Research Article  Open Access
Triangle Interception Scenario: A FiniteTime Guidance Approach
Abstract
Considering an aircraft threatened by an interceptor, one of the effective penetration strategies is to release a Defender from the aircraft to confront the interceptor. In this case, the aircraft, the Defender, and the interceptor constitute the threebody guidance relationship, and the cooperation of the aircraft and its Defender to achieve the best tactical effects turns into a concerned problem. This paper studies the triangle interception guidance problem via the finitetime theory. The paper presents linear system InputOutput FiniteTime Stabilization (IOFTS) method. The sufficient conditions of the linear system, being IOFTS, under FiniteTime Boundedness (FTB) constraint are proposed, by which the state feedback controllers design method is obtained, via Linear Matrix Inequalities (LMIs). The triangle interception guidance problems are studied in three different cases, where the proposed methods are applied to the guidance design. The simulation results illustrate the effectiveness of the proposed methods.
1. Introduction
A triangle interception scenario emerges if the aircraft launches a Defender antimissile as an active response while facing an enemy missile. The aircraft, the enemy missile, and the Defender antimissile constitute the triangle interception engagement. In the following parts of the paper, the Evader, Pursuer, and Defender are used to refer to these three aircrafts, as shown in Figure 1. Obviously, the Evader and the Defender are on one side, the Pursuer is on the other side.
Triangle interception problems are proposed by Boyell [1, 2], assuming constant bearing collision courses. Using the proportional navigation (PN) guidance law, closedform expression for the intercept point and conditions on speed ratios were also derived. PN guidance law always requires the missile to have stronger maneuver ability than the target; hence the methods did not fully exploit the position advantages of the Defender.
With the developments of missile performances and combat styles, the triangle interception problems have been a point of concern in the last several years. Ratnoo and Shima [3] and Yamasaki and Balakrishnan [4] studied this issue; in the battle scene it was assumed that the fighter launched a Defender while facing an enemy missile. Ratnoo and Shima [3] analyzed the threebody engagement geometry; different initial conditions and guidance laws were also considered in simulation studies. Yamasaki and Balakrishnan [4] proposed a Line of Sight (LOS) guidance law for the Defender, which caused a relatively straight trajectory. The threedimension condition was also considered in simulation study. Shaferman and Shima [5] proposed a multiple model adaptive guidance law, which had the advantages of considering detection errors and the nonlinear motion model, while it could also design cooperative guidance laws for both Defender and Pursuer.
Considering the engineering applications and classic guidance laws, LOS guidance law was concerned in triangle interception scenario. Yamasaki et al. [6] and Ratnoo and Shima [7] proposed the methods for this problem based on LOS guidance. Yamasaki et al. [6] modified the guidance laws considering different target maneuvers. Ratnoo and Shima [7] studied the strategies of Defender considering different Pursuer guidance laws. Ratnoo and Shima [8] also studied Defender guidance law based on LOS, which was compared with PN guidance law. The results show that the Defender adopting the proposed method had smaller acceleration demand than the Pursuer, which would further reduce while Evader is cooperating.
Moreover, scholars considered that the three aircrafts adjust guidance strategies online instead of an invariable guidance law. Shima [9] proposed the optimal cooperative guidance law, which assumed that the interceptor adopts some linear guidance law. Rubinsky and Gutman [10] proposed performance index considering missdistance and energy consumption simultaneously. Then the strategies of Evader and Defender were proposed based on optimal control and the influences of different timetogo and initial conditions were analyzed. Ratnoo and Shima [11] analyzed the consequences while PN, pure pursuit, and LOS guidance laws were taken by the Pursuer and Defender. Then the boundary conditions were presented.
The concept of FiniteTime Stability (FTS) was introduced in 1960s [12]. Up until now, much work has been done in the field of FTS. Given a bound on the initial condition, a system is said to be finitetime stable if the state does not exceed a certain threshold during a specified time interval. While external disturbances are considered, FTS is extended to FiniteTime Boundedness (FTB) [13, 14]. It worth noting that there is a different notion of FTS [15], which requires the system state to reach the system equilibrium in a finitetime interval, and the property is called finitetime attractiveness in some research [16, 17]. In the remainder of this paper, the FTS we mentioned refers to the former one.
InputOutput FiniteTime Stability (IOFTS) had been given in Amato et al. [18], which means that, given a class of norm bounded input signals over a specified time interval , the outputs of the system are also norm bounded over . Amato et al. [19–21] provided methods to solve IOFTS problem via static state feedback and output feedback with the disturbance considered as input of the system. Therefore, this contribution can be used for disturbance suppression.
It can be concluded that FTB considers the system state (not exceeding a given threshold) in a finitetime interval while IOFTS only considers the output (norm bounded with input disturbances satisfying some boundedness conditions) in a finitetime interval. However, in some applications, the system state stability and output should be considered comprehensively. Guo et al. [22] proposed IO finitetime stabilization method under FTB constraint for a class of linear timeinvariant (LTI) systems. In terminal guidance process, if the target maneuver was considered as external disturbance, the objective of state feedback control is to minimize the impact of the disturbance to the output, and some indices satisfy the physical constraints, such as acceleration and seeker’s view field of the missile. Therefore, the methods in [22] are suitable for the terminal guidance problem, but it should be further extended to linear timevariant (LTV) systems.
In the triangle interception scenario, terminal guidance processes of different flight vehicles are focused, which are all finished in finitetime intervals. The models describing the process of terminal guidance are also meaningful in some certain finitetime intervals. Moreover, the strategies of the three flight vehicles in the time interval of the Defender guidance to the Pursuer are different from the strategies of the Pursuer guidance to the Evader after escaping from the Defender. Therefore, the analysis and the guidance design of the triangle interception are typical finitetime control problem. If the infinitetime control methods were applied, the potential problems would include the closedlooped system performance with large conservative or not being able to achieve the requirements in the concerned finitetime interval. For a physical process finished in a finitetime interval, finitetime control theory may be more effective, since it only focuses on the concerned time interval and the performance indices are always defined in the concerned finitetime interval [23]. Furthermore, there have been some researches applying finitetime control theory for the terminal guidance problems in recent years [24–30]. Hence, the finitetime control methods will be used to solve the triangle interception problems, as dealt with in this paper.
2. Problem Statement
Consider the threebody engagement scenario, as shown in Figure 1, where the Evader launches a Defender missile to intercept the incoming Pursuer.
2.1. Assumptions
(a)The three vehicles are constantclosingspeed point masses moving in a plane.(b)The maneuvering dynamics of the three vehicles can be approximated by firstorder transfer functions with time constants , , and , respectively.(c)The Pursuer adopts augmented proportional navigation (APN) guidance law.(d)The Evader and the Defender have the Evader acceleration information and LOS angles information between themselves and the Evader, respectively. (A triangle can be determined by three easily available parameters, which contain two LOS angles () and one range (between Evader and Defender). Then, the PursuerEvader or PursuerDefender ranges can be determined. Furthermore, the Evader acceleration can be obtained by calculation.)
2.2. Dynamic Model
According to assumption (b), it can be obtained thatwhere is acceleration and is acceleration command. According to assumption (c), we assume that the Pursuer adopts APN guidance law, which iswhere and are constants.
In relative motion process between two vehicles, the rotation of the Line of Sight (LOS) satisfies and are LOS angles of PursuerEvader and PursuerDefender. Similarly, and are ranges of PursuerEvader and PursuerDefender. are accelerations of Pursuer, Evader, and Defender, respectively.
According to assumption (a), the three vehicles have constantspeed. Thus, after the respective collision triangles were reached and maintained, the PursuerEvader closing speed (denoted by ) and that of the PursuerDefender (denoted by ) are constants.
The PursuerEvader interception time can be approximated bySimilarly, Obviously, is required. For the purpose of intercepting Evader by Defender, we only consider the system behavior in the finitetime interval .
2.3. Problem Formulation
By (1)–(5), a state space description can be obtained, which iswhere system state vector is is external interference and is the corresponding matrix. and are defined as the following two cases.
(i) Due to the designated task of Evader, if we can only control the Defender to intercept the Pursuer, the input is ; then the corresponding is as follows:
(ii) If control authorities of both Defender and Evader are available, the input isThen the corresponding is as follows:In this case, the acceleration of Evader is considered with a physical constraint, described asThese two cases will be discussed in Sections 4 and 5, respectively.
Define ZeroEffortMiss distance of the Defender to the Pursuer aswhere is time of Defender going in to guidance blind zone (in general, is near ) and is the size of the blind zone, a constant. Therefore, is directly related to the missdistance of the Defender to the Pursuer.
Our purpose is to get a smaller by designing . Since the system has typical finitetime characteristic, system behavior concerned only with a finitetime interval, the problem will be solved via finitetime control theory.
3. FTB, IOFTS, and Stabilization Method of LTV Systems
Definition 1 ((IOFTS of LTV systems) [18]). Given a positive scalar , a positive definite matrix , and a positive definite matrixvalued function , system is said to be IOFTS with respect to if
Definition 2 ((FTB) [13]). Given a positive scalar , a positive definite matrix , and a positive definite matrixvalued function , such that , system (13) and (14) is said to be FiniteTime Bounded (FTB) with respect to , if
Remark 3. Compared with Lyapunov Asymptotic Stability (LAS) and BIBO stability defined in infinitetime interval, IOFTS is only concerned with the signals defined over a finitetime interval, which is more suitable for the guidance design problems.
Both FTB and IOFTS State Feedback Controller Design Problem. Consider a linear timevariant system (13) and (14), and a state feedback controllerwhere is a matrix to be determined later. Applying this controller to system (13) and (14) will result in the following closedloop system:whereBoth FTB and IOFTS state feedback controller design problems can be formulated as finding a state feedback controller in the form of (17) such that the closedloop system (13) and (14) is both FTB and IOFTS with respect to .
Theorem 4 ((necessary and sufficient conditions of IOFTS) [20]). Given system (13) and (14) and the time interval , the following statements are equivalent:(1)System (13) and (14) is IOFTS with respect to .(2)The coupled DLMI/LMIadmits a positive definite solution over .
Theorem 5 (see [18]). If there exists a positive definite matrixvalued function and a matrixvalued function such thatthe IOFTS state feedback controller design problem can be solved by .
Corollary 6 (both FTB and IOFTS condition). System (13) and (14) is both FTB and IOFTS with respect to , if there exists a symmetric positive definite matrixvalued function such that
Proof. We will first demonstrate that (22), (23), and (24) can ensure that system (13) and (14) is FTB with respect to . Let ; we have (In the following part of the proof, we omit whenever no ambiguity arises.)
By (22) and (13), we can obtainIntegrating (26) from to , we havewhich can be rewritten asConsidering (23) and (24), we can obtainTherefore, system (13) is FTB with respect to .
Now we shall prove that (22), (23), and (24) can ensure that system (13) and (14) is IOFTS with respect to . Considering , by (23), we can obtainBy (23) and (30), we can obtainwhich can be rewritten asTherefore, the proof follows.
Corollary 7. If there exists a symmetric positive definite matrixvalued function and a matrixvalued function such thatthen both FTB and IOFTS state feedback controller design problem can be solved by .
Proof. Replacing in (13) with , let . According to Corollary 6, then the proof follows.
4. Defender Intercepts Pursuer without Cooperation of Evader (Case i)
In some applications, an Evader has its own mission or target, which means that its trajectory will not change whether Pursuer exists or not. The state feedback is used to design controller with the object of minimizing the impact of the disturbance to the output (finitetime disturbance suppression). The acceleration command of Defender can be obtained by the following expression:where can be calculated by using IOFTS state feedback controller design method in Theorem 5.
According to the model in Section 2 and the proposed method in Section 3, using the parameters in Table 1, the controller can be obtained. Then the system performance and the trajectories of the aircrafts can be calculated, as shown in Figures 2–4.
(a)  
 
(b)  
 
(c)  
In the noncooperation case, the Evader follows a vertical upward flight path, the Pursuer guidance to the Evader. Meanwhile, the Defender is released from the Evader and guidance to the Pursuer. Compared with the PN guidance law, the Defender trajectory caused by the proposed method is more flat without significant turns, which means that the acceleration demands of the Defender are much smaller.
Figures 3 and 4 present the LOS angular rate and the Defender acceleration. It can be found that both methods can drive the LOS angular velocity tending to zero, but the acceleration demand by the proposed method is much smaller.
5. Evader and Defender Cooperate to Intercept Pursuer (Case ii)
While considering a cooperation case, the Evader can “lure” the Pursuer to help the Defender achieve an easier interception. However, the acceleration physical constraint of the Evader should be considered as presented in (11). So both the disturbance depression and finitetime state bounded should be satisfied comprehensively. We proposed both FTB and IOFTS controller design method (Corollary 7) to solve the problem.
According to the model in Section 2 and the proposed method in Section 3, using the parameters in Table 2, the controller can be obtained; then in the cases of cooperating and noncooperating Evaders, the system performances and the trajectories of the aircrafts can be calculated, as shown in Figures 5 and 6.
(a)  
 
(b)  

In the cooperative case, by mean of the Evader maneuvering to “lure” the Pursuer, the Defender more easily obtains better guidance effect. Figures 7–10 show the comparison of LOS angular rate and acceleration. In cooperative and noncooperative cases, the proposed method can drive the LOS angular rate which tends to zero, which means a small missdistance. However, in the cooperative case, the Defender requires smaller acceleration, which can be seen in the fact that the Evader “shares” some acceleration demand.
For some high speed targets, such as reentering ballistic missiles, a headon interception strategy will cause high closing speed. This imposes severe requirements on the interceptor systems such as precise detection of the target from a large distance by the onboard seekers and very fast response time of the missile subsystems. Head Pursuit (HP) guidance was introduced to solve this problem. The guidance law imposes a geometric relation in which the interceptor missile is positioned ahead of the target so that both fly in the same direction. The missile speed is planned to be lower than that of the target, and therefore the target closes on the interceptor missile [31, 32].
In our scenario, considering HP guidance adopted by the Pursuer to intercept the Evader, both FTB and IOFTS controller design method (Corollary 7) still can be used to solve the problem. The parameters are presented in Table 3. Then, in the cases of cooperating and noncooperating Evaders, the system performances and the trajectories of the aircrafts can be calculated, as shown in Figures 11–14.
(a)  
 
(b)  

It can be concluded that, considering HP guidance Pursuer, the closing speed between Evader and Pursuer is smaller than the PN guidance Pursuer case. With Evader and Defender cooperating, the acceleration demand of Defender is smaller than the noncooperative case.
6. MultiDefender versus MultiPursuer Case
In this case, we consider that, while facing multiple Pursuers, an Evader launches the corresponding number of Defenders as countermeasures. Figure 15 presents the two Defenders versus two Pursuers case. On one hand, if the Evader did not maneuver cooperatively, each Defender should be guided to the designated Pursuer. The guidance law of each Defender can be designed according to Case i in Section 4 (Defender intercepts Pursuer without cooperation of Evader). On the other hand, if the Evader cooperates with the Defenders, the Evader maneuvering would influence both Defender’s guidance processes.
For this problem, we defineAccording to the modeling method in Section 2, we can get the similar system as follows: , , and can be obtained by similar equations in Section 2. It is worth noting that the two Defender guidance processes are likely finished at different moments. We use , to denote the guidance finish moment of the two Evaders, respectively. Then, in time interval , one Defender guidance process has finished; we focus on the remaining three flight vehicles, so the problem turns into the same one in Section 5. In time interval , we focus on all five flight vehicles’ guidance processes.
In the time interval , we still can solve guidance design problems according to the method in Corollary 7. The difference is that contains two angular rates of Line of Sight, LOS, angular velocity, which reflects two Defenders’ guidance missdistance. The closedloop system should satisfy Thus, the chosen weighted matrix is crucial for the guidance design. The weights reflect attention degree of the corresponding LOS angular velocity. may have the formwhere and are both larger than zero. We set an appropriate larger weight for the corresponding Defender whose guidance timetogo is smaller than the other one. The purpose is to drive the corresponding LOS angular velocity convergence more rapidly. According to the parameters in Table 4, the previously mentioned models, and the methods, we design the corresponding controllers in different stages and add them to the original system. The five aircraft motion trajectories are shown in Figure 16.
(a)  
 
(b)  
 
(c)  
 
(d)  

It can be seen that, at 0 s, two Pursuers from different directions are guided to the Evader, which simultaneously release two Defenders. Since the distances between the Defender and Evader are not the same, the Evader tends to corporate with the Defender with less guidance timetogo. After one Defender finishes the guidance process, the Evader turns to fully cooperate with the other Defender.
Figure 17 shows the LOS angle velocities between the two Defenders and the two Pursuers. It can be established that the controller can drive the LOS angle velocities close to zero in the concerned finitetime interval, which means the missdistances satisfied the demand.
Figure 18 presents the Evader’s acceleration. It can be established that, before 5.95 s, the Evader is more inclined to cooperate with the Defender with the less guidance timetogo. After 5.95 s, one Defender has finished the guidance process; the Evader turns to fully cooperate with the other Defender. At the same time, the Evader acceleration does not exceed the physical capacity constraint.
7. Conclusion
This paper focuses on a triangle interception scenario; the guidance law design is deduced to a finitetime controller design problem. Sufficient conditions guaranteeing system of both FTB and IOFTS are proposed. Then a state feedback controller can be obtained via LMIs. The methods are applied to guidance design for triangle interception scenario. The results show that the proposed method is suitable for solving the problem in a finitetime interval. The method can be regarded as a finitetime stabilization method considering some finitetime performance constraints. Three cases in triangle interception scenarios are studied and the results show that better interception effect will be achieved while Evader and Defender cooperate. In multiDefender versus multiPursuer case, the Evader is more inclined to cooperate with the Defender with the smaller guidance timetogo. After one Defender finishes the guidance process, the Evader turns to fully cooperate with the other Defender.
Nomenclature
:  The acceleration of Pursuer, Evader, and Defender, respectively 
The acceleration command of Pursuer, Evader, and Defender, respectively  
:  The physical constraint of Evader acceleration 
:  The size of Defender’s blind zone 
:  A constant in augmented proportional navigation (APN) guidance law 
:  A constant in APN guidance law 
:  The Line of Sight (LOS) angle between Pursuer and Evader, the LOS angle between Pursuer and Defender 
:  The range between Pursuer and Evader, the range between Pursuer and Defender 
:  The final time of PursuerEvader interception process, the final time of PursuerDefender interception process 
:  The approaching speed between Pursuer and Evader, the approaching speed between Pursuer and Defender 
:  The ZeroEffortMiss distance of Defender to the Pursuer 
:  The guidance system time constant of Pursuer, Evader, and Defender, respectively. 
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgment
This work is supported by National Natural Science Foundation (NNSF) of China under Grants no. 61304239, no. 61304001, and no. 51475462.
References
 R. L. Boyell, “Defending a moving target against missile or torpedo attack,” IEEE Transactions on Aerospace and Electronic Systems, vol. 12, no. 4, pp. 522–526, 1976. View at: Publisher Site  Google Scholar
 L. R. Boyell, “Counterweapon aiming for defence of a moving target,” IEEE Transactions on Aerospace and Electronic Systems, vol. 16, no. 3, pp. 402–408, 1980. View at: Google Scholar
 A. Ratnoo and T. Shima, “Line of sight guidance for defending an aircraft,” in Proceedings of the AIAA Guidance, Navigation, and Control Conference, Toronto, Canada, 2010. View at: Google Scholar
 T. Yamasaki and S. N. Balakrishnan, “Triangle intercept guidance for aerial defense,” in Proceedings of the AIAA Guidance, Navigation, and Control Conference, Toronto, Canada, August 2010. View at: Google Scholar
 V. Shaferman and T. Shima, “Cooperative multiplemodel adaptive guidance for an aircraft defending missile,” Journal of Guidance, Control, and Dynamics, vol. 33, no. 6, pp. 1801–1813, 2010. View at: Publisher Site  Google Scholar
 T. Yamasaki, S. N. Balakrishnan, and H. Takano, “Modified CLOS intercept guidance for aircraft defense against a guided missile,” in Proceedings of the AIAA Guidance, Navigation, and Control Conference, Portland, Ore, USA, 2011. View at: Google Scholar
 A. Ratnoo and T. Shima, “Guidance laws against defended aerial targets,” in Proceedings of the AIAA Guidance, Navigation, and Control Conference, Portland, Ore, USA, August 2011. View at: Google Scholar
 A. Ratnoo and T. Shima, “Lineofsight interceptor guidance for defending an aircraft,” Journal of Guidance, Control, and Dynamics, vol. 34, no. 2, pp. 522–532, 2011. View at: Publisher Site  Google Scholar
 T. Shima, “Optimal cooperative pursuit and evasion strategies against a homing missile,” Journal of Guidance, Control, and Dynamics, vol. 34, no. 2, pp. 414–425, 2011. View at: Publisher Site  Google Scholar
 S. Rubinsky and S. Gutman, “Three body guaranteed pursuit and evasion,” in Proceedings of the AIAA Guidance, Navigation, and Control Conference, Minneapolis, Minn, USA, August 2012. View at: Publisher Site  Google Scholar
 A. Ratnoo and T. Shima, “Guidance strategies against defended aerial targets,” Journal of Guidance, Control, and Dynamics, vol. 35, no. 4, pp. 1059–1068, 2012. View at: Publisher Site  Google Scholar
 P. Dorato, “Shorttime stability in linear timevarying systems,” in Proceedings of the IRE International Convention Record Part 4, vol. 1, pp. 83–87, New York, NY, USA, 1961. View at: Google Scholar
 F. Amato, M. Ariola, and P. Dorato, “Finitetime control of linear systems subject to parametric uncertainties and disturbances,” Automatica, vol. 37, no. 9, pp. 1459–1463, 2001. View at: Publisher Site  Google Scholar
 F. Amato and M. Ariola, “Finitetime control of discretetime linear systems,” IEEE Transactions on Automatic Control, vol. 50, no. 5, pp. 724–729, 2005. View at: Publisher Site  Google Scholar  MathSciNet
 S. P. Bhat and D. S. Bernstein, “Finitetime stability of continuous autonomous systems,” SIAM Journal on Control and Optimization, vol. 38, no. 3, pp. 751–766, 2000. View at: Publisher Site  Google Scholar  MathSciNet
 W. Chen and L. C. Jiao, “Finitetime stability theorem of stochastic nonlinear systems,” Automatica, vol. 46, no. 12, pp. 2105–2108, 2010. View at: Publisher Site  Google Scholar  MathSciNet
 H. Du, C. Qian, S. Yang, and S. Li, “Recursive design of finitetime convergent observers for a class of timevarying nonlinear systems,” Automatica, vol. 49, no. 2, pp. 601–609, 2013. View at: Publisher Site  Google Scholar  MathSciNet
 F. Amato, R. Ambrosino, C. Cosentino, and G. De Tommasi, “Inputoutput finite time stabilization of linear systems,” Automatica, vol. 46, no. 9, pp. 1558–1562, 2010. View at: Publisher Site  Google Scholar  MathSciNet
 F. Amato, R. Ambrosino, M. Ariola, C. Cosentino, and G. De Tommasi, FiniteTime Stability and Control, vol. 453 of Lecture Notes in Control and Information Sciences, Springer, London, UK, 2014. View at: Publisher Site
 F. Amato, G. Carannante, G. De Tommasi, and A. Pironti, “Inputoutput finitetime stability of linear systems: necessary and sufficient conditions,” IEEE Transactions on Automatic Control, vol. 57, no. 12, pp. 3051–3063, 2012. View at: Publisher Site  Google Scholar  MathSciNet
 F. Amato, G. De Tommasi, and A. Pironti, “Necessary and sufficient conditions for inputoutput finitetime stability of impulsive dynamical systems,” in Proceedings of the American Control Conference (ACC '15), pp. 5998–6003, Chicago, Ill, USA, July 2015. View at: Publisher Site  Google Scholar
 Y. Guo, Y. Yao, S. Wang, K. Ma, K. Liu, and J. Guo, “Inputoutput finitetime stabilization of linear systems with finitetime boundedness,” ISA Transactions, vol. 53, no. 4, pp. 977–982, 2014. View at: Publisher Site  Google Scholar
 C. Li and Z. Qu, “Distributed finitetime consensus of nonlinear systems under switching topologies,” Automatica, vol. 50, no. 6, pp. 1626–1631, 2014. View at: Publisher Site  Google Scholar  MathSciNet
 M. Golestani, I. Mohammadzaman, and A. R. Vali, “Finitetime convergent guidance law based on integral backstepping control,” Aerospace Science and Technology, vol. 39, pp. 370–376, 2014. View at: Publisher Site  Google Scholar
 M. Golestani and I. Mohammadzaman, “PID guidance law design using short time stability approach,” Aerospace Science and Technology, vol. 43, pp. 71–76, 2015. View at: Publisher Site  Google Scholar
 Y. Guo, S. Wang, Y. Yao, and B. Yang, “Evader maneuver on consideration of energy consumption in flight vehicle interception scenarios,” Aerospace Science and Technology, vol. 15, no. 7, pp. 519–525, 2011. View at: Publisher Site  Google Scholar
 D. Zhou, S. Sun, and K. L. Teo, “Guidance laws with finite time convergence,” Journal of Guidance, Control, and Dynamics, vol. 32, no. 6, pp. 1838–1846, 2009. View at: Publisher Site  Google Scholar
 Y. Guo, Y. Yao, S. Wang, B. Yang, K. Liu, and X. Zhao, “Finitetime control with ${H}_{\infty}$infinity constraints of linear timeinvariant and timevarying systems,” Journal of Control Theory and Applications, vol. 11, no. 2, pp. 165–172, 2013. View at: Publisher Site  Google Scholar  MathSciNet
 Y. Guo, Y. Yao, S. C. Wang, B. Q. Yang, F. H. He, and P. Zhang, “Maneuver control strategies to maximize prediction errors in ballistic middle phase,” Journal of Guidance, Control, and Dynamics, vol. 36, no. 4, pp. 1225–1234, 2013. View at: Publisher Site  Google Scholar
 Y. Guo, X. X. Hu, S. C. Wang, J. Xie, and Q. H. Gao, “Inputoutput finitetime guidance laws in triangle interception scenario,” in Proceedings of the 33rd Chinese Control Conference (CCC '14), pp. 804–809, IEEE, Nanjing , China, July 2014. View at: Publisher Site  Google Scholar
 O. M. Golan and T. Shima, “Head pursuit guidance for hypervelocity interception,” in Proceedings of the AIAA Guidance, Navigation, and Control Conference and Exhibit, Providence, RI, USA, August 2004. View at: Google Scholar
 T. Shima and O. M. Golan, “Head pursuit guidance,” Journal of Guidance, Control, and Dynamics, vol. 30, no. 5, pp. 1437–1444, 2007. View at: Publisher Site  Google Scholar
Copyright
Copyright © 2016 Yang Guo 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.