Dissertation Overview

Four MAS systems are illustrated to achieve successful results using this new approach.

It is noted that this new proposed approach can also handle the insertion of random targets at any time instant and moving targets, which is illustrated in Cases 2 and 3 of Section VI.

This thesis is organized as follows. The problem of formulation of MAS and SOM algorithm for task assignment is first defined in Chapter 2. Chapter 3 presents the task assignment and path evolution (or planning) for MDS. The proposed FNN controller and Lyapunov stability analysis are also provided in Chapter 2 and Chapter 3. In Chapter 4, the HOHNN is proposed for nonlinear dynamical system identification. Finally, the discussions and future works of the proposed approach are given in Chapter 5.

The major contributions of this thesis are the successful developments of the following:

1) an adaptive fuzzy neural network (FNN) control system in which the Lyapunov stability theorem is used for on-line tuning of the missile guidance design parameters. 2) a monitoring controller is used to compensate the residual of the tracking error. 3) an online dispatching in multi-agent system (MAS) under the desired condition is adopted for the task assignment problem. 4) a battle scenario environment of the missile defense system (MDS) is constructed.

5) a novel high-order Hopfield-based neural network (HOHNN) is proposed for nonlinear dynamical system identification.

Chapter 2

Dynamic Task Assignment with Path Control for Multi-Agent System using Intelligent Adaptive SOM-based Fuzzy Neural Network

2.1 Background and Motivation

The traditional self-organizing map (SOM) aims to exclusively search the real-time shortest paths for all agents, thus allowing them to go to their targets. After this traditional task assignment, the weighting factors of our new SOM-based fuzzy neural network (FNN) controller are activated to force the agents toward their corresponding targets. The FNN controller is the main controller combining the fuzzy rules with the neural network. A monitoring controller is also designed to reduce the error between FNN controller and ideal controller. Using the Lyapunov constraints, the weighting factors for the proposed SOM-based FNN controller are updated to guarantee the stability of the path control system.

2.2 Problem Formulation

Consider a group of N agents in the M-dimensional workspace, it is desired to first perform task assignment by self-organizing map (SOM), after which the path control is activated so that all the agents are capable of going to their targets under the agent dynamics constraints. The dynamics for the i^th agent can be described by [37]

i i i

ia f u

M && + = , 1≤i≤N (2-1) where is the position; is the mass or inertia matrix;

represents the centripedal, Corriolis, gravitational effects and additive disturbances; and represents the control input. We assume that

M i∈ℜ a

M

M M

i∈ℜ ×

M f_i∈ℜ^M

ℜ

∈ u

u i k i

i F F

f = + (2-2) where and represent the known and the unknown vectors of the i^th agent, respectively. We also assume that the unknown vector is bounded by a known bound

k

Fi F_i^u

F . In i

other words, let

i u

i ≤F

F (2-3) for all the N agents. Moreover, it is assumed that, Mi is nonsingular and its lower and upper bounds are bounded by a known bound. In other words, the matrices Mi satisfy

2

2 α M (x )α M α

α

M_Li ≤ ^T _{i ni} ≤ _Ui (2-4) where MLi > 0 and MUi < ∞ are the known lower and upper bounds of the i^th agent, respectively, and is an arbitrary vector. Assume that the initial positions

of agents are located randomly in a given bounded space, and the initial positions of targets are distributed randomly in the same M-dimensional workspace. Then the main control objective is to find the best-matching pairs iteratively by SOM, such that agents can find their relatively shorter paths to the final chosen targets.

Therefore, the planning paths for all agents may have initial chattering (or transient) effects;

nevertheless, these disappear once the best-match time tb is reached. The stability of the closed-loop system can be guaranteed by adaptively adjusting the weighting factors for the proposed SOM-based FNN controller with the aid of a monitoring controller. Define the control inputs for all the agents, the overall concept proposed in this chapter can be illustrated in the following Fig. 2-1.

ℜM

∈ α

} ,...,

2 tD

,..., , {u₁ u₂ }

,..., ,

{a₁ a₂ a_N A=

, {t₁ T=

U t

= u_N}

Fuzzy Neural Network Controller Monitoring

Controller

Adaptive Fuzzy Neural Network (FNN) Controller

+ +

Adaptive Algorithm Tracking

Error

Multi-Agent System (MAS)

Self-Organizing Map (SOM)

A

T U

Fuzzy Neural Network Controller Monitoring

Controller

Adaptive Fuzzy Neural Network (FNN) Controller

+ +

Adaptive Algorithm Tracking

Error

Multi-Agent System (MAS)

Self-Organizing Map (SOM)

A

T U

Fig. 2-1. The closed-loop adaptive SOM-based FNN controller for MAS.

2.3 The Self-Organizing Map (SOM) 2.3.1 Description of SOM

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 the dynamic task assignment because the dimension of the targets can be simplified, and mapped to the relatively corresponding agents. Furthermore, the SOM can iteratively search the best-matching pairs if the targets and agents are dynamically inserted into the workspace. The overall MAS system can be considered a self-organizing system which can adjust its basic structure when its environment changes.

2.3.2 Major Works

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. Thus, no

prior index is imposed on the feature map. Once the network has been properly initialized, there are three essential processes involved in the formation of the SOM, as follows.

Random Indexed Process:

In order to prevent the dependence of an agent on the initial workspace configuration and the input data order, all targets are placed in random indexed patterns in iterations after the first sampling time TS. For each input pattern shown in Fig. 2-2, the random indexed input vectors chosen from the positions of targets are denoted as

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

{r₁ r₂ r_d r_D

R= , t>T_S (2-5) where TS is the sampling time. As long as an iteration starts, the target vector is transformed to the random indexed input vector. The neurons in the network compute their respective values of a discriminant function. This discriminant function then provides the basis for competition among the neurons. The particular neuron with the largest value of discriminant function is declared the winner of the competition. The synaptic weight vector of each neuron in the network has the same dimension as the input space. Let the synaptic weight vector corresponding to the input rd be denoted by

T d N d

d

d [p_1, p_2, p _, ]

P = L . (2-6) To find the best match for the input vector rd with the synaptic weight vectors , we compare the inner products to the N agents and select the largest. Based on

maximizing the inner product , the best-matching criterion is mathematically equivalent to minimizing the Euclidean distance. If we use the index iw to identify the neuron that best matches the input vector rd, we may then determine the index of winner neuron iw, which satisfies the following condition

d

pi,

d T d

i r

p_,

T d

pi_, r_d

d i i d

d

w i

i = (r )=argminr −p_, , i=1,2,...,N (2-7)

that sums up the essence of the random indexed process among the neurons. Depending on the application of interest, the response of the network could either be the index of the winning neuron or the synaptic weight vector closest to the input vector in a Euclidean sense.

⎥⎥

⎥⎥

⎦

⎤

⎢⎢

⎢⎢

⎣

⎡

=

M d

d d

d

r r r

, 2 ,

1 ,

r M

1 ,

p1

2 ,

p1 1 ,

p2

1 ,

pN

D ,

p1

D

pN ,

Pd

t wi

pi ,

⎥⎥

⎥⎥

⎦

⎤

⎢⎢

⎢⎢

⎣

⎡

=

M d

d d

d

r r r

, 2 ,

1 ,

r M

1 ,

p1

2 ,

p1 1 ,

p2

1 ,

pN

D ,

p1

D

pN ,

Pd

t wi

pi ,

Fig. 2-2. The structure of self-organizing map (SOM).

Competitive Process:

The winner neuron determines the spatial location of a topological neighborhood of excited neurons. In traditional SOM, the winning neuron locates the center of a topological neighborhood of cooperating neurons. In this chapter, the neighborhood of the winner is neglected since the agents move toward their corresponding targets without any cooperative process. For a given target as an 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 d ∉Ω (2-8)

where [iw, it] denotes that the pair 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. The distance Di,r

is given as

i d d

Di_, = r −a . (2-9) As long as the N winners are found in an iteration, the index of agents A are re-allocated to obtain a new corresponding to the targets to be used in the adaptive process. From the above two processes, the computational load for finding the best-matching pairs can be obtained as O(N²). 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.

} ,..., ,

{w₁ w₂ w_N W=

Adaptive Process:

The last process enables the excited neurons to increase the individual values of their discriminant functions in relation to the input patterns through suitable adjustments applied to their synaptic weights. In the competitive process, we define the group vector consisting of the winner agents defined as W. This group vector is then utilized to obtain the error matrix to update the weights of the winner. In comparison with incremental adjustment in traditional SOM, the proposed adaptive FNN controller can handle the overall path control for high-order nonlinear agents. This updating method is explained in the following section.

2.4 Design of Fuzzy Neural Network (FNN) Controller 2.4.1 Description of FNN

The FNN architecture in this thesis shown in Fig. 2-3 is a fully linked layer, in which the input layer accepts the input variables, the fuzzification layer calculates the Gaussian membership function and represents the fuzzy rules, and the output layer sums the output of the fuzzification layers. The fuzzy system in internal FNN is trained by the neural network

adaptive algorithm. Therefore, the fuzzy inference system and artificial neural network can complementarily operate for the controlling of nonlinear dynamical systems. For each layer in the following figure, the superscripted number represents each layer and the subscripted number represents the neuron in this layer. The detailed net input and net output are represented as follows.

Fuzzification layer Inference layer Input layer

1

x

1

1

x

k

1

x

K

y

^3P

4

y

O 2

y

1 1

y

1

1

y

k

1

y

K

2

y

H

Output layer

3

y

1 1

f

1

1

f

k

1

f

K

3

y

p

Fuzzification layer Inference layer Input layer

1

x

1

1

x

k

1

x

K

y

^3P

4

y

O 2

y

1 1

y

1

1

y

k

1

y

K

2

y

H

Output layer

3

y

1 1

f

1₁¹

f

1

f

k_k¹

f

1

f

K_K¹

f

3

y

p

Fig. 2-3. The architecture of fuzzy neural network (FNN).

Input layer:

An input vector is fed into the input layer of the i^th agent. The net input and output of the input layer are presented as follows.

1 1

k

k x

net = (2-10)

1 1

1

1_k f_k(net_k) net_k

y = = (2-11)

where represents the net input in this layer, and and are defined as the k^th input and output to the node of input layer, respectively. Each node in this layer represents an input linguistic variable. The presentations of notation in following layers are similar to those defined in this layer.

1

netk x¹_k y¹_k

Fuzzification layer:

Each node performs the fuzzification operation and acts as an element for membership degree calculation, in which the Gaussian function is adopted as the membership function of the IF-parts of the fuzzy rules given by

2 2 2

2 ( )

kh kh h k

v m

net =− x − (2-12)

) exp(

)

( ^{2 2}

2 2

h h

h

h f net net

y = = (2-13) where mkh and vkh are referred to as the mean and the standard deviation of the Gaussian function, respectively.

Inference layer:

Let A₁^h , A₂^h , …, A_K^h , and B denote the fuzzy sets characterized by their ^h corresponding membership function in (2-12) and (2-13) in the function layer, the h^th fuzzy rule can be defined as

Rule h: IF x₁² is A₁^h AND x₂² is A₂^h AND …, x_K² is A_K^h THEN x_h³ is B . ^h The inference layer implements the fuzzy AND aggregation operation which is chosen as the simple PRODUCT operation. Each node multiplies the incoming signals and outputs the result of this product as

∏

=

= ^H

h

h hp

p w x

net

1 3 3

3 (2-14)

∑

=

=

= _P

p p p p

p p

net net net

f y

1 3 3 3

3

3 ( ) (2-15)

where represents the rule weight of the h^th fired rule between the function layer and the inference layer.

3

whp

Output layer:

Each node multiplies the incoming signals and outputs the result of this product as follows

∑

=

= ^P

p

p po

o w x

net

1 4 4

4 (2-16)

4 4

4

4 _o ( _o) _o

o f net net

y = = (2-17)

where represents the output action strength of the o^th output associated with the p^th rule.

Finally, the overall representation is given by

4

wpo

∑ ∏

= =

−

−

=

= ^P

p

H h

kh kh k po

o

o y w x m v

y

1 1

2 2 1

4

4 exp( ( ) /( ) ). (2-18) In summary, the FNN output can be presented as

∑

=

= ^P

p

kh kh k p po

o w x m v

y

1

1

4 θ ( , , ) (2-19) where

∏

=

−

−

= ^H

h

kh kh k kh

kh k

p x m v x m v

1

2 2 1

1, , ) exp( ( ) /( ) )

θ ( . (2-20)

The above (2-20) represents the firing weight of the p^th neuron in the rule layer. The output in the output layer of FNN is adopted as the main controller to the MAS.

2.4.2 Major Works

The control problem of the MAS is to control the position of the winner target so that they can move to the desired target. The error matrix of MAS is defined as

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

{e₁ e₂ e_i e_N W

R

e= − = . (2-21) Considering the i^th agent, the tracking error vector defined as

T i i

Si [e e ]

e = & (2-22) represents the input vector fed into the input node of FNN controller. For the ease of notation, the tracking error vector for single agent is denoted as eS. Assuming all the system dynamics are well known and that there exists an ideal controller for i^th agent and d^th target based on the optimal control design, we then arrive at [39]:

) ( _{1 i 2 i}

i i d i

id Mr f M ke k e

u = && + + & + , 1≤i≤N, 1≤d≤D. (2-23) Applying (2-23) into (2-1), the following error dynamics in two-dimensional workspace can be given

S

S Ke

e& = (2-24) where

⎥⎦

⎢ ⎤

⎣

⎡

−

= −

1 2

1 0

k K k

is a Hurwitz matrix by choosing proper k1 and k2. However, the ideal controller uid is difficult to implement in practice since the system dynamics is highly nonlinear and sometimes unavailable. Therefore, in order to control the output state efficiently, the control law is assumed to take the following form:

m fnn

i u u

u = + (2-25) where ufnn is a FNN controller, and um is a monitoring controller. The FNN control ufnn is the main tracking controller used to imitate the ideal controller in (2-23), and the monitoring controller um is designed to recover the residual approximation error. The monitoring controller, which is similar to a hitting controller in a traditional sliding mode controller, is

derived in the sense of Lyapunov theorem to cope with all system uncertainties to guarantee the stability of the system. Fig. 2-4 illustrates the concept of (2-25) in our new approach. The FNN structure shown in Figs. 2-3 and 2-4 has been considered. For simplicity, the following m and v vectors are defined to collect all parameters in the hidden layer of Fig. 2-3 given as

m T

Adaptive Algorithm

1

FNN Controller

um

Adaptive Algorithm

1

FNN Controller

Fig. 2-4. The configuration of ufnn and um for the i^th agent.

Then, the output of FNN can be represented in vector form as

(2-28)

where , and . By the universal

approxim such that [40, 41]

(2-29)

tion theorem, there exists an ideal y_o

T

where E denotes the approximation error and we*, m^*, and v^* are the optimal parameter vectors of we, m, and v, respectively. In fact, the optimal parameter vectors needed to best approximate a given nonlinear function are difficult to determine. Thus, an estimate function is defined as

estimation error as wˆe mˆ vˆ on-line tune the parameters of the FNN to achieve favorable estimation. To achieve this goal, we use the linearization technique to transform the nonlinear Gaussian functions into partially linear form so that the Lyapunov theorem extension can be applied [40] as follows

H

and H is the higher-order term, and

∂m

∂θ_h and

∂v

∂θ_h

are defined respectively as

⎥⎦

Substituting (2-32) into (2-31) gives

d uncertainty bound Δ is difficult to determine, it is on-line estimated in the following section.

2.5 The Lyapunov Stability Analysis

The proposed control system is comprised of an FNN identifier and an optimal controller defined in (2-25), in which ufnn is used to mimic the ideal controller uid, and the compensation tangent controller um is used to compensate for the difference between the FNN controller and the ideal controller. Substituting (2-25) into (2-1) and using (2-23), the error dynamic equation becomes

ˆ )

where is a bounded matrix. Since K is a Hurwitz matrix, given a symmetric positive-definite matrix , there exists a symmetric positive-definite matrix

, such that the following Lyapunov equation [39, 42]

−1 is satisfied.

Theorem 2-1: Consider the dynamic nonlinear system represented by (2-1) with the control law in (2-25), where the FNN identifier is designed as (2-30). Then, the weighting vectors , , and will remain bounded, and the performance errors will approach zero. The parameters are updated by the following learning rules:

wˆe mˆ vˆ can be guaranteed.

Proof:

Let the Lyapunov-like function candidate be

v

Taking the derivative of V in (2-42) with respect to time and using (2-36) and (2-37), yields

)

Substituting the learning rules (2-38)–(2-41) into (2-43), (2-43) becomes

Δ proven [39]. In addition, the right hand side of (2-45) is bounded, that is, . Using Barbalat’s Lemma [39], we can prove that

L2 the overall approximation scheme is guaranteed based on the above results and the Lyapunov stability theorem. Based on (2-38)–(2-40), 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.

2.6 Illustrated Examples

In this section, four numerical simulation cases are presented in order to illustrate the effectiveness of the proposed new intelligent SOM-based FNN controller discussed in previous section. For ease of plotting, we only consider agents and targets in a bounded two-dimensional space; however, qualitatively, the results are expected to be the same for higher dimensions. We consider agents with point-mass dynamics with unknown mass and additive sinusoidal disturbances. In other words, we consider the model

Without loss of generality, we assume that unity mass Mi = 1 are for all the N agents, and that there exists the following unknown uncertainties.

Case 2-1: N = D = 6 and sin(

) , and sin(

) , satisfies the bounded assumption ||fi|| 1, 1≦ ≦i 64. As controller parameters in the ≦ simulations below, we choose

N the closed-loop configuration of SOM-based FNN for MAS dynamic task assignment and path control.

Fuzzy Neural Network Controller dt

d

Monitoring Controller

A

T

− + um

e

Adaptive Fuzzy Neural Network (FNN) Controller

MAS

ufnn

++

U Competitive

process

W

e&

Adaptive Algorithm

R

Tracking Error

eS

Self-Organizing Map (SOM) Ts t>

Random indexed process

v m w&ˆ_e, &ˆ,&ˆ

Fuzzy Neural Network Controller dt

d

Monitoring Controller

A

T

− + um

e

Adaptive Fuzzy Neural Network (FNN) Controller

MAS

ufnn

++

U Competitive

process

W

e&

Adaptive Algorithm

R

Tracking Error

eS

Self-Organizing Map (SOM) Ts t>

Random indexed process

v m w&ˆ_e, &ˆ,&ˆ

Fig. 2-5. The closed-loop SOM-based FNN for MAS dynamic task assignment and path control.

The best-match time tb is added and shown in the simulation results. In the following figures, the agents marked as ‘Χ’ are randomly located in the grey circle, and the points marked as

‘ ’ are the positions of targets which are◇ the input to the SOM for finding the winner agents.

When the number of agents and targets are different, our proposed approach can also be applied and implemented in the MAS. In this chapter, the agent-target matching pairs of SOM are assumed to be completed before the control inputs are fed to the MAS for path control. In other words, time delay is not considered in the proposed SOM. Moreover, we assume that the number of agents and targets are the same at any time moment in the following cases in order to construct the agent-target matching pair. Therefore, if there is one target inserted into the MAS, there should be one more agent inserted.

Case 2-1: Static random targets

Consider 6 agents and 6 targets in the same two-dimensional (2D) workspace shown in Fig.

2-6. The targets find their matching agents that are then forced to their corresponding targets via SOM-based FNN controller. In order to ensure and check whether or not the

and targets. It is apparent that the transient chattering effects in all paths disappeared after the time is larger than tb. In Fig. 2-7, it can be seen that tb = 1.91 seconds is the best-match time;

after tb, the errors and change of errors will no longer chatter because the best-matching pairs are found. Finally, the best match is completed and the tracking error converges to a

after tb, the errors and change of errors will no longer chatter because the best-matching pairs are found. Finally, the best match is completed and the tracking error converges to a

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