respect to time is given byThe real control is constructed as the following form: where is the gain matrix with symmetric positive definite form; is introduced to compensate for the coupling between and ; is the robust function compensating for the uncertainty , which is given bywhere , , , and are positive design parameters.
3.2. Nonlinear Tracking Differentiator
It should be emphasized that, as a common drawback of classical backstepping method, the derivative of the virtual control (e.g., and ) is computationally complex, which intensifies the “computer explosion” of backstepping method. To overcome such drawback, a nonlinear tracking differentiator is introduced, by which the derivative of the virtual control can be well approximated. The discrete form of the nonlinear tracking differentiator is as follows [29]:where is the input; is the approximation of ; is the approximation of the derivative of ; is the sampling step; is the gain parameter that determines the speed of tracking; is the timeoptimal control synthesis function, which is given byUsing the nonlinear tracking differentiator of (18), we have and . The differentiator is not only of high accuracy in the approximation of the input and its derivative, but also capable of filtering the noise within the input signal.
To analyze the performance of this differentiator, a numerical simulation is conducted, in which the gain parameter and sampling step are set as and . The tracking results of sinusoidal input and its derivative are presented in Figure 2, from which we can see that the differentiator is of high speed and accuracy in the approximation of the input and its derivatives.
3.3. Stability Analysis
Define the candidate Lyapunov function as where are the tracking errors of ; that is, Under Assumption 2, are constants; then the derivatives of satisfyDifferentiating (20) with respect to time, we haveDefine the tracking error of asThen, we get Substituting (7) and (22) into (25), we have Sincehence, we haveSimilarly, we getIt is easy to verify that Substituting (28)(29) into (23), we have Furthermore, we getwhere Hence, we have whereConsequently, the following conclusions can be obtained.
(1) For arbitrary , by choosing proper gain parameters , , and , the state tracking errors (e.g., , , and ) and the unknown parameter tracking errors (e.g., , , and ) can converge to the neighborhoods of the origin exponentially; that is, The rate and domain of convergence can be regulated by adjusting the values of , , , and . What is more, the algorithm is designed with no information of the upper bound of the uncertainties, which is easier to realize.
4. Simulation and Analysis
4.1. Simulation Conditions
To verify the performance of proposed integrated guidance and control algorithm, numerical simulation is conducted for threedimensional near space interception.
The initial position and velocity vectors of the interceptor with respect to inertial coordinate system are set as MeanwhileThe initial position and velocity vectors of the target with respect to inertial coordinate system are set as For the purpose of comparison, three guidance and control schemes are carried out in the simulation, details of which are as follows.
(1) The proposed integrated guidance and control algorithm is denoted by IGC. The gain parameters of the IGC algorithm are set as
(2) The proportional navigation guidance law combined with PID control law is denoted by PN+PID. Using PN law, we can get the commanding normal overload with respect to half velocity coordinate system; that is [30],where the gain parameter is set as .
The PID law is used to track the commanding normal overload, which is given by the following [30]:where , , , , , and .
(3) The sliding mode guidance law combined with active disturbance rejection control law is denoted by SMG+ADRC. Using SMG law, we can get the commanding normal acceleration with respect to lineofsight coordinate system [31]:where the gain parameters are set as , , , , , and . Furthermore, we can get the commanding angle of attack and sideslip, that is, and .
The ADRC law is used to track the guidance command, which is given by the following [29]:where , , , ; , , ; , , , ; , , and .
4.2. Results and Analysis
Under conditions above, numerical simulations are carried out in two typical engagement scenarios. For better illustration of proposed methods performance when there are aerodynamic uncertainties, 100 MonteCarlo runs are performed in each engagement scenario, in which the random variables are the deviation of the aerodynamic parameters from their nominal design values, uniformly distributed in the interval [−20%, +20%].
Scenario 1 (nonmaneuvering target). Suppose that the target does not maneuver; the mean miss distance and interception time of the three schemes resulting from MonteCarlo runs are summarized in Table 1. Simulation results of the three schemes are presented in Figures 3–6. (To minimize the article length, only one representative result from MonteCarlo runs is presented, in which the aerodynamic parameters deviate from their nominal design values by +20%.)

Scenario 2 (step maneuvering target). Assume that the target escapes with acceleration of and . The mean miss distance and interception time of the three schemes resulting from MonteCarlo runs are summarized in Table 2. Simulation results of the three schemes are presented in Figures 7–10. (To minimize the article length, only one representative result from MonteCarlo runs is presented, in which the aerodynamic parameters deviate from their nominal design values by +20%.)

Simulation results above indicate that, comparing to conventional design methods, the integrated guidance and control algorithm is more effective with the following advantages:(1)less miss distance and interception time, which is not obvious when attacking nonmaneuvering target while it is rather remarkable when attacking maneuvering target;(2)smaller attack angle and sideslip angle, hence reduced required normal overload and less fluctuation of trajectory; therefore, the integrated design excels in maneuvering target interception especially when the available normal overload is limited;(3)higher flight stability, with the attack angle, sideslip angle, roll angle, rotational angular velocity, and deflection of aerodynamic fins remaining stable during the whole flight, while those of conventional design schemes tend to diverge when the interceptor approaches the target.
5. Conclusion
Aiming at threedimensional near space interception of maneuvering target, a novel integrated guidance and control method considering the coupling among three channels is proposed in this paper. Firstly, considering the coupling among the pitch, yaw, and roll channels, threedimensional integrated guidance and control model is employed by combining the interceptortarget relative motion model with the nonlinear dynamics of the interceptor, in which the direct relationship between the interceptortarget relative motion and the deflections of aerodynamic fins is established. Secondly, regarding the acceleration of the target as bounded uncertainty of the system, an integrated guidance and control algorithm is designed based on robust adaptive backstepping method, with the upper bound of the uncertainties unknown. Moreover, a nonlinear tracking differentiator is introduced to reduce the “compute explosion” caused by backstepping method. It is proved that tracking errors of the state and the upper bound of the uncertainties converge to the neighborhoods of the origin exponentially. Finally, simulation and comparison are carried out under three guidance and control schemes. In contrast to conventional design methods, the integrated guidance and control algorithm leads to higher flight stability, less miss distance, and required normal overload, especially when attacking the highmaneuvering target.
Nomenclature
:  Lineofsight distance (m) 
:  Velocity of interceptor (m/s) 
:  Elevation angle of the lineofsight (rad) 
:  Azimuth angle of the lineofsight (rad) 
:  Flight path angle (rad) 
:  Pitch angle (rad) 
:  Angle of attack (rad) 
:  Angle of sideslip (rad) 
:  Roll angle (rad) 
, , :  Roll, yaw, and pitch rates (rad/s) 
, , :  Aileron, rudder, and elevator deflections (rad) 
:  Mass of interceptor (kg) 
:  Dynamic pressure (Pa) 
:  Reference area (m^{2}) 
:  Reference length (m) 
, , :  Roll, yaw, and pitch moments of inertia (kg·m^{2}) 
:  Partial derivative of lift force coefficient with respect to 
:  Partial derivative of lateral force coefficient with respect to 
:  Partial derivatives of rolling moment coefficient with respect to and 
:  Partial derivatives of yawing moment coefficient with respect to , and 
:  Partial derivatives of pitching moment coefficient with respect to , and . 
Competing Interests
The authors declare that there are no competing interests regarding the publication of this paper.
Acknowledgments
This work was partially supported by the National Natural Science Foundation of China (Grant no. 11572097).
References
 R. H. Chen, J. L. Speyer, and D. Lianos, “Optimal intercept missile guidance strategies with autopilot lag,” Journal of Guidance, Control, and Dynamics, vol. 33, no. 4, pp. 1264–1272, 2010. View at: Publisher Site  Google Scholar
 S. S. Moosapour, G. Alizadeh, S. Khanmohammadi, and H. Moosapour, “A novel robust proportional navigation guidance law design for missile considering autopilot dynamic,” Transactions of the Institute of Measurement and Control, vol. 35, no. 5, pp. 703–710, 2013. View at: Publisher Site  Google Scholar
 W. Shang, J. Guo, S. Tang, Y. Ma, and Y. Zhang, “Impact time guidance law considering autopilot dynamics based on variable coefficients strategy for maneuvering target,” Mathematical Problems in Engineering, vol. 2015, Article ID 815149, 10 pages, 2015. View at: Publisher Site  Google Scholar
 D. Chwa and J. Y. Choi, “Adaptive nonlinear guidance law considering control loop dynamics,” IEEE Transactions on Aerospace and Electronic Systems, vol. 39, no. 4, pp. 1134–1143, 2003. View at: Publisher Site  Google Scholar
 B. Xu and D. Zhou, “Three dimensional adaptive dynamic surface guidance law accounting for autopilot lag,” in Proceedings of the American Control Conference (ACC '14), pp. 578–583, IEEE, Portland, Ore, USA, June 2014. View at: Publisher Site  Google Scholar
 P. P. Qu and D. Zhou, “A dimension reduction observerbased guidance law accounting for dynamics of missile autopilot,” Proceedings of the Institution of Mechanical Engineers, Part G: Journal of Aerospace Engineering, vol. 227, no. 7, pp. 1114–1121, 2013. View at: Publisher Site  Google Scholar
 G.L. Li, H. Yan, and H.B. Ji, “A guidance law with finite time convergence considering autopilot dynamics and uncertainties,” International Journal of Control, Automation and Systems, vol. 12, no. 5, pp. 1011–1017, 2014. View at: Publisher Site  Google Scholar
 S. M. He and D. F. Lin, “Continuous composite finitetime convergent guidance laws with autopilot dynamics compensation,” ISA Transactions, vol. 58, pp. 270–278, 2015. View at: Publisher Site  Google Scholar
 P. Qu, C. Shao, and D. Zhou, “Finite time convergence guidance law accounting for missile autopilot,” Journal of Dynamic Systems, Measurement and Control, Transactions of the ASME, vol. 137, no. 5, Article ID 051014, pp. 1–8, 2015. View at: Publisher Site  Google Scholar
 Z. X. Zhang, S. H. Li, and S. Luo, “Composite guidance laws based on sliding mode control with impact angle constraint and autopilot lag,” Transactions of the Institute of Measurement and Control, vol. 35, no. 6, pp. 764–776, 2013. View at: Publisher Site  Google Scholar
 S. He, D. Lin, and J. Wang, “Robust terminal angle constraint guidance law with autopilot lag for intercepting maneuvering targets,” Nonlinear Dynamics, vol. 81, no. 12, pp. 881–892, 2015. View at: Publisher Site  Google Scholar
 D. Zhou, P. P. Qu, and S. Sun, “A guidance law with terminal impact angle constraint accounting for missile autopilot,” Journal of Dynamic Systems, Measurement and Control, vol. 135, no. 5, Article ID 051009, pp. 1–10, 2013. View at: Publisher Site  Google Scholar
 H. Yan, X. Wang, B. Yu, and H. Ji, “Adaptive integrated guidance and control based on backstepping and inputtostate stability,” Asian Journal of Control, vol. 16, no. 2, pp. 602–608, 2014. View at: Publisher Site  Google Scholar  MathSciNet
 M. F. Jegarkandi, A. Ashrafifar, and R. Mohsenipour, “Adaptive integrated guidance and fault tolerant control using backstepping and sliding mode,” International Journal of Aerospace Engineering, vol. 2015, Article ID 253478, 7 pages, 2015. View at: Publisher Site  Google Scholar
 S. Xingling and W. Honglun, “Backstepping active disturbance rejection control design for integrated missile guidance and control system via reducedorder ESO,” ISA Transactions, vol. 57, pp. 10–22, 2015. View at: Publisher Site  Google Scholar
 A. Zhurbal and M. Idan, “Effect of estimation on the performance of an integrated missile guidance and control system,” IEEE Transactions on Aerospace and Electronic Systems, vol. 47, no. 4, pp. 2690–2708, 2011. View at: Publisher Site  Google Scholar
 H. Yan and H. Ji, “Integrated guidance and control for dualcontrol missiles based on smallgain theorem,” Automatica, vol. 48, no. 10, pp. 2686–2692, 2012. View at: Publisher Site  Google Scholar  MathSciNet
 K. Chen, B. Fu, Y. Ding, and J. Yan, “Integrated guidance and control method for the interception of maneuvering hypersonic vehicle based on high order sliding mode approach,” Mathematical Problems in Engineering, vol. 2015, Article ID 648231, 19 pages, 2015. View at: Publisher Site  Google Scholar
 S. Shamaghdari, S. K. Nikravesh, and M. Haeri, “Integrated guidance and control of elastic flight vehicle based on robust MPC,” International Journal of Robust and Nonlinear Control, vol. 25, no. 15, pp. 2608–2630, 2015. View at: Publisher Site  Google Scholar  MathSciNet
 M.Z. Hou and G.R. Duan, “Adaptive dynamic surface control for integrated missile guidance and autopilot,” International Journal of Automation and Computing, vol. 8, no. 1, pp. 122–127, 2011. View at: Publisher Site  Google Scholar
 X. L. Liang, M. Z. Hou, and G. R. Duan, “Adaptive dynamic surface control for integrated missile guidance and autopilot in the presence of input saturation,” Journal of Aerospace Engineering, vol. 28, no. 5, Article ID 04014121, pp. 1–8, 2015. View at: Publisher Site  Google Scholar
 F. Wang, G. Liu, and X.G. Liang, “Integrated guidance and control design for the near space interceptor,” International Journal of Aeronautical and Space Sciences, vol. 16, no. 2, pp. 278–294, 2015. View at: Publisher Site  Google Scholar
 H. T. Song, T. Zhang, G. L. Zhang, and C. J. Lu, “Integrated interceptor guidance and control with prescribed performance,” International Journal of Robust and Nonlinear Control, vol. 25, no. 16, pp. 3179–3194, 2015. View at: Publisher Site  Google Scholar  MathSciNet
 M. Z. Hou, X. L. Liang, and G. R. Duan, “Adaptive block dynamic surface control for integrated missile guidance and autopilot,” Chinese Journal of Aeronautics, vol. 26, no. 3, pp. 741–750, 2013. View at: Publisher Site  Google Scholar
 H. Song and T. Zhang, “Fast robust integrated guidance and control design of interceptors,” IEEE Transactions on Control Systems Technology, vol. 24, no. 1, pp. 349–356, 2016. View at: Publisher Site  Google Scholar
 C. Guo and X.G. Liang, “Integrated guidance and control based on block backstepping sliding mode and dynamic control allocation,” Proceedings of the Institution of Mechanical Engineers Part G: Journal of Aerospace Engineering, vol. 229, no. 9, pp. 1559–1574, 2015. View at: Publisher Site  Google Scholar
 H. T. Song, T. Zhang, and G. L. Zhang, “L_{1} adaptive state feedback controller for threedimensional integrated guidance and control of interceptor,” Proceedings of the Institution of Mechanical Engineers Part G: Journal of Aerospace Engineering, vol. 228, no. 10, pp. 1693–1701, 2014. View at: Publisher Site  Google Scholar
 Y. A. Hu, Y. Q. Jin, and H. Y. Li, Robustness of Adaptive Backstepping Control for Nonlinear Systems, Publishing House of Electronics Industry, Beijing, China, 2010.
 J. Q. Han, Active Disturbance Rejection Control TechniqueThe Technique for Estimating and Compensating the Uncertainties, National Defense Industry Press, 2009.
 X. F. Qian, R. X. Lin, and Y. N. Zhao, Flight Dynamics of Missile, Beijing Institute of Technology Press, Beijing, China, 2008.
 D. Zhou, New Guidance Laws for Homing Missile, National Defense Industry Press, 2002.
Copyright
Copyright © 2016 Changsheng Gao 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.