The Design of SOM for Task Assignment of MDS

→ S

t e when

∫

0^{t dt}<∞

Δ 2 . The stability of the overall approximation scheme is guaranteed based on the above results and the Lyapunov stability theorem. Based on (3-36)–(3-38), the adaptive law of weighting factors in an element form can be obtained. Thus, the Lyapunov stability theorem is guaranteed under the optimal approximation model with no modeling error. Q.E.D.

3.5 The Design of SOM for Task Assignment of MDS

After the single agent-target control system is constructed, the overall MDS (or MAS) consists of (N + D) numbers of agent-target matches will be discussed in this section. Suppose that D≥N , it takes P(D,N)=D!/(D−N)! computation steps to find the total distances or damaging cost in traditional exhaustive method. In real-time MDS environment, this pre-computation before the targets are lunched is time-consuming. Therefore, the principal goal of SOM is to transform an input pattern of arbitrary dimension into a one- or two-dimensional discrete map as well as to perform this transformation adaptively in a topologically ordered fashion [19, 38]. The SOM is suitable for dealing with task assignment because the dimension of the targets can be simplified, and mapped to the corresponding agents. The overall MAS system can be considered a self-organizing system which can adjust its basic structure when its environment changes. The algorithm of the SOM proceeds first by initializing the synaptic weights in the network, such that it can be done by assigning them in random indexed patterns. Considering a multi-agent-multi-target scenario, the positions and angles of i^th agent and d^th target can be further defined as

A a_i =[x_a,i y_a,i z_a,i x_&a,i y_&a,i z_&a,i ψ_a,i θ_a,i]^T ∈ ,

d

, T

t_d =[x_t,d y_t,d z_t,d x_&t y_&t,d z_&t,d ψ_t,d θ_t,d]^T∈ , respectively. The proposed control inputs of the i^th agent can also be defined as

U u_i =[a_yc,i a_zc,i]^T =[u_1,i u_2,i]^T∈

where ayc,i and azc,i are the yaw and pitch acceleration commands of the i^th agent, respectively.

Define the positions of agents A={a₁,a₂,...,a_N} where a_i =[x_a,i y_a,i z_a,i]^T is the position of the i^th agent, and denote the random indexed input vectors chosen from the positions of targets as

} ,..., ,..., ,

{r₁ r₂ r_d r_D R=

where r_d =[x_t,i y_t,i z_t,i]^T ∈T is the position of d^th target. Once the positions of agents and targets are initialized, the competitive process of SOM can start to find the winner neurons. In traditional SOM, the winning neuron locates the center of a topological neighborhood of excited neurons. In this thesis, the neighborhood of the winner is neglected since the agents move toward their corresponding targets without any cooperative process. In the traditional competitive process, the total Euclidean distances and total damaging cost have to be considered. Therefore, we first assume that all the asset values are neglected, and the competitive mechanism will choose the Euclidean distance between the i^th agent and the d^th target defined as

i d d

Di_, = r −a . (3-44) In traditional SOM, this Euclidean distance is the parameter for the competitive process.

However, the other parameters, like total Euclidean distances in [23], should also be considered with the distance expression. The values of assets in MDS is more important than their corresponding Euclidean distances and the motivation of SOM in this chapter is to minimize the total damaging cost, therefore, a new distance expression from (3-44) with the equitable distribution of workload can be defined as

d i d total d

i D

V D_, V ⎟⎟⎠ _,

⎜⎜ ⎞

⎝

⎛ +

= + δ

δ (3-45)

where

∑

=

= ^D

d d

total V

V

1

is the total value of all the assets; δ is the adjustment parameter defined by the user to determine the importance of asset value. The smaller the δ is, the more important of asset value is. As shown in Fig. 3-5, the new distance expression constructed by the input of δ, a_i, and rd forms a new N-by-D distance matrix

⎥⎥

⎥

⎦

⎤

⎢⎢

⎢

⎣

⎡

=

D N N

D

D D

D D

, 1

,

, 1 1

, 1

L M O M

L

D . (3-46)

For some given δ, agent, and target as input, the output neurons compete to be the winner according to a specified criterion described as

} } , { ; ,..., 2 , 1 ; ,..., 2 , 1 , min{

] ,

[i_w i_t = D_i,d i= N d = D i r ∉Ω (3-47) where [iw, it] denotes that the match in which the itth target from the iwth agent is the winner, and Ω is the set of neurons in which the winner has been chosen in an iteration. From (3-47), the N winners can be found to obtain a new W={w₁,w₂,...,w_N} which is the re-allocated index of agents A that corresponds to the random indexed targets R to be used in the adaptive process.

⎥ ⎥

winner neuron

winner neuron

D

Fig. 3-5. The structure of self-organizing map (SOM).

Algorithm 3-1:

Step 1 N agents are created in A

D targets with random indexed are created in R D damaging costs are created in V

calculate

∑

=

= ^D

d d

total V

V

1

define the adjustment parameter δ Step 2 for agent a_i,i =1,2,...,N in A

it is d, find the winner neuron with index iw

nd the index iw to the interception list L tep 4 agents are dispatched to hit the targets with orders in L

] ,

[i_w i_t is obtained in the iwth row, d^th column appe

end S

From the above two processes, the number of computation steps for finding the minimal total damaging cost can be obtained, which is equal to N∗D. In comparison with traditional SOM method, the new adaptive SOM method eliminates the time consuming tuning in neighborhood function and is able to reduce the computational load in the task assignment of MAS. Note that in this thesis, the hit probability of agent is assumed as 100 %. However, the SOM mechanism can also find the minimal total damaging cost even if the leakage of agents is considered. By arranging the winner agents W, the interception list L can be obtained which is ordered by the index of agent. The list is a useful command or decision for MDS to determine which target should be intercepted by which agent in the future. The last mechanism of SOM is the adaptive process which enables the winner agents W to update the

ositions of the winners.

le 3-1: SOM-based dispatching

agents p

Examp Step 1

Figure 3-6 shows four steps for SOM example in MDS in which there are the positions of A

a a a a

A={ ₁, ₂, ₃, ₄}⊂ } , , , , ,

(N = 4), and the positions of targets with random indexed (D = 6), and three surviving assets

{r₁ r₂ r₃ r₄ r₅ r₆

R= S={s₁,s₂,s₃} with their values

1 ) (s₁ =

V , V(s₂)=2, and V(s₃)=3. The damaging costs caused from the attacking targets in MDS can be formed as V={V₁, ₂, ₃,V₄,V₅,V₆}={1,3,1,2,3,2} which means t1 will attack s1, t2 will attack s3, t3 will attack

V V

s1, t4 will attack s2, etc. Before the beginning of attack of targets, the total damaging cost

12

total V

V can be obtained.

Step 2

In the SOM mechanism, the positions of agents A and random indexed targets R will first be used to calculate the Euclidean distance as

}

If δ is chosen as 0.1, the new distance matrix from (3-45) can be obtained for all the agents as:

.

Because we are focusing on the task assignment to minimize the damaging cost, we can assume that the distances between the agents and targets are the same and are normalized to one. This implies that

}

Therefore, the minimum for each agent can be found from the following new distance matrix:

⎥⎥ matching pair is {a1, t2}. Therefore, after repeating from the first row to the fourth row, the winner-target pairs {a1, t2}, {a2, t5}, {a3, t4}, and {a4, t6} can be obtained by using the

competitive process in (3-47) list.

Step 4

Picking the indexes of the targets in the matching pairs, the interception list

can further be constructed as a MDS command which shows that the t2, t5, t4, and t6 targets should be intercepted by the a1, a2, a3, and a4 agents, respectively. Defended by the agents, all the assets after this attacking wave have the remaining damaging costs

}

the final total damaging cost becomes ⁶ 2

1

′=

′ =

∑

=

d d

total V

V which is the minimal value. In this example, although the Euclidean distances are almost neglected, the situation for the two or more assets have the same value and there exists relatively short Euclidean distance from some agent to its corresponding target should be taken into consider.

}

Agents are input into SOM

Targets with damaging costs

Assets are under attacks Interceptions by the list

Random

Agents are input into SOM

Targets with damaging costs

Assets are under attacks Interceptions by the list

Random indexed process

R

Fig. 3-6. The self-organizing map (SOM) example in MDS.

In traditional exhaustive method to find the minimal total damaging cost, it will take computation steps. However, it only takes at most

)

computation steps by the proposed SOM. Consider N agents and D targets in MDS, if we have to find the optimal (or worst) matching pairs under some condition such as the minimal

total paths, the computational load of traditional exhaustive method is , however, the computational load of the proposed SOM is

)!

/(

! D N

D −

D

N ∗ . Therefore, as shown in the following Table 3-1, the SOM has relatively smaller computational load than exhaustive method if there are more number of agents and targets.

Table 3-1 Comparisons of computational loads for the task assignment using exhaustive methods and the proposed SOM.

Optimal (or worst) match by

the exhaustive method SOM

N = D = 6 720 36

N = 6, D = 8 20160 48

N = D = 8 40320 64

N = 8, D = 64 1.7846×10¹⁴ 512

N = D = 64 1.2689×10⁸⁹ 4096

In comparison with incremental adjustment in traditional SOM, the proposed adaptive SOM with the proposed FNN controller can handle the overall TAPE problems in MDS. The closed-loop configuration of SOM with FNN controller for MDS TAPE is shown in Fig. 3-7.

FN Cont

N roller dt

d

Monit Contr

eural Ne oring oller

um

e

Adaptive Fuzzy N twork (FNN) controller ufnn

++

U

e&

Ada algo

ptive rithm Tracking

Error eS

v m w&ˆ_e,&ˆ,&ˆ

Limiter

MDS

Self-Organizing Map (SOM)

A W

R Random indexed

process

T

Competitive process

V

A

L

FN Cont

N roller dt

d

Monit Contr

eural Ne oring oller

um

e

Adaptive Fuzzy N twork (FNN) controller ufnn

++

U

e&

Ada algo

ptive rithm Tracking

Error eS

v m w&ˆ_e,&ˆ,&ˆ

Limiter

MDS A

Random indexed process

T

Self-Organizing Map (SOM)

A W

R

Competitive process

V

L

Fig. 3-7. The closed-loop configuration of SOM with FNN controller for MDS TAPE.

3.6 Illustrated Examples

In this section, computer simulations are performed to illustrate the effectiveness of the proposed FNN guidance law. In order to assess the performance characteristics in a closed-loop engagement scenario, it is necessary to specify target dynamics. The target motion model is assumed to produce no axial acceleration or roll motion. Then, the simplified dynamics of target motion can be represented in the inertial frame as follows [44]:

⎪⎪

where aty and atz are the y-axial and z-axial acceleration of target, respectively. For all the scenarios, assume that the target maneuvers with aty = 0 g, atz = –1 g for all the time. To limit the missile’s maneuverability, a 30 g (g = 9.8 m/s²) maneuvering limiter is considered for simulations. The pitch and yaw autopilot dynamics are chosen as second-order time invariant linear systems and the ground tracker as a simplified differential tracking system with damping ration 0.6 and natural frequency 6π rad/s, as shown in Fig. 3-8. The ground tracker provides the estimated values of σ , _t γ_t, σ&_t, and γ& , as well as the measurement data of _t

σ

Δ and Δ . In the following, the estimated value is distinguished from its true value by γ inserting the symbol to the corresponding variable. To evaluate the influence of measurement noise, random noises with magnitude between

ˆ

3 .

±0 degrees are included. The controller parameters in the simulations below are chosen as follows [46]:

⎥⎥

In Case 3-1, the one-to-one agent-target missile guidance laws using CMAC and FNN

controller are discussed to show the capability and efficiency of the proposed FNN controller.

Followed by the tow control algorithm comparisons, the proposed SOM is adopted to handle the task assignment for MAS in MDS in Case 2.

σt

σa

γa

σ Δ

γt

γ Δ

γˆt

(3-3)

s 3 . 355

064 . 0

s 1

s 1

+ +

+ +

+ +

+ +

− −

− −

− −

− − 0.064

s 3 . 355

Rˆa

e1

e2

σt

σa

γa

σ Δ

γt

γ Δ

γˆt

(3-3)

s 3 . 355

064 . 0

s 1

s 1

+ +

+ +

+ +

+ +

− −

− −

− −

− − 0.064

s 3 . 355

Rˆa

e1

e2

Fig. 3-8. Block diagram representation of estimation algorithm for guidance information [44, 46].

Case 3-1:

The closed-loop configuration of FNN controller for missile guidance law is shown in Fig.

3-9. For large scale simulation purpose, we have generated 25 initial positions with angles for target in Fig. 3-10, which shows an XI-YI coordinate system. The subscripted number of target in Fig. 3-10 represents the number of scenario, and the position of target in the i^th scenario is defined as

)) 30 sin(

), sin(

) 30 cos(

), cos(

) 30 cos(

(

: _t ° _{t t} ° _{t t} °

i R R R

i

i σ

σ

t ,

)) 60 sin(

), sin(

) 60 cos(

), cos(

) 60 cos(

(

12: ° ° °

+ t t t t t

i R R R

i

i σ

σ

t ,

) , 0 , 0 (

25: Rt

t , i=1 L,2, ,12. (3-49) In fact, our agent is at (10, 10, 6), which is very close to the ground base in XI-YI-ZI coordinate system. There are only three initial positions/angles of target in [4, 6], which were specifically chosen to guarantee the success of convergence. Therefore there are 25 battle scenarios in which Rt is set as 7000 meters from the ground base, and each initial position and angle of target is chosen from the 25 locations as shown in Fig. 3-10. Once the position of target is chosen, the initial angle of target will also be applied as the heading angle to the ground base.

FNN controller dt

d

Monitoring controller

− +

um

e

Adaptive Fuzzy Neural Network (FNN) controller ufnn

++

U

e&

Adaptive algorithm Tracking

error eS

v m w&ˆ_e, &ˆ,&ˆ

Limiter

Defending missile

Attacking missile

FNN controller dt

d

Monitoring controller

− +

um

e

Adaptive Fuzzy Neural Network (FNN) controller ufnn

++

U

e&

Adaptive algorithm Tracking

error eS

v m w&ˆ_e, &ˆ,&ˆ

Limiter

Defending missile

Attacking missile

Fig. 3-9. The closed-loop configuration of FNN controller for missile guidance.

‧: I

-8000 -6000 -4000 -2000 0 2000 4000 6000 8000

-8000

nitial position of targets ---: Contour line

□: Asset region

XI

-8000 -6000 -4000 -2000 0 2000 4000 6000 8000

-8000

nitial position of targets ---: Contour line

□: Asset region

XI

Fig. 3-10. The initial positions and angles of targets in the XI-YI plane.

The errors and change of errors of FNN controller in 25 battle scenario simulations are shown in Fig. 3-11. From Fig. 3-11, it can be obviously seen that our proposed adaptive FNN controller is capable of performing missile guidance. The control input u of FNN controller shown in Fig. 3-12 contains the yaw and pitch acceleration commands which are denoted as dyc and dzc, respectively. In comparison with the CMAC used for missile guidance in [4, 6], all the 25 guidance results under the same scenario are listed in Table 3-2 for computational load (CL) and miss distance (MD) of DM.

0 1 2 3 4 5 6 7 8 9 -20

-10 0 10 20

0 1 2 3 4 5 6 7 8 9

-60 -40 -20 0 20 40

Time (second) e

e&

0 1 2 3 4 5 6 7 8 9

-20 -10 0 10 20

0 1 2 3 4 5 6 7 8 9

-60 -40 -20 0 20 40

Time (second) e

e&

Fig. 3-11. The errors and change of errors of FNN controller in 25 battle scenario simulations.

0 1 2 3 4 5 6 7 8 9

-200 -100 0 100 200

0 1 2 3 4 5 6 7 8 9

-400 -200 0 200 400

Time (second) dyc

dzc

0 1 2 3 4 5 6 7 8 9

-200 -100 0 100 200

0 1 2 3 4 5 6 7 8 9

-400 -200 0 200 400

Time (second) dyc

dzc

Fig. 3-12. The control inputs of FNN controller in 25 battle scenario simulations.

Table 3-2 The 25 guidance results (MD: Miss Distance; CL: Computational Load).

CMAC FNN controller

controller

scenario MD (m) CL (sec) MD (m) CL (sec)

1 2.7600 6.641 4.7312 0.641

2 2.9284 6.594 1.9743 0.625

3 3.1054 6.578 3.7066 0.640

4 5.8901 6.594 5.9765 0.641

5 0.1236 6.594 10.6379 0.765

6 6.0915 6.594 9.1060 0.766

7 11.4619 6.625 2.6917 0.766

8 7.7567 6.672 2.1204 0.641

9 8.1750 6.609 0.7921 0.641

10 8.6591 6.578 2.0889 0.625

11 3.8047 6.562 8.3483 0.625

12 0.9872 6.578 9.7337 0.750

13 6.6148 8.610 4.0755 0.813

14 0.6873 8.625 4.8888 0.672

15 0.9697 8.625 4.7711 0.672

16 4.6191 8.609 5.0199 0.671

17 2.0204 8.641 5.9066 0.688

18 1.9740 8.625 7.6901 0.688

19 4.5209 8.641 5.8192 0.656

20 4.8498 8.609 2.5172 0.796

21 2.1815 8.625 2.9209 0.672

22 5.8322 8.609 6.2649 0.687

23 4.8678 8.625 7.0819 0.687

24 5.1705 8.609 4.6043 0.672

25 42.5002 9.500 3.6836 0.671

In Table 3-2, the average MD of CMAC is 5.9421 meters which is larger than that of FNN, which is 5.0861 meters. In Table 3-2, it is obvious that the CMAC will fail in 25^th scenario due to its MD equals to over 40 meters, whereas the agent in using our FNN controller is only 3.6836 meters. It can be seen that the scenarios for CMAC should be chosen carefully to

prevent the divergence of missile guidance. For the comparison of computational load, Table 3-2 also shows the running time under the same Windows XP system. It is amazing to see that the average CL (in seconds) of CMAC is ten times larger than that of FNN. In real-time control system, the larger CL is not preferred, especially when the intelligent missile guidance law is applied to agent. From the above simulation results, the proposed adaptive FNN controller is capable of efficiently maneuvering the agent toward the target in finite time, and the MD of agent can also be reduced to a satisfactory level. Moreover, the proposed FNN controller has much smaller CLs than those of CMAC in all the simulation results.

Case 3-2:

The adjustment parameter is chosen as δ = 0.1 to emphasize the importance of asset value. For ease of simulation in the following scenarios, the asset value will be randomly chosen as an integer from 1 to 3; the numbers of assets, agents, and targets are also randomly chosen from 1 to 15. The initial positions and angles of targets in [44, 46] are chosen without any reason, which is very sensitive during simulation. If more scenarios are needed, it is time-consuming for the user to determine these initial parameters, even for simulation purpose. In this case, the ground tracker is set at the original point and the initial LOS ranges of targets are also set as 7000 meters. Also, the positions of assets and initial agents are randomly located in the bounded asset region. For ease of simulation, a random number of targets are chosen from the 25 initial positions to represent the attacking targets, which are incoming from different degrees as shown in Fig. 3-10. Once an attacking target is moving toward some asset, the initial angle of this attacking target can be decided by calculating the heading angle to the asset. The hit probability of all the targets for assets is also assumed as 100 % such that the assets will not be destroyed if the agents are forced to intercept the corresponding targets.

Scenario 1:

Consider 9 agents and 9 targets in the same three-dimensional workspace shown in Fig. 3-13.

not considered, scenario 1 can be regarded as the extension of single agent-target guidance problem. In this scenario, the asset damaging cost by all the targets is listed as

} 2 , 3 , 3 , 2 , 1 , 1 , 3 , 2 , 1

={

V without interception. After the task assignment, the 9 agents are forced to their corresponding targets. The interception list L={3,7,8,9,6,2,1,4,5} can be obtained via SOM with the consideration of the physical distances between the agents and targets. The interception list of will be obtained if the physical distances between the agents and targets are not considered in our SOM. The agents will find their matching targets by L, and then be forced to the incoming targets via adaptive SOM with FNN controller. From the simulation result, it shows that the total damaging cost is reduced to 0 which is the minimal total damaging cost. The error and change of error of all the agents also approaches to zero as shown in Fig. 3-14.

} 9 , 8 , 7 , 6 , 5 , 4 , 3 , 2 , 1

={ L

-6000 -4000 -2000 0 2000 4000 6000 -5000

0 5000

0 1000 2000 3000 4000 5000 6000 7000

t7 t5

t4 t9 t3 t6

X t2

t10 t8

t1

Y

Z

Hit point

XI

YI

ZI

-6000 -4000 -2000 0 2000 4000 6000 -5000

0 5000

0 1000 2000 3000 4000 5000 6000 7000

t7 t5

t4 t9 t3 t6

X t2

t10 t8

t1

Y

Z

Hit point

XI

YI

ZI

Fig. 3-13. The trajectories of 9 agents with 9 targets.

0 1 2 3 4 5 6 7 8 9 -100

-50 0 50 100

0 1 2 3 4 5 6 7 8 9

-100 -50 0 50 100

e

Time (second) e&

0 1 2 3 4 5 6 7 8 9

-100 -50 0 50 100

0 1 2 3 4 5 6 7 8 9

-100 -50 0 50 100

e

Time (second) e&

Fig. 3-14. The errors and change of errors of MAS.

Scenario 2:

Consider 3 agents and 9 targets in the same three-dimensional workspace shown in Fig. 3-15.

In this scenario, the asset damaging cost by all the targets is listed as V ={1,2,1,3,2,1,3,3,2} without interception. The interception list L={7,8,4}

! 9

can be obtained via SOM. It can be obviously seen that the interception list L command to the FNN for the 1^st, 2^nd, and 3^rd agent to intercept the 5^th, 8^th, and 4^th target. Therefore, from the simulation result it shows that the total damaging cost is reduced to 9 which is the minimal total damaging cost. The error and change of error of all the agents also approaches to zero as shown in Fig. 3-16. In this scenario, the traditional exhaustive method will take /(9−3)!(=504) computation steps to find the minimal total damaging cost.

-6000 -4000 -2000 0 2000 4000 -2000

0 2000 0 1000 2000 3000 4000 5000 6000

t2 t9

t6 t5 t7

X t1 t8

t4

t3

Y

Z

XI

YI

ZI

-6000 -4000 -2000 0 2000 4000

-2000 0 2000 0 1000 2000 3000 4000 5000 6000

t2 t9

t6 t5 t7

X t1 t8

t4

t3

Y

Z

XI

YI

ZI

Fig. 3-15. The trajectories of 3 agents with 9 targets.

0 1 2 3 4 5 6 7 8

-100 -50 0 50

0 1 2 3 4 5 6 7 8

-100 -50 0 50 100

e

Time (second) e&

0 1 2 3 4 5 6 7 8

-100 -50 0 50

0 1 2 3 4 5 6 7 8

-100 -50 0 50 100

e

Time (second) e&

Fig. 3-16. The errors and change of errors of MAS.

Scenario 3:

Consider 4 agents and 12 targets in the same three-dimensional workspace shown in Fig. 3-17.

In this scenario, the asset damaging cost by all the targets is listed as }

V without interception. The interception list can be

obtained via SOM. From the simulation result, it shows that the total damaging cost is reduced to 18 which is the minimal total damaging cost. The error and change of error of all the agents also approaches to zero as shown in Fig. 3-18. In this scenario, the traditional exhaustive method will take more time-consuming

} than that of SOM to find the minimal total damaging cost.

-6000

Fig. 3-17. The trajectories of 4 agents with 12 targets.

0 1 2 3 4 5 6 7 8 9 10 -100

-50 0 50 100

0 1 2 3 4 5 6 7 8 9 10

-100 -50 0 50 100

e

Time (second) e&

0 1 2 3 4 5 6 7 8 9 10

-100 -50 0 50 100

0 1 2 3 4 5 6 7 8 9 10

-100 -50 0 50 100

e

Time (second) e&

Fig. 3-18. The errors and change of errors of MAS.

From the above simulation results, the adaptive SOM with FNN control method is capable of handling task assignment of the agents and then efficiently maneuvering the agents toward the corresponding targets in finite time. The control errors of all the agents can also be reduced to a satisfactory level. Moreover, the total damaging cost can be minimized and the number of

From the above simulation results, the adaptive SOM with FNN control method is capable of handling task assignment of the agents and then efficiently maneuvering the agents toward the corresponding targets in finite time. The control errors of all the agents can also be reduced to a satisfactory level. Moreover, the total damaging cost can be minimized and the number of

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