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Research on Integrated Guidance and Control of Distributed Cooperation of Multi-Interceptor with State Coupling.

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 strategywas 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:

[mathematical expression not reproducible] (1)

In the equation, [q.sub.[beta]] and [q.sub.[epsilon]] denote the elevation angle and horizontal sight angle of the "leader" and target, respectively; [a.sub.m4[epsilon]] and [a.sub.m4[beta]] denote the longitudinal and lateral motion acceleration of the "leader," respectively; [a.sub.t[epsilon]] and [a.sub.t[beta]] denote the longitudinal and lateral motion acceleration of the target, respectively; r represents the relative distance between the "leader" and target.

The kinetic model of the interceptor "leader" can be indicated as follows:

[mathematical expression not reproducible] (2)

[a.sub.m3[epsilon]] = qS[C.sup.[alpha].sub.y] / m

[a.sub.m3[beta]] = qS[C.sup.[beta].sub.z] / m (3)

In the equation, S is the reference area of the "leader;" [bar.L] is the reference length of the "leader;" m is the mass of the "leader;" [alpha] and [beta] are the attack angle and sideslip angle, respectively; [[omega].sub.z] and [[omega].sub.y] are the pitch angular velocity and yaw rate, respectively; and [mathematical expression not reproducible] are the disturbance and uncertain disturbance of the various links of the system. [J.sub.z] and [J.sub.y] are the rotational inertia; [mathematical expression not reproducible] and are the related aerodynamic force and torque coefficient, respectively; [M.sub.z] and [M.sub.y] are the pitch moment and yawing moment of the "leader," respectively; and [a.sub.m3[epsilon]] and [a.sub.m3[beta]] 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 [a.sub.m3[epsilon]] = [a.sub.m4[epsilon]] and [a.sub.m3[beta]] = [a.sub.m4[beta]]. By defining [mathematical expression not reproducible],the nonlinear model of the "leader" IGC of the interceptor with state coupling can be obtained, according to (1), (2), and (3).

[mathematical expression not reproducible] (4)

In the equation,

[mathematical expression not reproducible] (5)

The unknown disturbances [d.sub.2] and [d.sub.3] in the "leader" IGC model are assumed to be continuously differentiable and the first-order derivative is bounded. di < N, i = 2, 3, N is a positive constant.

3. Design of Interceptor "Leader" Controller

3.1. Design of Finite-Time Disturbance Observer. Aiming at the uncertainty [a.sub.t], [d.sub.2], and [d.sub.3] 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 [mathematical expression not reproducible] and

[mathematical expression not reproducible] (6)

[mathematical expression not reproducible] (7)

Define [w.sub.1] = [[[v.sub.[epsilon]], [V.sub.[beta]]].sup.T], and according to (6) and (7),

[[??].sub.1] = [h.sub.1] ([w.sub.1]) + [h.sub.2] ([w.sub.1]) [x.sub.2] + [a.sub.t] (8)

In the equation,

[mathematical expression not reproducible] (9)

The following FTDO is designed to estimate the acceleration [a.sub.t] of the target, and

[mathematical expression not reproducible] (10)

In the equation [w.sub.1] = [[[w.sub.11], [w.sub.12]].sup.T], [z.sub.10] = [[[z.sub.101], [z.sub.102]].sup.T], [Z.sub.11] = [[[Z.sub.111], [Z.sub.112]].sup.T], [Z.sub.12] = [[[Z.sub.121], [Z.sub.122]].sup.T], [V.sub.10] = [[[V.sub.101], [V.sub.102]].sup.T], [V.sub.11] = [[[v.sub.111], [V.sub.112]].sup.T], [V.sub.12] = [[[v.sub.121], [V.sub.122]].sup.T], [L.sub.1] = [[[L.sub.11], [L.sub.12]].sup.T], [[??].sub.t] is the estimated acceleration at of the target, the estimated value [w.sub.1] and [[??].sub.t] is [[??].sub.1] and [[??].sub.t], [[lambda].sub.10], [[lambda].sub.11], and [lambda].sub.12] are the coefficients to be designed for the disturbance observer, [q.sub.1] and [p.sub.1] are the terminal coefficients, respectively, and 0< [p.sub.1] < [q.sub.1].

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 at of the target is defined as [e.sub.11] = [z.sub.11] - [a.sub.t].

Similarly, the disturbance [d.sub.2] and [d.sub.3] of the second subsystem and third subsystem is estimated, and

[mathematical expression not reproducible] (11)

In the equation, the estimated value of disturbance [d.sub.2] and [d.sub.3] is [[??].sub.2] and [[??].sub.3], respectively, and the estimated error is [e.sub.21] = [z.sub.21] - [d.sub.2] and [e.sub.31] = [z.sub.31] - [d.sub.3], 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.

(1) The command signal of the first subsystem of (4) is defined as [x.sub.1d]. 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:

[s.sub.1] = [g.sup.-1.sub.11] ([x.sub.1])([x.sub.1] - [x.sub.1d]) (13)

Taking the derivative of s1, the dynamic equation of error is given by

[mathematical expression not reproducible] (14)

According to the dynamic surface design method and FTDO estimated value [[??].sub.t] in (10), the virtual control amount of the first dynamic surface can be obtained as follows:

[mathematical expression not reproducible] (15)

In the equation, [k.sub.1] = diag{[k.sub.11], [k.sub.12]} is the positive definite matrix. In the design process, differential blast would occur, while the differential of the virtual control amount [x*.sup.2] is taken. In order to avoid the complicated computation process owing to item inflation, [x*.sup.2] must be obtained through the first-order low-pass filter, and the virtual control amount of the filter can be obtained as follows:

[mathematical expression not reproducible] (16)

In the equation, [[tau].sub.2] = diag{[[tau].sub.21], [[tau].sub.22]} is the time constant of the filter, and the differential of the virtual control after the error surface filter can be obtained.

[[??].sub.2] = [[tau].sub.2-.sup.1] ([[bar.x*].sub.2] - [x*.sub.2]) (17)

(2) The second dynamic error surface is defined as

[s.sup.2] = [x.sup.2] - [[bar.x*].sup.2] (18)

Taking the derivative of [s.sub.2], the dynamic equation of error can be obtained as follows:

[mathematical expression not reproducible] (19)

Similar to the first dynamic surface design method, the estimated FTDO value [[??].sub.2] is substituted in (11). Thus, the virtual control of the second dynamic surface can be obtained as follows:

[mathematical expression not reproducible] (20)

In the equation, [k.sub.2] = diag{[k.sub.21], [k.sub.22]} is the positive definite matrix. Similarly, by obtaining [x*.sub.3] through the first-order low-pass filter, the virtual control amount of the filter can be obtained as follows:

[mathematical expression not reproducible] (21)

In the equation, [[tau].sub.3] = diag{[[tau].sub.31], [r.sub.32]} is the time constant of the filter. The differential of virtual control after the error surface filter can be obtained as follows:

[[??]*.sub.2] = [[tau].sub.3.sup.-1] ([[bar.x*].sub.3] - [x.sub.3]) (22)

(3) The third dynamic error surface is defined as follows:

[s.sub.3] = [x.sub.3] - [[bar.x*].sub.3] (23)

Taking the derivative of [s.sub.3], the dynamic equation of error can be obtained as follows:

[mathematical expression not reproducible] (24)

To guarantee the convergence velocity of the interceptor "leader," an adaptive sliding-mode reaching law is designed:

[mathematical expression not reproducible] (25)

In the equation, [k.sub.a] >0, [k.sub.b] > 0, and r denotes the change in relative distance between the "leader" and target.

According to (24) and (25) and estimated FTDO [[??].sub.3] of (12), the adaptive dynamic surface sliding-mode control law of the interceptor "leader" is given by

[mathematical expression not reproducible] (26)

In the equation, [k.sub.3] = diag{[k.sub.31], [k.sub.32]}, [k.sub.4] = diag{[k.sub.41], [k.sub.42]}, and [partial derivative] = diag{[[partial derivative].sub.11], [[partial derivative].sub.11]} are positive definite matrices, and 0 < [partial derivative]<1.

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

[mathematical expression not reproducible] (27)

In the equation, [N.sub.1], [N.sub.2], and [N.sub.3] are positive constants. The filter error is defined as follows:

[y.sub.2] = [[bar.x*].sub.2] - [x*.sub.2],

[y.sub.3] = [[bar.x*].sub.3] - [x*.sub.3] (28)

Taking the derivative of [y.sub.2] and [y.sub.3], the dynamic error of the filter can be obtained as follows:

[mathematical expression not reproducible] (29)

According to (13)--(23) and (28),

[mathematical expression not reproducible] (30)

According to (4), (11)-(21), and (26)-(28),

[mathematical expression not reproducible] (31)

In the equation, [[??].sub.11] = [g.sup.-1.sub.11] ([x.sub.1])[g.sub.12] ([x.sub.1])[e.sub.11]. Let us assume that [absolute value of [e.sub.11]] < [[??].sub.1], where [[??].sub.1] is a positive constant.

[mathematical expression not reproducible] (32)

[mathematical expression not reproducible] (33)

According to Young's equation and (30)-(33),

[mathematical expression not reproducible] (34)

[mathematical expression not reproducible] (35)

[mathematical expression not reproducible] (36)

It can be learnt that variables and their differential in

the system model are bounded, and there are continuous functions [[??].sub.2] and [[??].sub.3], where [[??].sub.2] >0 and [[??].sub.3] > 0, enabling variables [[??]*.sub.2] and [[??].sub.3] to meet

[mathematical expression not reproducible] (37)

According to Young's equation and (28)-(29) and (37):

[mathematical expression not reproducible] (38)

[mathematical expression not reproducible] (39)

According to the state coupling IGC nonlinear system model (4), a Lyapunov function is selected:

V= 1/2 ([S.sup.T.sub.1] [S.sub.1] + [S.sup.T.sub.2] [S.sub.2] + [S.sup.T.sub.3] [S.sub.3] + [S.sup.y.sub.2] [y.sub.2] + [S.sup.y.sub.3] [y.sub.3]) (40)

Taking the derivative of (40),

[mathematical expression not reproducible] (41)

The design parameters meet the following rules:

[mathematical expression not reproducible] (42)

In the equation, k is a constant, and k >0. Therefore,

[mathematical expression not reproducible] (43)

In the equation, [mathematical expression not reproducible].

According to (43),

[mathematical expression not reproducible] (44)

[s.sub.1], [s.sub.2], [s.sub.3], [y.sub.2], and [y.sub.3] are consistent and eventually bounded. Thus, large parameters [k.sub.1], [k.sub.2], [k.sub.3], and [k.sub.4], as well as small parameters [[tau].sub.2] and [[tau].sub.3] are selected, to make the value of [kappa] sufficiently large and A/k sufficiently small to ensure control precision.

Remark 2. Theoretically, the final boundaries of error surfaces [s.sub.1], [s.sub.2], and [s.sub.3] and filter errors [y.sub.2] and [y.sub.3] will become smaller with the increasing design parameters [k.sub.1], [k.sub.2], [k.sub.3], and [k.sub.4] and the decreasing [[tau].sub.2] and [[tau].sub.3]. This change leads to a higher controlling precision. However, in reality, using too large parameters ([k.sub.1], [k.sub.2], [k.sub.3], and [k.sub.4]) and too small parameters ([[tau].sub.2] and [[tau].sub.3]) 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 [[tau].sub.2] and [[tau].sub.3] 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 [mathematical expression not reproducible], where [bar.V] = {[[bar.v].sub.i], i = 1, 2, ..., n} denotes the set of interceptor nodes and [bar.E] denotes the lines between the interceptor nodes. The weighted coefficient matrix is indicated as [mathematical expression not reproducible]; [[bar.a].sub.ij] > 0 implies that the interceptor node i and node j can exchange information. However, if [[bar.a].sub.ij] = 0, information cannot be exchanged. [bar.L] denotes the Laplace matrix of the undirected graph [bar.G], among which the elements satisfy

[mathematical expression not reproducible] (45)

[mathematical expression not reproducible] denotes whether the interceptors can obtain the state information of the leader, [[bar.b].sub.i] >0, i [member of] {1, 2, 3, ..., j} indicates that the interceptors can obtain the state information of the leader, and [[bar.b].sub.i] = 0, i [member of] {1, 2, 3, ..., j} 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:

[mathematical expression not reproducible] (46)

In the equation, [[zeta].sub.0] = [[[x.sub.0], [y.sub.0], [z.sub.0]].sup.T] denotes the position of the "leader," [[zeta].sub.i] = [[[x.sub.i], [y.sub.i], [z.sub.i]].sup.T], i [member of] {1, 2, 3, ..., j} denotes the position of the interceptors, [v.sub.i] = [[??].sub.i] i [member of] {1, 2, 3, ..., j} denotes the velocity of the interceptors, [mathematical expression not reproducible] is a constant, and [[bar.k].sub.i1] > 0.

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 [bar.M] has the following properties:

(1) If [bar.G] is connected, the characteristic value of [bar.M] is [[lambda].sub.min] ([bar.M]) > 0, and it is called the algebraic connectivity of the network-connected graph. The larger the value of [[lambda].sub.min] ([bar.M]), the more connected the network.

(2) One of the characteristic values of [bar.M] is 0, and its corresponding characteristic vector is 1.

The error variable is defined as [e.sub.i] = [[zeta].sub.i] - [[zeta].sub.0], and

[mathematical expression not reproducible] (47)

The Lyapunov function is defined as

V = 1/2 [e.sup.T] ([bar.M] + [bar.B])e (48)

In the equation, e = [[[e.sub.1], [e.sub.2], ..., [e.sub.n]].sup.T].

[[bar.k].sub.i] = min{[[bar.k].sub.i1]}, and taking the derivative of the above equation,

[mathematical expression not reproducible] (49)

Let V(e) [not equal to] 0. Then, according to the above equation,

[mathematical expression not reproducible] (50)

According to (49)-(50),

[??](t) [less than or equal to] -[[bar.k].sub.i], [2[[lambda].sub.min] ([bar.M] + [bar.B])] (51)

Therefore, V(t) is convergent in finite time, namely, the convergence state from the "follower" to the "leader"; it can realize cooperative guidance and control of the multiinterceptor.

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:

[mathematical expression not reproducible] (52)

In the equation, [mathematical expression not reproducible] are the velocity components of the ith interceptor in the inertial frame, and [[theta].sub.mi] and [[[phi].sub.mvi] 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

[mathematical expression not reproducible] (53)

According to (52), the total velocity, trajectory inclination angle, and trajectory deflection angle of the interceptor can be obtained as follows:

[mathematical expression not reproducible] (54)

To obtain the differential of the total velocity and trajectory inclination angle of the tractor [bar.[??]], the signal is obtained through the filter. Let [bar.x] and [[bar.x*].sub.1] be the postfiltering instruction and prefiltering instruction, respectively. Then,

[mathematical expression not reproducible] (55)

In the equation, [[zeta].sub.n] and [[omega].sub.n] 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:

[[??].sub.m] = cos [[alpha].sub.i] - cos [[beta].sub.i] / m Pi - g sin [[theta].sub.mi] (56)

In the equation, [P.sub.i] is the motor power, m is the quality of "follower", [[alpha].sub.i] and [[beta].sub.i] are the attack angle and sideslip angle, respectively, [[theta].sub.mi] is the trajectory inclination angle, and g is the gravitational acceleration.

According to (56), the error surface is defined as follows:

[s.sub.v] = [V.sub.m] - [[bar.V].sub.mi] (57)

In the equation, [[bar.V].sub.mi] is the reference velocity command of the "follower" after filtering. Taking the derivative of [s.sub.V],

cos [[alpha].sub.i] - cos [[beta].sub.i] / m (58)

In the equation, [[bar.[??]].sub.mi] 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:

[??] = -[k.sub.a] s -[k.sub.b] [[absolute value of s].sup.[partial derivative]] sgn (s) (59)

According to (56)-(59), the thrust of the "follower" can be obtained as follows:

[mathematical expression not reproducible] (60)

In the equation, [k.sub.v1] >0, [k.sub.v2] >0, 0< [[partial derivative].sub.V] < 1.

By defining [mathematical expression not reproducible], the state coupling kinetic equations of the ith "follower" can be indicated as follows:

[mathematical expression not reproducible] (61)

In the equation,

[mathematical expression not reproducible] (62)

According to (10)-(12), the following form of FTDO is designed for evaluating the disturbance [d.sub.i1], [d.sub.i2], and [d.sub.i3] to (61),

[mathematical expression not reproducible] (63)

In the equation, the estimated value of [d.sub.i1] is [[??].sub.i1], and the estimated error is [e.sub.i11] = [z.sub.i11] - [d.sub.i1].

Similarly, the estimated values of [d.sub.i2] and [d.sub.i3] are [[??].sub.i2] and [[??].sub.i3], respectively, and the estimated errors are [e.sub.i12] = [z.sub.i12] - [d.sub.i2] and [e.sub.i13] = [z.sub.i13] - [d.sub.i3].

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).

(1) The first dynamic error surface is defined as follows:

[s.sub.i1] = [g.sup.-1.sub.i1] ([x.sub.i1] - [x.sub.i1d]) (64)

In the equation, [x.sub.i1d] = [[[[bar.[theta]].sub.mi], [[bar.[phi]].sub.vmi]].sup.T] is the instruction of the trajectory inclination angle and trajectory deflection angle after filtering. Taking the derivative of [s.sub.i1],

[mathematical expression not reproducible] (65)

According to the dynamic surface sliding-mode control method and FTDO estimated value [[??].sub.i1], the virtual control of the first dynamic surface is selected as

[mathematical expression not reproducible] (66)

In the equation, [[??].sub.i1d] is the differential of the trajectory inclination angle and trajectory deflection angle after filtering, and [k.sub.i1] = diag{[k.sub.i11], [k.sub.i12]} is the positive definite matrix. The value of [x*.sub.i2] is obtained through the first-order low-pass filter, and the virtual control after filtering and its differential are given by

[mathematical expression not reproducible] (67)

In the equation, [[tau].sub.i2] = diag{[[tau].sub.i21], [[tau].sub.i22]} is the time constant of the filter.

(2) The second dynamic error surface is defined by

[s.sub.i2] = [x.sub.i2] - [[bar.x*].sub.i2] (68)

In the equation, [[bar.x*].sub.i2] is the command signal after filtering. Taking the derivative of [s.sub.i2],

[mathematical expression not reproducible] (69)

According to the dynamic surface sliding-mode control method and FTDO estimated value [[??].sub.i2], the virtual control of the first dynamic surface is given by

[mathematical expression not reproducible] (70)

In the equation, [[bar.[??]*].sub.i2] is the instruction differential after filtering, and [k.sub.i2] = diag{[k.sub.i21], [k.sub.i22]} is the positive definite matrix. The value of [x*.sub.i3] is obtained through the first-order low-pass filter, and the virtual control after filtering and its differential can be obtained as follows:

[mathematical expression not reproducible] (71)

In the equation, [[tau].sub.i3] = diag{[[tau].sub.i31], [[tau].sub.i32]} is the time constant of the filter.

(3) The third dynamic error surface is defined as follows:

[s.sub.i3] = [x.sub.i3] - [[bar.x*].sub.i3] (72)

In the equation, [[bar.x*].sub.i3] is the command signal after filtering. Taking the derivative of [s.sub.i3],

[mathematical expression not reproducible] (73)

According to the sliding-mode reaching law (see (59)), and FTDO estimated value [[??].sub.i3], the "follower" dynamic surface sliding-mode control law is designed as follows:

[mathematical expression not reproducible] (74)

In the equation [[bar.[??]*].sub.i3] is the instruction differential after filtering, [k.sub.i3] = diag{[k.sub.i31], [k.sub.i32]} and [k.sub.i4] = diag{[k.sub.i41], [k.sub.i42]]} are the positive definite matrices, [[partial derivative].sub.i3] = diag{[[partial derivative].sub.i31], [[partial derivative].sub.i32]}, and 0 < [[partial derivative].sub.i3] < L

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 [d.sub.2] = [d.sub.3] = [d.sub.i1] = [d.sub.i2] = [d.sub.i3] = 0.02 sin(f), and the intercepted target has a linear acceleration of [a.sub.t[epsilon]] = [a.sub.t[beta]] = 5m/[s.sup.2]. 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.

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.

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 multiinterceptor, with certain engineering values.

Notations

[V.sub.m], [V.sub.mi]: Interceptor velo city

[V.sub.t]: Target velocity

[alpha], [[alpha].sub.i]: Attack angle

[beta], [[beta].sub.i]: Sideslip angle

[[omega].sub.y], [[omega].sub.yi]: Yaw rate

[[omega].sub.z], [[omega].sub.zp]: Pitch angular velocity

[[phi].sub.vm], [[phi].sub.vmi]: Trajectory deflection angle of interceptor

[rho]: Air density

[[theta].sub.m], [[theta].sub.mi]: Trajectory inclination angle of interceptor

[[theta].sub.t]: Trajectory inclination angle of target

[mathematical expression not reproducible]: Contribution to lift due to angle of attack [alpha]

[mathematical expression not reproducible]:: Contribution to yaw force due to sideslip angle [beta]

g: Acceleration due to gravity

[J.sub.y]: Moment of inertia around the yaw axis

[J.sub.z]: Moment of inertia around the pitch axis

[bar.L]: Reference length

m: Interceptor mass

[mathematical expression not reproducible]: Contribution to pitch moment due to angle of attack [alpha]

[mathematical expression not reproducible]: Contribution to yaw moment due to sideslip angle [beta]

[mathematical expression not reproducible]: Contribution to yaw moment due to yaw rate [[omega].sub.y]

[mathematical expression not reproducible]: Contribution to pitch moment due to pitch rate [[omega].sub.y]

[M.sub.y], [M.sub.yi]: Yaw moment

[M.sub.z], [M.sub.zi]: Pitch moment

q: Dynamic pressure

q[beta]: Elevation angle

q[epsilon]: Horizontal sight angle

r: Relative distance S: Reference area

[v.sub.r]: Relative velocity

[v.sub.[beta]]: Tangential relative velocity normal to yaw line-of-sight(YLOS)

[v.sub.[epsilon]]: Tangential relative velocity normal to pitch line-of-sight(PLOS)

[P.sub.i]: Motor power

[a.sub.m4[epsilon]], [a.sub.m4[beta]]: Longitudinal and lateral motion acceleration

[a.sub.t[epsilon], [a.sub.t[beta]]: Longitudinal and lateral motion acceleration of the target

[a.sub.m3[epsilon]], [a.sup.m3[beta]]: Longitudinal and lateral acceleration.

Data Availability

The data used to support the findings of this study are available from the corresponding author upon request.

https://doi.org/10.1155/2018/3193405

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].

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Xiang Liu and Xiaogeng Liang

School of Automation, Northwestern Polytechnical University, Xi'an 710072, China

Correspondence should be addressed to Xiang Liu; 1x861002@163.com

Received 17 March 2018; Revised 8 June 2018; Accepted 3 July 2018; Published 1 August 2018

Academic Editor: Ai-Guo Wu

Caption: FIGURE 1: Global communication topology structure of interceptor "leader" and "follower".

Caption: FIGURE 2: Local communication topology structure of the interceptor "leader" and "follower".

Caption: FIGURE 3: Motion trails of the interceptor "leader," "follower," and target in the global communication topology: X denotes the horizontal motion trail of the interceptor and target, Y denotes the longitudinal motion distance of the interceptor and target, and Z denotes the lateral motion trail of the interceptor and target.

Caption: FIGURE 4: Motion trails of the interceptor "leader," "follower," and target in the local communication topology. X denotes the horizontal motion trail of the interceptor and target, Y denotes the longitudinal motion distance of the interceptor and target, and Z denotes the lateral motion trail of the interceptor and target.

Caption: FIGURE 5: Interceptor velocity curve in the global and local communication topologies: (a) velocity curve in the global communication topology; (b) velocity curve in the local communication topology.

Caption: FIGURE 6: Trajectory inclination angle curve in the global and local communication topologies: (a) trajectory inclination angle curve in the global communication topology; (b) trajectory inclination angle curve in the local communication topology.

Caption: FIGURE 7: Trajectory deflection angle curve in the global and local communication topology: (a) trajectory deflection angle curve in global communication topology; (b) trajectory deflection angle curve in local communication topology.

Caption: FIGURE 8: Attack angle curve in the global and local communication topologies: (a) attack angle curve in the global communication topology; (b) attack angle curve in the local communication topology.

Caption: FIGURE 9: Sideslip angle curve in the global and local communication topologies: (a) sideslip angle curve in the global communication topology; (b) sideslip angle curve in the local communication topology.

Caption: FIGURE 10: Pitch angular velocities curve in the global and local communication topologies: (a) pitch angular velocities curve in the global communication topology; (b) pitch angular velocities curve in the local communication topology.

Caption: FIGURE 11: Yaw rate curve in the global and local communication topologies: (a) yaw rate curve in the global communication topology; (b) yaw rate curve in the local communication topology.

Caption: FIGURE 12: Comparison of the interceptor's attack angle curves in the design methods with and without state coupling: (a) attack angle curve in the design method with state coupling; (b) attack angle curve in the design method without state coupling.

Caption: FIGURE 13: Comparison of the interceptor's sideslip angle curves in the design methods with and without state coupling: (a) sideslip angle curve in the design method with state coupling; (b) sideslip angle curve in the design method without state coupling.
TABLE 1: Initial conditions of the leader, follower, and target.

No.                Parameter    Value    Parameter    Value

 1     Leader     [x.sub.m0]     0m     [y.sub.m0]     0m
 2    Follower1   [x.sub.m10]   600m    [y.sub.m10]    0m
 3    Follower2   [x.sub.m20]   800m    [y.sub.m20]    0m
 4    Follower3   [x.sub.m30]   400m    [y.sub.m30]    0m
 5     Target     [x.sub.t0]    4000m   [y.sub.t0]    5000m

No.                Parameter    Value    Parameter     Value

 1     Leader     [z.sub.m0]     0m     [V.sub.m0]    800 m/s
 2    Follower1   [z.sub.m10]   500m    [V.sub.m10]   800 m/s
 3    Follower2   [z.sub.m20]   1000m   [V.sub.m20]   800 m/s
 4    Follower3   [z.sub.m30]   -500m   [V.sub.m30]   800 m/s
 5     Target     [z.sub.t0]    3000m   [V.sub.t0]    400 m/s
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Title Annotation:Research Article
Author:Liu, Xiang; Liang, Xiaogeng
Publication:Journal of Control Science and Engineering
Article Type:Report
Date:Jan 1, 2018
Words:7257
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