Abstract
With the aim of achieving cooperative target interception by using multi-interceptor, a distributed cooperative control algorithm of the multi-interceptor with state coupling is proposed based on the IGC (integrated guidance and control) method. Considering the coupling relationship between the pitch and ya w channels, a state coupling “leader” IGC model is established, an FTDO (finite-time disturbance observer) is designed for estimating the unknown interference of the model, and the “leader” controller is designed according to the adaptive dynamic surface sliding-mode control law. Secondly, the cooperative control strategy of the multi-interceptor is designed with the “leader-follower” distributed network mode to obtain the speed in the three directions of the interceptor in air and transform them to the general flight speed, trajectory inclination angle, and trajectory deflection instruction by using the transformational relation of kinematics. Finally, the “follower” controller is designed with the FTDO and dynamic surface sliding-mode control. The designed multi-interceptor distributed cooperative IGC algorithm with state coupling has good stability according to the simulation results of two different communication topologies.
1. Introduction
With the rapid development of antimissile technology, it is getting difficult for a single interceptor to break through the defense and intercept targets efficiently, thus making it hard for an interceptor to adapt to the demands of future battlefield scenarios. Therefore, an interceptor with cooperative target interception capability would be more suitable for the future. Multi-interceptor can realize functional complementation through information interaction and sharing, which can not only considerably enhance defense penetration and counterforce of the interceptor but also finish tasks that cannot be achieved by a single interceptor [1].
With respect to cooperative guidance and control of multi-interceptors, the authors in [2, 3] proposed a guidance law with controllable attack time and angle-of-attack constraint and applied it to the salvo attack of anti-ship missiles. Based on this idea, researchers subsequently introduced some other guidance and control methods, including sliding-mode control [4, 5], optimal control [6], differential game [7], and dynamic surface control [8]. This group of methods relies on specifying the attack time before launching to achieve coordination. No information exchange occurs between missiles during flight; hence, these methods apparently have temporal limitations. With the progress in consensus of multiagent systems, researchers have begun to use the consensus theory to study the cooperative guidance and control of multi-interceptors. Using the coordination strategy under the cooperative guidance framework, the authors in [9] adjusted missile trajectories, such that the coordination variable of each missile can approach the expected coordination variable and realize cooperative guidance. The authors in [10] applied the “leader-follower” formation control to cooperative guidance of multi-interceptors by putting forward an analogous “leader-follower” cooperative guidance framework. The authors in [11, 12] explored the guidance and control law of the “leader-follower” topology with angle constraint and topology switch present. By constructing an integrated cost function for multiple missiles, the authors in [13] designed a cooperative guidance law for multiple missiles intercepting a maneuvering target. However, the application of this integrated cost function was faced with multiple constrains because each missile required the global information of all the participating partners.
The interceptor guidance and control system is a highly dynamic, strong-coupling, varying, and uncertain multivariant system featuring complicated dynamic characteristics. Therefore, the integrated guidance and control (IGC) design method can allocate the control ability of interceptors more properly. It mainly generates control power according to the relative motive between the interception targets and the interceptors and then drives the interceptors to chase the targets. Moreover, it cannot only stabilize flight attitude but also enhance guidance precision [14]. In recent years, many researchers worldwide conducted studies focusing on the design method of IGC. The authors in [15, 16] designed an IGC control law with the sliding control mode and back-stepping control algorithm. The sliding-mode control method has been widely used in the design of IGC of aircraft [17], missile [18], and unmanned helicopter [19, 20]. In existing papers, most of them were designed in a single channel [21, 22], regardless of the coupling between channels. The authors in [23, 24] designed IGC algorithms in three dimensions, but designing the controller is difficult when establishing the model with a high order.
According to the above literature review, the guidance loop and control loop of the multi-interceptor cooperative guidance and control have been studied separately by experts. However, external disturbance and its coupling relation during the multi-interceptor flight cannot be ignored. In the meantime, multi-interceptor needs to communicate during its flight to finish cooperative control. Thus, unsmooth local communication should be considered while designing the controller. In light of this, a distributed cooperative control strategy was introduced on top of the integrated guidance and control (IGC) method by considering the coupling between the interception pitch and yaw channels, a design method of cooperative IGC of the multi-interceptor with state coupling of the “leader-follower” distributed topology structure is proposed. The design of the “leader” and “follower” control algorithm using the dynamic surface sliding-mode and finite-time disturbance observer can effectively enhance the stability of the interceptor during the flight and furthermore ensure the target to be hit by the “leader” and “follower” simultaneously following the distributed cooperative controlling strategy. The proposed method can enhance the stability of cooperative guidance and control of the multi-interceptor.
2. “Leader” IGC Model of Interceptor with State Coupling
According to the relative motion relation between the interceptor “leader” and target [25, 26], the relative motion model of the interceptor “leader” and target is established as follows:
In the equation, and denote the elevation angle and horizontal sight angle of the “leader” and target, respectively; and denote the longitudinal and lateral motion acceleration of the “leader,” respectively; and denote the longitudinal and lateral motion acceleration of the target, respectively; represents the relative distance between the “leader” and target.
The kinetic model of the interceptor “leader” can be indicated as follows:
In the equation, is the reference area of the “leader;” is the reference length of the “leader;” is the mass of the “leader;” and are the attack angle and sideslip angle, respectively; and are the pitch angular velocity and yaw rate, respectively; and , , , and are the disturbance and uncertain disturbance of the various links of the system. and are the rotational inertia; , , , , , and are the related aerodynamic force and torque coefficient, respectively; and are the pitch moment and yawing moment of the “leader,” respectively; and and are the longitudinal and lateral acceleration of the “leader.”
Assuming that the sight angle of the interceptor in the terminal guidance stage changes slightly and the included angle of the sight angle and velocity direction of the interceptor are relatively small, let and . By defining , , and , the nonlinear model of the “leader” IGC of the interceptor with state coupling can be obtained, according to (1), (2), and (3).
In the equation,
The unknown disturbances and in the “leader” IGC model are assumed to be continuously differentiable and the first-order derivative is bounded. is a positive constant.
3. Design of Interceptor “Leader” Controller
3.1. Design of Finite-Time Disturbance Observer
Aiming at the uncertainty , , and included in the system model (4), an FTDO (finite-time disturbance observer) is designed for estimating unknown disturbances, in order to eliminate the impact of unknown disturbances on the leader system of the interceptor. Define , , , and
Define , and according to (6) and (7),
In the equation,
The following FTDO is designed to estimate the acceleration of the target, and
In the equation , , , , , , , , is the estimated acceleration of the target, the estimated value and is and , , , and are the coefficients to be designed for the disturbance observer, and are the terminal coefficients, respectively, and .
It can be learnt according to [27] that appropriate parameters can guarantee that the FTDO error system is steady in finite time. The estimated error of the acceleration of the target is defined as .
Similarly, the disturbance and of the second subsystem and third subsystem is estimated, andIn the equation, the estimated value of disturbance and is and , respectively, and the estimated error is and , respectively.
3.2. Design of Adaptive Dynamic Sliding-Mode Controller
Because the interceptor IGC model is an unmatched and uncertain system, and aiming at the state coupling IGC model (4) and FTDO estimated value (10)–(12), the “leader” control algorithm is designed by taking advantage of the adaptive dynamic sliding-mode control law.
The command signal of the first subsystem of (4) is defined as . In order to realize the guidance goal, the sight angular velocity should be removed. According to the design method of dynamic surface sliding-mode control, the first dynamic error surface is defined as follows:
Taking the derivative of , the dynamic equation of error is given by
According to the dynamic surface design method and FTDO estimated value in (10), the virtual control amount of the first dynamic surface can be obtained as follows:
In the equation, is the positive definite matrix. In the design process, differential blast would occur, while the differential of the virtual control amount is taken. In order to avoid the complicated computation process owing to item inflation, must be obtained through the first-order low-pass filter, and the virtual control amount of the filter can be obtained as follows:
In the equation, is the time constant of the filter, and the differential of the virtual control after the error surface filter can be obtained.
The second dynamic error surface is defined as
Taking the derivative of , the dynamic equation of error can be obtained as follows:
Similar to the first dynamic surface design method, the estimated FTDO value is substituted in (11). Thus, the virtual control of the second dynamic surface can be obtained as follows:
In the equation, is the positive definite matrix. Similarly, by obtaining through the first-order low-pass filter, the virtual control amount of the filter can be obtained as follows:
In the equation, is the time constant of the filter. The differential of virtual control after the error surface filter can be obtained as follows:
The third dynamic error surface is defined as follows:
Taking the derivative of , the dynamic equation of error can be obtained as follows:
To guarantee the convergence velocity of the interceptor “leader,” an adaptive sliding-mode reaching law is designed:
In the equation, , , and denotes the change in relative distance between the “leader” and target.
According to (24) and (25) and estimated FTDO of (12), the adaptive dynamic surface sliding-mode control law of the interceptor “leader” is given by
In the equation, , , and are positive definite matrices, and .
3.3. Stability Analysis
Theorem 1. Consider the integrated guidance and control system for the “leader” (equation (4)). If the convergence rate is calculated using (25), the disturbance values of the system (see (4)) are estimated using (10)-(12), and filter equations (16) and (21) are implemented; then finally under the dynamic surface sliding-mode control law (see (26)), imposing the constraint for ensuring the system (see (4)) output error converging into the adjacent area of the origin, an arbitrary adjacent area of the origin can be obtained with the appropriate design parameter determined.
To prove:
Let us assume that the estimated error of FTDO system meets
In the equation, , , and are positive constants.
The filter error is defined as follows:
Taking the derivative of and , the dynamic error of the filter can be obtained as follows:
According to (13)–(23) and (28),
According to (4), (11)–(21), and (26)–(28),
In the equation, . Let us assume that , where is a positive constant.
According to Young’s equation and (30)–(33),
It can be learnt that variables and their differential in the system model are bounded, and there are continuous functions and , where and , enabling variables and to meet
According to Young’s equation and (28)–(29) and (37):
According to the state coupling IGC nonlinear system model (4), a Lyapunov function is selected:
Taking the derivative of (40),
The design parameters meet the following rules:
In the equation, is a constant, and . Therefore,
In the equation, .
According to (43),
, , , , and are consistent and eventually bounded. Thus, large parameters , , , and , as well as small parameters and are selected, to make the value of sufficiently large and sufficiently small to ensure control precision.
Remark 2. Theoretically, the final boundaries of error surfaces , , and and filter errors and will become smaller with the increasing design parameters , , , and and the decreasing and . This change leads to a higher controlling precision. However, in reality, using too large parameters (, , , and ) and too small parameters ( and ) will result in an input saturation for the interceptor control system. The nonlinear behavior of the saturated system results in a higher requirement of overload exceeding the available overload. Therefore, the angle of attack and the sideslip angle of the interceptor exceed the allowable range leading to a reduced controlling performance of the system. Furthermore, the physical constraints of the low-pass filter prohibit parameters and from being too small. Therefore, the parameters of the control algorithm should be properly selected by combining practical situations.
4. Distributed Network Synchronization Strategy
4.1. Design of Cooperative Control Strategy Based on the Distributed Network
Based on the principle of time consistency of a multiagent system, the multi-interceptor cooperative control strategy is designed to ensure that all interceptors hit the targets at the same time. In the cooperative system of the multi-interceptor, the state information of other interceptors can be obtained through information interaction, for realizing time consistency, and such information interaction can be described using the graph theory. Assuming that each interceptor is a communication node, the information exchange among interceptors can be indicated as , where denotes the set of interceptor nodes and denotes the lines between the interceptor nodes. The weighted coefficient matrix is indicated as ; implies that the interceptor node and node can exchange information. However, if , information cannot be exchanged. denotes the Laplace matrix of the undirected graph , among which the elements satisfy
denotes whether the interceptors can obtain the state information of the leader, indicates that the interceptors can obtain the state information of the leader, and indicates that the interceptors cannot obtain the state information of the leader.
Based on the “leader-follower” topology structure, the distributed cooperative control strategy of the multi-interceptor is designed as follows:
In the equation, denotes the position of the “leader,” denotes the position of the interceptors, denotes the velocity of the interceptors, is a constant, and .
Theorem 3. If the “follower” state of the interceptor can converge to the “leader” state following the cooperative control strategy (see (46)), such cooperative strategy is then considered to be successful.
To prove:
Lemma 4. Laplace matrix has the following properties:
(1) If is connected, the characteristic value of is , and it is called the algebraic connectivity of the network-connected graph. The larger the value of , the more connected the network.
(2) One of the characteristic values of is 0, and its corresponding characteristic vector is 1.
The error variable is defined as , and
The Lyapunov function is defined as
In the equation, .
, and taking the derivative of the above equation,
Let . Then, according to the above equation,
Therefore, is convergent in finite time, namely, the convergence state from the “follower” to the “leader”; it can realize cooperative guidance and control of the multi-interceptor.
4.2. Implementation of Distributed Network Cooperative Control Strategy
In order to implement the distributed network synchronization strategy, instructions provided to the synchronization strategy should be traced for each interceptor “follower.” The motion relation of the interceptors involved in cooperative interception is given as follows:
In the equation, , , and are the velocity components of the ith interceptor in the inertial frame, and and are the trajectory inclination angle and trajectory deflection angle of the ith interceptor.
According to the distributed network synchronization strategy (46), the velocity reference instruction of the interceptor is given by
According to (52), the total velocity, trajectory inclination angle, and trajectory deflection angle of the interceptor can be obtained as follows:
To obtain the differential of the total velocity and trajectory inclination angle of the tractor , the signal is obtained through the filter. Let and be the postfiltering instruction and prefiltering instruction, respectively. Then,
In the equation, and are the damping and bandwidth of the filter, respectively.
5. Design of Interceptor “Follower” Controller
The instructions provided by the cooperative control strategy can be transformed into velocity, trajectory inclination angle, and trajectory deflection angle instruction. In order to track the command signal of interceptor “follower” in the cooperative network, the “follower” controller adopts the dynamic surface sliding-mode control algorithm. Assuming that the velocity of the interceptor “follower” is controllable, the flight velocity can be indicated as follows:
In the equation, is the motor power, is the quality of “follower”, and are the attack angle and sideslip angle, respectively, is the trajectory inclination angle, and is the gravitational acceleration.
According to (56), the error surface is defined as follows:
In the equation, is the reference velocity command of the “follower” after filtering. Taking the derivative of ,
In the equation, is the differential of total velocity after filtering.
To ensure that the velocity of the “follower” can track the system command rapidly, the following sliding-mode reaching law is adopted:
According to (56)–(59), the thrust of the “follower” can be obtained as follows:
In the equation, , , .
By defining , , and , the state coupling kinetic equations of the ith “follower” can be indicated as follows:In the equation,
According to (10)–(12), the following form of FTDO is designed for evaluating the disturbance , , and to (61),
In the equation, the estimated value of is , and the estimated error is .
Similarly, the estimated values of and are and , respectively, and the estimated errors are and .
To ensure that the “follower” can track the command signal of the cooperative control strategy rapidly and guarantee steady flight attitude, the “follower” controller is designed with the dynamic surface sliding-mode control law, and according to the FTDO estimated value and state coupling kinetic equation (61).
The first dynamic error surface is defined as follows:
In the equation, is the instruction of the trajectory inclination angle and trajectory deflection angle after filtering. Taking the derivative of ,
According to the dynamic surface sliding-mode control method and FTDO estimated value , the virtual control of the first dynamic surface is selected as
In the equation, is the differential of the trajectory inclination angle and trajectory deflection angle after filtering, and is the positive definite matrix. The value of is obtained through the first-order low-pass filter, and the virtual control after filtering and its differential are given by
In the equation, is the time constant of the filter.
The second dynamic error surface is defined by
In the equation, is the command signal after filtering. Taking the derivative of ,
According to the dynamic surface sliding-mode control method and FTDO estimated value , the virtual control of the first dynamic surface is given by
In the equation, is the instruction differential after filtering, and is the positive definite matrix. The value of is obtained through the first-order low-pass filter, and the virtual control after filtering and its differential can be obtained as follows:
In the equation, is the time constant of the filter.
The third dynamic error surface is defined as follows:
In the equation, is the command signal after filtering. Taking the derivative of ,
According to the sliding-mode reaching law (see (59)), and FTDO estimated value , the “follower” dynamic surface sliding-mode control law is designed as follows:
In the equation is the instruction differential after filtering, and are the positive definite matrices, , and .
It can be learnt by referring to (27)–(44) that the stability of the control algorithm of the interceptor “follower” can be guaranteed by selecting appropriate parameters.
6. Simulation Verification
To verify the effectiveness of the distributed cooperative IGC algorithm of the multi-interceptor with state coupling designed in this study, it is assumed that the flight velocity of the interceptor “leader” remains the same. According to the global communication topology structure shown in Figure 1 and the local communication topology structure shown in Figure 2, Figure 1 assumes that the “leader” can communicate with the remaining three “followers,” while the “followers” can communicate with each other. Figure 2 assumes that the “leader” can only communicate with “follower 1,” while “followers” can communicate with each other. The initial conditions of interceptor “leader,” “follower,” and target are listed in Table 1.
Focusing on the two communication topology structures shown in Figures 1 and 2, a simulation study is conducted for the cooperative IGC algorithm of the multi-interceptor with state coupling designed in this study. It is assumed that the disturbance of the system is , and the intercepted target has a linear acceleration of . The comparison of the simulation results of the cooperative IGC algorithm of the multi-interceptor with state coupling in the global and local communication topology is given, as shown in Figures 3–11.
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Figures 3 and 4 show the motion trail of the interceptor “leader,” “follower,” and target in the local and global communication topologies. It can be seen that the motion trail of the interceptor “follower” in two different communication topologies is gradually consistent with that of the “leader.” Eventually, the “leader” and “follower” hit the target at the same time. The motion trail curve is smooth, showing short interception duration, fast convergence speed, and good stability.
Figures 5(a) and 5(b) show the velocity curve of the interceptor in the global and local topologies, from which it can be seen that the interceptor “follower” features a considerable overstriking property in the initial stage due to the lack of “leader.” The convergence rate is slower than that in the global topology, but it can eventually reach the steady state of the “leader.” The convergence process changes smoothly, and it shows good robustness to external disturbance. Similarly, it is clear from Figures 6–11 that the interceptor “follower” in the global and local communication topologies realizes the tracking of control instruction of the “leader.” The convergence process is relatively smooth, and it also shows good robustness to external disturbance.
It can be seen from the simulation results of the global and local communication topologies that the distributed cooperative IGC algorithm of the multi-interceptor with state coupling designed in this study completes the tracking the instructions of the cooperative control strategy in the two different topologies and eventually achieves cooperative target interception.
The cooperative IGC algorithm of the multi-interceptor with state coupling proposed in this study and the traditional method for multi-interceptor cooperation without state coupling are compared through numerical simulations. Figures 12 and 13 show the comparison results.
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Figures 12 and 13 present the comparison of the attack and sideslip angle curves associated with the cooperative IGC algorithm of the multi-interceptor with state coupling and the traditional method for the multi-interceptor cooperation without state coupling. The convergence process associated with the cooperative IGC algorithm of the multi-interceptor with state coupling appeared to be smoother compared to that associated with the traditional method for the multi-interceptor cooperation without state coupling. The section of the curve after t = 4s particularly revealed more stable angle and sideslip angle curves using the state-coupled design methods. In contrast, fluctuations in the attack and sideslip angle curves were observed for the traditional design methods without state coupling. In other words, the cooperative IGC algorithm of the multi-interceptor with state coupling exhibited better resistance to interference and allowed for a more stable control of the interceptor compared to the traditional method without state coupling.
7. Conclusions
This study focused on the cooperative target interception by multi-interceptor and designed cooperative IGC algorithm of the multi-interceptor with state coupling “leader-follower” structure. The algorithm is designed by considering the coupling relation between the pitch and yaw channels of the interceptor. Further, this study combines the IGC method and introduces the distributed cooperative control strategy. The interceptor “leader” and “follower” control algorithm is designed separately by employing the dynamic surface sliding-mode control law and FTDO. The distributed cooperative control strategy guarantees that the “leader” and “follower” can hit the targets at the same time. The algorithm displays ideal trajectory characteristics in the simulation verification, and it can realize the cooperative interception of targets in both the global and local communication topologies. Furthermore, the study provides a design method for the cooperative target interception of the multi-interceptor, with certain engineering values.
Notations
| : | Interceptor velocity |
| : | Target velocity |
| : | Attack angle |
| : | Sideslip angle |
| : | Yaw rate |
| : | Pitch angular velocity |
| : | Trajectory deflection angle of interceptor |
| : | Air density |
| : | Trajectory inclination angle of interceptor |
| : | Trajectory inclination angle of target |
| : | Contribution to lift due to angle of attack |
| : | Contribution to yaw force due to sideslip angle |
| : | Acceleration due to gravity |
| : | Moment of inertia around the yaw axis |
| : | Moment of inertia around the pitch axis |
| : | Reference length |
| : | Interceptor mass |
| : | Contribution to pitch moment due to angle of attack |
| : | Contribution to yaw moment due to sideslip angle |
| : | Contribution to yaw moment due to yaw rate |
| : | Contribution to pitch moment due to pitch rate |
| : | Yaw moment |
| : | Pitch moment |
| : | Dynamic pressure |
| : | Elevation angle |
| : | Horizontal sight angle |
| : | Relative distance |
| : | Reference area |
| : | Relative velocity |
| : | Tangential relative velocity normal to yaw line-of-sight(YLOS) |
| : | Tangential relative velocity normal to pitch line-of-sight(PLOS) |
| : | Motor power |
| : | Longitudinal and lateral motion acceleration |
| : | Longitudinal and lateral motion acceleration of the target |
| : | Longitudinal and lateral acceleration. |
Data Availability
The data used to support the findings of this study are available from the corresponding author upon request.
Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this paper.
Acknowledgments
The authors gratefully acknowledge the suggestions and help by the Professor Xiaogeng Liang and Northwestern Polytechnical University. This work was supported by the Aeronautical Science Foundation of China [Grant no. 2016ZC12005].