Abstract
In the descent phase of the hypersonic target flight trajectory, the hypersonic speed of the target makes the reaction time shorter. Due to overload restrictions, it is challenging for the interceptor to achieve successful interception. To address this problem, a predictive differential game guidance approach based on dynamic optimization algorithm is proposed to relax the interceptor acceleration requirements. Two-dimensional kinematics and dynamics engagement mode for hypersonic target interception is formulated as a nonlinear differential control model in the presence of matched uncertainties. The nonlinear differential model is transformed into a differential game model by combining a performance index. The performance index is positively correlated with the Line-Of-Sight (LOS) rate and control effort of interceptor and negatively correlated with the target maneuver. The Chaotic Quantum Particle Swarm Optimization (CQPSO) algorithm is utilized to address the nonlinear differential game problem to generate the instantaneous and predicted guidance commands for the interceptor and target. Numerical examples are given to verify the performance of the proposed guidance law in various engagement scenarios, including different target maneuvers, initial heading angles and target overload. Moreover, the performance of the algorithm is validated comparing with traditional and state-of-the-art guidance laws.
1. Introduction
The guidance algorithm for interceptors is a key technology for ensuring accurate interceptions. Proportional navigation (PN) and its variants [1, 2] are commonly used guidance methods for interceptors because of their simplicity, effectiveness, and ease of implementation [3].
Owing to the increasing guidance requirements and application of modern advanced control theories, guidance methods have developed rapidly, and state-of-the-art guidance methods continue to arise. These guidance methods not only decrease the miss distance but also study the interception of maneuvering targets, terminal angle constraints, energy optimization, and other issues. For example, the sliding mode guidance laws [4] solve the uncertainty of the system model and external disturbance suppression problem. Optimal guidance laws [5] solve the control problem in terms of time and energy optimization. Differential game guidance regards the target as an intelligent agent in which the target executes the best maneuver strategy to avoid interception [6]. Moreover, heuristic algorithms such as genetic algorithms, ant colony algorithm, and particle swarm optimization (PSO) algorithm [7–13] have been utilized to calculate the guidance commands, which further improve the performance of both traditional and modern guidance laws.
The engagement of hypersonic targets has recently become a new challenge. When the hypersonic target reenters the atmosphere and reaches the descent stage of the hypersonic flight trajectory, its speed is so fast that the remaining time for the defense system is so short that the interceptor no longer has an advantage in speed compared to the target [14, 15]. Traditional guidance methods cannot intercept hypersonic targets to lag behind the target in some engagement scenarios [16]. Thus, the application and improvement of modern guidance laws for hypersonic target interception has been investigated. Suboptimal midcourse guidance with a terminal-angle constraint for interceptors against near-space hypersonic targets was introduced in [17], which can provide handover conditions for intercepting hypersonic targets. The results of [18] exploited a fourth-order sliding mode controller in an augmented integrated guidance and control model using an arbitrary order robust exact differentiator to estimate the high-order derivatives of the sliding manifold. An integrated guidance and control algorithm based on the integral backstepping method was presented in [19], which eliminates the target maneuver on the miss distance and improves the robustness of the control system. A partially integrated guidance and control method that provides a quick and stable response to the target maneuver was introduced in [20]. It should be noted that when using these technologies, the normal acceleration required by the interceptor is much greater than the target acceleration in some combat situations, which results in reduced guidance accuracy or even misses the target in some engagement scenarios, especially at the descent stage of the hypersonic flight trajectory when the interceptor’s response time is shorter. It is well known that the best trajectory for a missile to intercept a hypersonic target is the inverse trajectory or nearby head-on [15]. Owing to the low closing speed with reduced energy requirements, the so-called head-pursuit guidance [21, 22] was developed for hypersonic target interception. However, in the orbit change stage of head-pursuit interception, the rail control engine of the interceptor is used to change the direction of the trajectory by approximately 180°. This guidance method is unachievable because of the short reaction time and overload limitation during the descent phase of the hypersonic target flight trajectory.
To date, extensive studies have been conducted on differential game guidance for subsonic and supersonic target interception. In [23], a guidance law based on the pursuit-evasion game was formulated for a maneuvering target, where the performance index is the intercept time. To achieve a zero missdistance, a new guidance law based on a differential game with bounded controls was derived [24]. Another differential game guidance law considering the estimation error in the target maneuvering acceleration was proposed in [25]. The paper [26] presents a new concept for deriving improved differential-game-based guidance laws that make use of information about the target orientation. However, there have been no studies on the application of differential game guidance to the interception of hypersonic targets in the descent phase of the flight trajectory.
In summary, the above techniques used to intercept hypersonic targets may be ineffective at the descent stage of the hypersonic flight trajectory because of the short reaction times and overload saturation. Moreover, there has been no research on the application of differential game guidance to the interception of hypersonic targets. Consequently, the above contents motivate this study to provide a heuristic guidance algorithm called predictive differential game guidance based on chaotic quantum particle swarm optimization (CQPSO-PDGG) for hypersonic target interception. The main contributions of this study can be summarized as follows: (1)To the best of our knowledge, the guidance algorithm proposed in this study is the first to apply the differential game guidance algorithm to intercept the hypersonic target in the descent phase of the flight trajectory(2)The differential game guidance model is solved by CQPSO to obtain instantaneous and predicted guidance commands, which reduces the overload requirements of the interceptor and computational cost(3)CQPSO-PDGG can achieve lateral interception instead of tail-chase in different engagement scenarios, so that the interceptor will not miss the target owing to speed disadvantages
The remainder of this paper is organized as follows. The two-dimensional kinematics and dynamic model of engagement are described in Section 2. The differential game problem and mathematical foundations are presented in Section 3. The CQPSO-PDGG algorithm is presented in Section 4. The numerical simulation results are presented in Section 5. Finally, the conclusions are summarized in Section 6.
2. Kinematics and Dynamics Engagement Model
The interception geometry in two-dimensional space is shown in Figure 1; hypersonic target tends to strike in a top-down manner at the descent stage of the flight trajectory. is the inertial reference frame. The engagement kinematic equations are written as follows [27]. where the relative distance and LOS angle are represented as and and and are the heading angle of the interceptor and the target. and represent the velocity; and represent the acceleration Assuming a first order lag for the interceptor and target, we can obtain that where and represent the time-constant and the guidance command of the interceptor and and represent the time-constant and the guidance command of the target. Differentiating Equations (1) and (2) with respect to time produces,
Equations (1)–(7) are formulated as nonlinear control model of the guidance system that can be rewritten as where is the state of the guidance system and , , and are functions of with the following form:
In real combat, the velocity of interceptor is affected by drag and thrust ; the time derivative of interceptor velocity can be calculated as [13] where is the acceleration due to gravity and is the interceptor mass. The aerodynamic drag is modeled as [28] where is the dynamic pressure, is the air density, and is the reference area. The zero-lift coefficient and the induced drag coefficient are where is Mach number, modeled as where variation of temperature with altitude is related as
The thrust and the mass of the interceptor are
The atmosphere density is given by where the atmospheric pressure, , is calculated as where represents latitude.
Definition 1. The quantity Zero Effort Miss (ZEM) distance at an instant is defined as the closest distance between the interceptor and the target that would result if both the interceptor and the target do not maneuver from that instant onwards. The value of ZEM, , comes out to be
From the expression of in Equation (18), it can be seen that regulating to zero implies regulation of to zero value, and regulation of to zero while keeping leads to interception. Therefore, the LOS rate will be considered in the performance index; the detail will be shown in the following sections.
3. Differential Game Guidance Algorithm of the Guidance Problem
Consider the nonlinear system with uncertainties: where represents the system state, represents the control variable, and denotes the uncertainties. The uncertainties should satisfy the following inequality constraint: where denotes a function of state . In the presence of uncertainties, the control variable should make the system reach an asymptotically stable, that is, the nonlinear system with uncertainties can be regarded as the robust control system. Moreover, the robust control system can be transformed into the optimal control systems as follows [29]:
Here, Equation (22) is the performance index that considers the uncertainties; and represent the state and control weighting matrices.
Considering the uncertainties, Equation (8) can be rewritten as a nonlinear differential guidance model with uncertainties as follows:
In Equation (23), and denote the uncertainties of interceptor and hypersonic target’s guidance commands, which satisfy the following conditions: where and denote a function of state . Also, we can transform the robust control model, described in Equation (23), into an optimal control model as follows: where , has to remain positive semidefinite for all , and and have to remain positive-definite for all as well. is the prediction horizon, and is the start time.
The optimal control model defined by Equation (25) and Equation (26) is essentially a differential game model. The objective of interceptor and target is to find one control input minimizing the infinite time performance index and another control input maximizing performance index, that is, the saddle-point of the performance index.
4. Differential Game Guidance Algorithm Based on the CQPSO
As can be seen from the previous section, the differential game guidance model described by Equations (25) and (26) can be regarded as a dynamic optimization problem. In this section, we hope to find solutions to the dynamic optimization problem based on the CQPSO in the prediction horizon. CQPSO has faster convergence speed and higher convergence accuracy compared with Adaptive Particle Swarm Optimization (APSO) Algorithm [30] and Quantum Particle Swarm Optimization (QPSO) algorithm [31]. The general structure of the CQPSO-PDGG algorithm is shown in Figure 2. CQPSO-PDGG has two loops: the external loop iterates when the interceptor encounters the hypersonic target, and the internal loop iterates to calculate the instantaneous guidance commands and predicted guidance commands. More explanation is provided as follows:
Step 1. Initialization
The parameters of the CQPSO-PDGG are set. These parameters include the initial state of the interceptor and the target, the prediction horizon, the initial value and initial velocity of the particles, and the parameters of the CQPSO. The initial value of particle satisfies the uniform distribution, and the elements of particle are mutually independent. The particle of CQPSO in this paper is defined as
Here, the , , and are the number of particles, start time of the prediction horizon, and iteration number.
Step 2. Prediction of the future states
From Equation (25), the future states can be predicted as where is the sampling interval.
Step 3. Computation of the performance index
The performance index of the particle can be calculated as
As can be seen form Equations (29)–(33), the interceptor aims to reduce the state tracking error, control effort, and uncertainty of interceptor, and increase the target’s control effort, so as to obtain a smaller performance index value.
Step 4. Updating local and global optimal particles
The local optimal particles and global optimal particle are updated according to the performance index value of each particle, and the corresponding local performance index value and global performance index value are also updated.
Step 5. Generation of chaotic mapping sequence
The weighted average of the local optimal particles and the global optimal particles can be calculated as: where . Also, taking the first element of particle as an example, the chaotic mapping relationship is established by calculating the distance between and , and the chaotic search range is dynamically adjusted during iterations. The search range of is stated as where . By normalizing , the initial value of the chaotic sequence is given as
This paper exploits the Cat mapping function as chaotic sequence generator [32, 33]. A new chaotic sequence can be generated through this mapping function, where is the length of the chaotic sequence. The same operation can be performed on the other elements of the particle and on the other particles, so that we can get new particles.
Step 6. Computation of the performance index
For each new particle, compare its current performance index value with the value of and . If current value is better (with smaller value of performance index), then update the particle and its performance index value.
Step 7. Updating the velocity and position of particles
The velocity and position of particles are updated as
Here, is the velocity, is the inertia weight, and are constants, and and satisfy uniform distribution between [0,1].
Step 8. Stopping criteria of inner loop
The internal loop of the CQPSO-PDGG is terminated after the maximum number of iterations ().
Step 9. Calculation of the guidance commands
After iterations, the instantaneous guidance commands and predicted guidance commands can be estimated based on optimal particles as follows: where represents the set of subscripts of the top particles after the iterations. and are the instantaneous guidance commands; and are the predicted guidance commands. The average value of the guidance commands from to , input to the autopilot as the actual guidance command, can be calculated as where , , and are the actual guidance commands input to the autopilot at for . Also, is the initial value of particle at .
Step 10. Stopping criteria of outer loop
The external loop is terminated as soon as the interceptor encounters the hypersonic target.
5. Simulation and Discussion
In this section, the effectiveness of CQPSO-PDGG law proposed in this paper would be investigated through numerical simulation. In the first part, the simulation results of the algorithm are presented in various engagement scenarios. In the second part, the performance of the CQPSO-PDGG is compared with traditional and state-of-the-art guidance laws. In the following sections, the parameters of CQPSO-PDGG are set as follows: , , , , , , , , , , , , , , and seconds. Therefore, and are met; the stability and convergence of the CQPSO are guaranteed [34, 35]. Moreover, the guidance command uncertainties are modeled according to [27], define and , and Equation (7) can be further written as where , , , and .
5.1. Hypersonic Target Interception in Various Engagement Scenarios
The initial coordinates of the interceptor and the target are and , respectively. The initial velocity of the interceptor and the target are and . The initial heading angles of the interceptor and the target are and . Figures 3–5 show the engagement scenario against the nonmaneuvering target, step-maneuvering target, and sinusoidal-maneuvering target, respectively.
(a) LOS rate
(b) Relative velocity
(c) Engagement trajectory
(a) LOS rate
(b) Relative velocity
(c) Engagement trajectory
(a) LOS rate
(b) Relative velocity
(c) Engagement trajectory
Simulation results are depicted in Figures 3–6. The LOS rate is regulated to zero within a period of 4 to 6 seconds as shown in Figures 3(a), 4(a), and 5(a), which leads the ZEM to zero according to Equation (17). Meanwhile, Figures 3(b), 4(b), and 5(b) show that a negative is maintained to ensure that the relative distance between the interceptor and the target decreases. Lateral interceptions are achieved successfully in Figures 3(c), 4(c), and 5(c) in the present of the model uncertainties. The acceleration of interceptor is presented in Figure 6. Figures 6(a) and 6(b) show that the acceleration of interceptor tends to zero when the target is nonmaneuvering or perform a step maneuver. In Figure 6(c), the target performs a sinusoidal maneuver; the acceleration of interceptor is stable within an interval.
(a) Nonmaneuvering target
(b) Step-maneuvering target
(c) Sinusoidal-maneuvering target
Table 1 illustrates the interceptor overload requirements of methods studied in other references for hypersonic target interception. The interceptor overload requirements is greater than the target acceleration, and the ratio between the two is greater than 1.5. Figure 7 describes the lateral acceleration of the interceptor when the target performs a sinusoidal maneuver with different overload. When the hypersonic target is maneuvering with 5 g, 10 g, 20 g, and 40 g overloads, the maximum acceleration of is 5.5 g, 10.7 g, 21.8 g, and 44.4 g, respectively. The maximum acceleration of is slightly greater than the target acceleration, and the ratio between the two is less than 1.2. Therefore, the results manifest that the overload capability requirements of CQPSO-PDGG is much less compared to the techniques studied in the other literature.
As shown in Figure 6, the initial guidance commands are large at the beginning of interception. Figure 8 depicts that the missile intercepts the hypersonic target at different heading angles. It can be observed from these figures that when the deviation between the heading angle of interceptor and the line-of-sight angle is large, the initial overload required by the interceptor is large, when the deviation is small, the required overload is small, that is, the two variables are positively correlated.
(a) Engagement trajectories
(b) Lateral acceleration of the interceptor
5.2. Comparison with Multiple Guidance Law
In this section, the proposed CQPSO-PDGG algorithm is compared with LQDG [6], SDRE-DG [6], ADP [27], PSO-DGG [13], PN-IPSOG [12], and MCACC [7]. The parameters, required for the simulation, are selected according to the corresponding references as given in Table 2. The simulation results are shown in Figures 9–13 and Table 3.
(a) Lateral acceleration of interceptor
(b) Engagement trajectories
(a) Lateral acceleration of interceptor
(b) Engagement trajectories
(a) Lateral acceleration of interceptor
(b) Engagement trajectories
(a) -axis
(b) -axis
(a) LOS rate
(b) Lateral acceleration of interceptor
Compared with the SDRE-DG [6], LQDG [6], PSO-DGG [13], and ADP [27], the lateral acceleration of interceptor and engagement trajectories are illustrated in Figures 9–11 and Table 3. Figures 9–11 are investigated with step-maneuvering target, sinusoidal-maneuvering target, and nonmaneuvering target, respectively. The values of the parameters used in simulation are shown in Table 2, which are the same as Ref. [6, 13, 27]. As seen in Table 3, the maximum acceleration and control effort of CQPSO-PDGG are less than the other algorithm. It can be observed from Figures 9(a), 10(a), and 11(a) that the CQPSO-PDGG can generate stable lateral acceleration and tend to zero, but the lateral acceleration of the SDRE-DG, LQDG, and ADP is unstable in final engagement, and the overload requirement of CQPSO-PDGG is lower than the values reported in [13, 27] in the initial stage.
Moreover, CQPSO-PDGG is compared with PN-IPSOG [12] and MCACC [7] as shown in Figures 12 and 13. The lateral acceleration of interceptor is presented in Figure 12. It can be seen from the figures that the CQPSO-PDGG is more stable than the PN-IPSOG guidance at the initial stage. Figure 13 reports the LOS rate and lateral acceleration of interceptor. One can see from Figure 13(a), the CQPSO-PDGG makes the LOS rate approach 0 faster than the MCACC. Also, the maximum acceleration and control effort of CQPSO-PDGG are less than the other algorithm as shown in Table 3. As a brief summary, the simulation results reveal that the CQPSO-PDGG had a better performance than the above guidance techniques.
5.3. Computational Requirement
The simulations of CQPSO-PDGG are implemented in a MATLAB environment, and the main configuration of the computer is 2.1 GHz CPU and 16 GByte RAM. The calculation cycle of each guidance command is approximately 0.08 seconds. Therefore, using C++ programming language will greatly improve the real-time performance of the algorithm.
6. Conclusion
In this paper, a CQPSO-PDGG method is proposed for hypersonic target interception in the descent phase of the hypersonic target flight trajectory. The kinematics and dynamics engagement model, namely, nonlinear guidance models, is established in the presence of uncertainties. Then, the nonlinear differential guidance problem is transformed into a differential game guidance problem by introducing a performance index. Finally, the differential game guidance model is solved by CQPSO to obtain the instantaneous guidance commands and predicted guidance commands. The performance of the proposed algorithms is verified by the numerical simulations in different engagement scenarios such as various target maneuvers, heading angles of interceptor, and target acceleration. Moreover, comparisons with LQDG, SDRE-DG, ADP, PSO-DGG, PN-IPSOG, and MCACC show that the CQPSO-PDGG method can intercept the target with lower overload requirements and less control effort. Also, the CQPSO-PDGG is more stable than these guidance algorithm at the initial stage and final stages of interception. In the future, we will investigate the cooperative guidance law with multiple interceptors based on DG to further improve the performance of guidance system for hypersonic target interception.
Data Availability
All data and models generated or used during the study appear in the submitted article.
Conflicts of Interest
The authors declare that they have no conflict of interest.