Missile Cooperative Engagement Formation Configuration Control Method

In traditional engagement model, there is only one single weapon or platform which resists to another single one, and the connections between members are few, so the complete and global information in engagement space can’t be utilized sufficiently, which directly results in traditional model is confronting with more and more disadvantages especially as the high-tech and information-tech are developing faster and faster. In view of those issues, the concepts and technology of missile formation cooperative are presented, developed and expanded recently(Cui et al., 2009). Compared to traditional model, weapon or platform with cooperative manner manifests great advantages in aspects of ability of penetration, electronic countermeasures and ability of searching moving targets etc, furthermore the synthetical engagement efficacy is developed greatly. Many new cooperative weapon systems are established and developed fast recently, such as Cooperative Engagement Capability (CEC) system, Net Fire System and LOw Cost Autonomous Attack System (LOCAAS) etc, wherein the LOCAAS is most relevant to our topic, so we will introduce it more detailed. In order to meet the requirements of future aerial warfare, United States Force has developed a series of high technical and high accurate airborne guided weapon systems, such as Joint Common Missile (JCM), Joint Direct Attack Missile (JDAM), Wind Corrected Munitions Dispenser (WCMD) and LOw Cost Autonomous Attack System (LOCAAS). These weapons are paid much attention because their great capabilities of high-precision, all-weather engagement and attacking beyond defence area etc. Especially, LOCAAS also has other great advantages besides those aspects mentioned above, such as low cost, general utilization and attacking multi-targets simultaneously, so it is paid more attention than other weapons, and it becomes an outstanding representative of high-tech weapon and new engagement model. LOCAAS is developed from an previous weapon named as Low Cost Anti-Armor Submunitions (also called LOCAAS for short) which was a kind of short-range unpowered airborne and air-to-ground guided weapon developed in 1998. At a later time, researchers added thrust system to this old LOCAAS, so it had capability of launching beyond defence


Introduction
In traditional engagement model, there is only one single weapon or platform which resists to another single one, and the connections between members are few, so the complete and global information in engagement space can't be utilized sufficiently, which directly results in traditional model is confronting with more and more disadvantages especially as the high-tech and information-tech are developing faster and faster. In view of those issues, the concepts and technology of missile formation cooperative are presented, developed and expanded recently (Cui et al., 2009). Compared to traditional model, weapon or platform with cooperative manner manifests great advantages in aspects of ability of penetration, electronic countermeasures and ability of searching moving targets etc, furthermore the synthetical engagement efficacy is developed greatly. Many new cooperative weapon systems are established and developed fast recently, such as Cooperative Engagement Capability (CEC) system, Net Fire System and LOw Cost Autonomous Attack System (LOCAAS) etc, wherein the LOCAAS is most relevant to our topic, so we will introduce it more detailed. In order to meet the requirements of future aerial warfare, United States Force has developed a series of high technical and high accurate airborne guided weapon systems, such as Joint Common Missile (JCM), Joint Direct Attack Missile (JDAM), Wind Corrected Munitions Dispenser (WCMD) and LOw Cost Autonomous Attack System (LOCAAS). These weapons are paid much attention because their great capabilities of high-precision, all-weather engagement and attacking beyond defence area etc. Especially, LOCAAS also has other great advantages besides those aspects mentioned above, such as low cost, general utilization and attacking multi-targets simultaneously, so it is paid more attention than other weapons, and it becomes an outstanding representative of high-tech weapon and new engagement model. LOCAAS is developed from an previous weapon named as Low Cost Anti-Armor Submunitions (also called LOCAAS for short) which was a kind of short-range unpowered airborne and air-to-ground guided weapon developed in 1998. At a later time, researchers added thrust system to this old LOCAAS, so it had capability of launching beyond defence The background and significance were presented first, and then the technique frame of missile formation cooperative control system is showed, which can clearly elaborate how missiles in a formation work together in a cooperative manner. The specific relationships between each loop are analyzed, which are very necessary to clearly explain how to design missile formation control system. Followed are the main contents of this chapter, which are the detailed processes of establishing and designing missile formation control system. In this part, we will only consider the external loop of missile formation control system because the design method on individual missile inner controller is easy to be found in many literatures, and then we just assumed it is closed-loop and stable. The detailed process is divided into two steps, first is using proportion-differential control method to design missile formation keeping controller, the other explains how to design optimal keeping controller of missile formation. Some simulations are made to compare these two formation control systems proposed in this chapter at last.

Frame of missile formation control system
The missile formation control system mentioned in this chapter only has one leader missile (showed as Fig.2), and it mainly consists of the following subsystems: cooperative engagement mission planning subsystem, formation configuration describing subsystem, leader inner-loop control subsystem, follower inner-loop control subsystem and formation configuration control subsystem. The sketch map illustrates relationships between each subsystem in one missile formation system is showed in Fig.3. ,, . Relationships between subsystems of missile formation control system.
The cooperative mission planning subsystem supplies real-time status of mission space and distribution situation of targets, and restricts the flight status of leader at the same time; the formation configuration describing system receives the information supplied by cooperative engagement subsystem and establishes associated function of engagement efficacy according to different missile formation configurations, and then makes the decision set about formation configurations based on the returned value of this efficacy function. After optimizing the values of efficacy function according to the specific mission space, the optimal formation configuration can be calculated within this decision set. The optimal formation configuration will be different when the missile formation is flying in different segments of mission space, that is, the optimal formation configuration will be decided by specific engagement requirement. The formation configuration describing subsystem will also restrict flight status of leader together with cooperative mission planning subsystem at the same time; The leader inner-loop control subsystem receives the information sent from the cooperative engagement planning subsystem and controls the leader to fly stably, and then the real flight states of leader can be obtained; The required states of follower missile can be calculated through the states in relative spatial dimension i.e. the distance between missiles which is obtained from formation configuration describing subsystem; The follower inner-loop control subsystem receives the command states, which is similar to that of leader mentioned above, and then control follower to fly stably. The actual flight states of follower missiles are fed back to the formation configuration control subsystem and used to achieve the expected formation configuration which is decided by the specific cooperative engagement mission. This chapter focus on missile formation keeping control problems. First, we consider the actual flight states of leader as perturbation variables acting on the controller, and assume the follower inner-controller is closed and stable loop, that is, followers can track the required commands of velocity, flight path angle and flight deflection angle rapidly and stably. We will make further assumption that those three channels referred above are oneorder systems expressed as : These inertial time constants will be used in the subsequent sections for designing and simulating missile formation keeping controller. After missile formation achieving four-dimensionally rendezvous, this formation should carry out some subsequent missions, such as relative navigation and location, cooperative searching targets, locating and recognizing targets and cooperative penetration etc., however, these missions require missile formation keeping its configuration or relative www.intechopen.com

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dimensional states for some periods of time. Next sections will present some further and detailed researches on how to keep missile formation optimally and robustly. The main thought of the following parts is showed as below: basing on the kinematics relationships between missiles in inertial coordinate frame, the missile formation proportional-derivative (PD) controller via feeding back full states will be present firstly; Second, the relative motion will be established in relative coordinates frame, which can indicate the characteristics of relative motion directly. Based on this direct relative motion, the optimal controller that has non-zero given point and restrains slowly variant perturbations is designed in the third part.

PD controller of missile formation keeping
3.1 Definition of coordinate frame system 1. Relative coordinate frame rr r r ox y z  The origin r o of this coordinate coincides with the mass centre of leader, and axis rr ox points to the velocity direction of leader, rr oy is perpendicular to rr ox and points to up direction, rr oz composes right-hand coordinate frame together with other two axes mentioned above. 2. Inertial coordinate frame system II I I OX Y Z  The origin I O of this coordinate frame is fixed to an arbitrary point on ground, axis II OX lies in horizontal plane and points to target, axis II OY is perpendicular to II OX and points to up direction, the last axis II OZ also composes right-hand coordinate frame system together with those two axes mentioned above. The relationship between these two coordinate frames is showed in Fig.4. 3. Trajectory coordinate frame system 12 2 2 ox y z  The origin of this coordinate 1 o coincides with the mass centre of missile, and axis 12 ox points to the velocity direction of missile, 12 oy is perpendicular to 12 ox and points to up direction, 12 oz composes right-hand coordinate frame together with other two axes mentioned above. The spatial directions are parallel to those in relative coordinate frame system.

Model of relative motion between missiles 3.2.1 Basic assumptions
It is assumed that the control system of missile self is closed-loop and stable, that is, missile can track the required velocity, flight path angle and flight deflection angle rapidly and stably. Further, we consider these tracking channels are one-order inertial loops described as section 2. In order to express the relationships conveniently, we will change the one-order inertial loops showed in Equ.(1) to the following forms:

Establishing the control model
The kinematics equations of missile in inertial coordinate frame can be expressed as: cos cos sin cos sin The kinematics relationship of two missiles in inertial and relative coordinate frame is showed in Fig.5, so the relative positions of two missiles can be obtained from these sketch maps: where ** , x y and * z are the distances relative to leader in relative coordinate frame, and the transformation matrices in Equ.(5) are: and making derivatives to last equation yields: ** 21 ** 21 11 21 1 1 1 21 1 11 ** 21 making further derivatives gets: www.intechopen.com

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so we can get the following vector equation: where ** 1 2 2 2* * 21 21 11 2 1 1 1 1 1 1 2 1 1 ** 12 The control variables of follower are: As for the expression of 1 f , we also need to know: In order to eliminate the deviations of relative positions, that is, make the deviation e be asymptotically equivalent to zero, we will choose the following PD control laws: so there will be: 12 1r kk     ee f Gu further, the required control quantity can be expressed as below: What should be done next is to analyze the existence conditions of expression (14). The condition is: iff matrix r G is non-singular, the control quantity exists. The determinant of r G can be calculated by Equ.(11), it is: Through checking this expression, it is obvious that r G is non-singular when the missile formation is flying, that is, 2 0 V  , so we can choose proper coefficients 12 , kkto guarantee expression (13) to be converged, which is lim

Establishing model of relative motion
Because relative coordinate frame is rotating, the relationship between relative derivative and absolute derivative should be considered during the process of establishing relative motion equations in relative coordinate frame. The relationship between these two derivatives is: where fr V and lr V are the absolute velocities of follower and leader in relative coordinate frame respectively; r V is the relative velocity from follower to leader in relative coordinate frame;  represents the rotating angular velocity of relative coordinate frame relative with respect to inertial space, and it is described in relative coordinate frame; r is the position vector of follower relative to leader in relative coordinate frame. We can get the absolute velocities of follower and leader in relative coordinate frame by the transformations as below: at the same time, the relative velocity between two missiles in relative coordinate frame: www.intechopen.com

Missile Cooperative Engagement Formation Configuration Control
where , xyand z are components of relative position vector r in relative coordinate frame, and the rotating angular velocity of relative coordinate frame with respect to inertial space can be expressed as: further, we can get the following expression directly from equation (16): After expanding all the terms from Equ. (17)

Establishing the optimal control model of missile formation keeping 4.2.1 Linearized method
The formation motion equations (22) are nonlinear, we can treat them by linearized method to get linear forms that can be utilized and analyzed more conveniently. During the process of missile formation flight, some variables can be considered as small quantities, such as f  , l  and elf    , and the states of leader can be considered as inputs, then Equ.(22) can be transformed to: Dealing with this expression by small perturbation linearized method yields: (1 ) , which can be considered as slowly variant variables; the system matrix A is described as: and output matrix C is:

Method of substituting variables
As analyzed in last section, we can get linear control model showed as Equ.(24) via small perturbation linearized method, however, this linear model requires the actual flight path is closely around feature points. Although we can design the missile formation keeping controller at those points within the whole flight scope by gain-scheduled method, it will increase amount of work greatly. Next we will re-deal with the nonlinear equations of relative kinematics showed in Equ.
the meanings of other variables are same as those in Equ.(24). It can be stated through analyzing this section that the indirect control model of relative motion based on the relationships of relative motion between missiles has a direct mapping relationship between inputs and outputs. However, this mapping just can be indicated by a mathematical form rather than some intuitive physical meanings, so we can't give the linear expressions just through modelling directly. As mentioned above, we should transform the coordinate space of input variables and then establish the indirect linear model (27). After the missile formation controller system is described by the Equ. (27)

Establishing the optimal control model of missile formation keeping
As for the missile formation keeping control problem, the aim is to keep the distances between members within one formation on a non-zero states. After considering equation (27), we can further describe this formation keeping problem to another problem that how to regulate non-zero given value of output affected by slowly variant perturbations. So we can design this optimal controller by two steps: first, we need to design an optimal output regulator that can overcome slowly variant perturbations; second, we should further design optimal controller based on the first step which can maintain the missile formation on a nonzero desired relative states.

Optimal proportional-integral (PI) controller of missile formation
As for the system with invariant or slowly variant perturbations showed in expression (27), we can choose PI control law to overcome these perturbations, which is similar to classical control method. In order to clearly explain the principle of how this integral feed-back controller eliminates stable errors, we will make a hypothesis that the stable outputs of system are zeros. Because there has an integrator in the controller, outputs can be constants although inputs are zeros, if these values are just rightly equivalent to the perturbations that are effecting on the input points but the signs are inverse, then the inputs of this control system will be eliminated to zeros, so the final outputs of system can be kept to zeros (Xie, 1986). First, we transform the perturbations of system to control inputting ports, and then system (27) can be changed to: where  W is transformation of initial perturbations, it has following form after comparing to the Equ. (27) Q is state regulating weight matrix of augmented system; 1 R is control energy weight matrix of augmented system. When augmented system (30) is controllable, the optimal control quantities for minimizing the performance index (31) should be: where P is the solution of Riccati Equation: where Q is the states regulating weight matrix of original system (27), R is the control energy weight matrix of original system. From original system (27), there will be expression showed as below:

 X QX X C Q CX Y Q Y
where Y Q is the outputs regulating weight matrix, then expression(31) that describes the problem of quadratic optimal states regulating can be transformed to the problem of quadratic optimal output regulating, which is: besides, because perturbations  W is slowly variant variables, following expression can be noted: www.intechopen.com

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At last the quadratic optimal performance index of outputs regulating can be expressed as: As for the optimal control variables(32), it can be transformed from Equ.(35) to: Expanding equation (37) which is the expression of optimal control quantities that can minimize the performance index (34) for zero-given points.

Optimal controller for non-zero given points
In order to keep the output variables [ , , ] T xyz  Y on non-zero points, the final system states and control inputs should also be non-zero, and then the optimal control quantities will be transformed from Equ.(37) to the following form: where 0  U is additional control quantities for non-zero states. Considering the output equations of augmented system: 11  YC X (40) and the expansions of augmented states 1 X : we should choose the following output matrix of augmented system to make sure the outputs of augmented system accord with those of original system: 1 100000 010000 001000 Substituting the control values(39) into the state equation(30) of augmented system yields: and then the asymptotically stable system can be expressed as: where 10 X is the stable value of state 1 X . If all the eigenvalues of matrix 11  AB K lie in the left complex plane, then matrix is 11  AB K non-singular, and further we can get the expression showed as below from Equ.(42): and the relationship between non-zero points and stable value of states will meet: * 10 1 10  YC X further the relationship required between non-zero states * 10 Y and the stable value 10 X of state 1 X should be expressed as: At last we can implement optimal control to system(27) that describes the relative kinematics of missiles by the following optimal control quantity: and this control quantity can keep missile formation on the desired relative dimensional states.

Stability analysis
In order to analyze the stability of optimal controller of missile formation, we need to transform the model of missile formation relative motion to tracking error model. The tracking error of state is chosen by state equation (27) as: ˆd  X XX (46) where is X the actual flight states of missile formation; d X is the expected states. Substituting last equation into Equ.(27) obtains the error state equation: whereŴ is the invariant perturbations of tracking error state equation, the value of this term is: ˆd  WW A X Because invariant perturbations do not affect the dynamic performance of control system no more than affect the stable tracking performance of this system, we can ignore this term when just analyzing the stability of control system, that is, we can just analyze the following control system: Taking Lyapunov function to this system gets: The derivative of this Lyapunov function is: Substituting Equ. (26) to expression(50) yields:

BR B PX = PX
and then the derivative of Lyapunov function can be changed to: where P is the solution of Riccati Equation, which has the following characters (Xie, 1986) , P is non-negative matrix.
So Equ.(52) is non-negative, and it also states that we can't get the conclusion that this system is asymptotical until now, and need some more stronger conditions. Original system is time-variant system, so we can complement the stability conditions of system (48) (Slotine et al, 1991).
After analyzing that Lyapunov function appointed in the previous section, we can find that this function satisfies the former two aspects of Barbalat Lemma. If we want to prove expression(52) also satisfies the third point, we just need to prove   exists and is bounded.

Simulation results and analysis
After simulating, we can get the result curves Fig.6-Fig.12, where Fig.6 is the 3-dimensional motions of missile formation; Fig.7 and Fig.8 are the position components of follower 1 and follower 2 in relative coordinate frame respectively; Fig.9 is the distances between members in missile formation; Fig.10-Fig.11 are the control quantities in velocity, flight path angle and flight deflection angle channels of followers respectively.   www.intechopen.com

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It can be found from those result curves showed above that PD keeping controller of missile formation can implement keeping control at about 25s, and control quantities are feasible; the stable error in r x direction is about 5m, and those in r y and r z directions are 1m and 6m respectively. Because three channels of PD controller are coupled seriously, it is very difficult to find the obvious relationships between the inertial time constants of each channel and control performance; From the change of each curve, we can find that the maneuver of leader affect the formation keeping control distinctly; Besides, we can find that there are evident regulating processes from initial states to expected states, and the maximum control quantity exists in two followers, its value is about 60m.

Simulations for Optimal Controller of missile formation keeping 5.2.1 Initial Conditions
We will choose the following conditions for the simulations to optimal controller: 1. weight matrices of optimal controller are:  output regulating weight matrix: diag(4.0,6.0,6.0) y  Q ;  control energy weight matrix: diag(1.0,1.0,1.0)  R ;  weight matrix of control energy changing: 1 diag(1.0,1.0,1.0)  R .
2. other conditions are same as section 5.1.

Simulation results and analysis
After simulating, we can get the result curves Fig.13-Fig.18, where Fig.13 is the 3dimensional motion of missile formation; Fig.14 and Fig.15 are the position components of follower 1 and follower 2 in relative coordinate frame respectively; Fig.16-Fig.18 are the control quantities in velocity, flight path angle and flight deflection angle channels of followers respectively. There are following conclusions after analyzing above result curves: 1. Optimal keeping controller of missile formation can implement keeping on desired relative states at about 20s, it has faster response speed than PD controller; 2. The maneuver motion of leader also disturbs missile formation keeping control, and there also exists regulation process when the states of missile formation changing from initial states to final states, but the disturbance amplitude is smaller that of PD controller; 3. The stable error of relative motion in r x direction is about 0.5m, and those in other two directions are 3m and 4m respectively, which are decreased by comparing to PD controller. Especially the improvement in r x direction is more evident, which is because the velocity of leader is considered as slowly variant perturbation that will affect evidently to the relative motion in r x direction, however, the optimal controller designed in this chapter can restrict this perturbation well, so there will be more higher tracking precision; 4. We can further decrease the stable errors by enhancing output regulating weight matrix y Q , but this manner will increase the changing rate at the same time, so we should coordinate the values of weight matrix y Q , R and 1 R to obtain proper control quantities, and achieve the minimum stable tracking error.

Conclusion
In this chapter, we mainly focus on some control problems of missile formation engagement. Significance of cooperative engagement was first presented by taking LOCAAS as instance, which showed the synthetical efficacy can be greatly increased by adopting cooperative engagement manner. Following the significance was analysis of frame of cooperative engagement system, which supplied the train of thought of how to research on missile formation control problem. Missile formation keeping control system design is the main content of this chapter. In this part, we established the model of relative motion in two ways firstly, and then designed missile formation control system based on PD control law and optimal control method respectively. Finally, the comparisons were made by a series of simulations, the conclusion that optimal controller is better from the points of view of stability and rapidity can be obtained from the result curves.

Acknowledgment
Thank Prof. Sang-Young Park and Korean Industry-Academic Cooperation Foundation very much. They supply plenteous financial support and good working conditions.