AGENTS COORDINATION IN FLAT HIERARCHICAL

SOCIETY-ORIENTED SYSTEMS

Mihaela R. Cistelecan

Technical University of Cluj-Napoca, str. Daicoviciu 15, RO-3400, Cluj-Napoca, Romania

Keywords:

agents, distributed system, sliding manifold

Abstract:

The paper proposes a framework for engineering a coherent society-oriented system using the concept of

sliding-mode control from systems theory. The agents are coordinated through local control laws using a

switching logic related to the sliding manifolds. The sliding manifolds stand for sub-plans and therefore

they are updated every time a change in planning is required. The agents have the ability of self-adaptation

by adjusting the structure of the sliding manifolds. The design is performed using tools from the algebraic

framework. We also address the problem of deriving the sliding manifolds from the given speciﬁcations.

1 INTRODUCTION

Nowadays, a topic of great interest for many differ-

ent research areas is the study of the distributed sys-

tems. The outstanding feature of these systems is that

their architecture looks like a network consisting of

many different interacting entities. The control sys-

tems and computer science communities pursue dif-

ferent ways in their study of the distributed systems.

The approaches from the perspective of the computer

science emphasize the agent behaviour and give an

analysis of the outcome of the system. The (decen-

tralized) decision making system and interaction pro-

tocols are often given in a very complex form and

the outcome is a descriptor of the ”adequate guess”

of the decision. These approaches are important be-

cause they always deal with complex agents - ratio-

nal agents - that make decisions based on logical as-

sertions or plans. Mathematically, the behaviour of

the rational agents, may be described by either dis-

crete event systems (DES) or hybrid systems, (Ortiz

et al., 2001), (Antsaklis and Passino, 1993). Since

the interaction among entities is based on social rules

these systems are most often called either social sys-

tems or society-based systems (SoS). Aside from the

fact that the interaction is based on social rules, of-

ten the agent is not the only type of entity that is en-

countered in SoSs. In SoSs the interaction may be

represented through different kinds of action, such as

cooperation, collaboration, competition, negotiation,

communication. Sometimes, the interaction histories

are also important, (Sen and Dutta, 2002). While

the analysis problem for SoS is quite well developed,

the synthesis problem still poses many challenges.

On the other hand, the approaches from the perspec-

tive of the control systems community deal with not

so sophisticated agents, no planning is implemented

by agents, but the synthesis problem is successfully

solved using tools from control theory. The systems

that these approaches deal with are generally called

multi-agent systems. The features revealed by the two

kinds of approaches may also be noticed by compar-

ing (Fisher and Wooldridge, 2003) and (Yan et al.,

2003).While the technological systems that obey so-

cial laws are dynamical systems and therefore they

should deal with traditional concepts like equilibrium

and system trajectory still, embedding the planning

action into their dynamics is not a trivial matter.

In this paper, we do not dwell on the philosophi-

cal aspects of the social systems, but stick to the idea

of engineering a system that eventually accomplishes

the given task. Therefore, we are interested only in

the synthesis problem. Our research tries to develop

a design procedure for SoSs by using concepts from

the perspective of dynamical systems theory. The ba-

sic idea of the proposed framework, essentially com-

ing from (Cistelecan, 2004a), is that the coordina-

tion in a SoS should be made dependent on the dy-

namics of the interaction among entities. Therefore,

we are in favor of a systematic off-line development

Cistelecan M. (2004).

AGENTS COORDINATION IN FLAT HIERARCHICAL SOCIETY-ORIENTED SYSTEMS.

In Proceedings of the First International Conference on Informatics in Control, Automation and Robotics, pages 76-83

DOI: 10.5220/0001141400760083

 SciTePress

of a coordination mechanism implying the following:

1. a decentralized control law that take into account

the interaction among entities 2. formalization of the

coherence problem as a stability problem; 3. a dis-

tributed planning induced by the sliding manifolds. In

our research we transform the hybrid systems that de-

scribe the agents behavior into dynamic polynomial

systems. The purpose of this paper is to extend the

analysis of the framework already introduced in (Cis-

telecan, 2004b) by investigating thoroughly the prob-

lem of deriving the sliding manifolds from the prob-

lem speciﬁcations.

This paper is organized as follows. In Section 2 we

investigate the issues that are important for the SoSs

design. In Section 3 we show the mathematical model

of the agent. In Section 4 we introduce the idea of a

nominal control system for the proposed framework.

In Section 5 we introduce the control law as a sliding

mode control law. In the next two sections we explain

the basic ideas by using an example. In Section 8

we give some preliminaries for the problem of self-

adaptation, leaving for a next paper a more detailed

mathematical formalization. We conclude the paper

by Section 9.

2 COHERENCE AND GOAL

CODING

The ﬂat hierarchical (one-level) SoSs do not beneﬁt

from an external supervisor, instead, some agents, the

”leaders” or ”managers”, have the authority to control

indirectly the whole dynamics by directly controlling

the entities they interact with. A ”leader” may in-

ﬂuence ”ordinary” agents but remains insensitive to

them. A leader agent has its own dynamics that can be

prescribed or modiﬁed only by an external source. For

example, for a task of collecting objects, if the work-

ing area is to be changed a leader should decide when

and where to move next the whole group of agents.

Obviously, by concatenating many ﬂat hierarchies a

multi-level hierarchy could be obtained. If the ”lead-

ers” embedded on a level of the hierarchy are given

references from a higher level, then the decision from

the higher level is reﬁned at the lower level.

When designing SoSs the following four issues are

important: 1. How to represent the agent behaviour

by a comprehensive model;2. How to model the goal

to be accomplished by the SoS; 3. How to model the

interactions;4. How to infer the appropriate decision

so that eventually the SoS achieves the goal.

Two observations are in place here. First, inside

a SoS the agents functionality is often different. An

example taken from the RoboCup game shows that a

goalkeeper and a player obey different assignments

of the defend-attack protocol. Secondly, a SoS is

composed of many different entities, some of them

- the active entities or agents - are able to take deci-

sions, some of them - the passive entities - just inﬂu-

ence the decisions taken by the active entities. Ex-

amples of passive entities are: landmarks, obstacles,

queues, (non-intelligent) targets on the military ﬁeld,

ball, gate (RoboCup), interrogated global data-bases.

Despite the difference in their protocol and the type

of passive entities the SoS consists of, the agents have

to implement a (decentralized) decision making pro-

cess that guarantees the SoS coherence. The (coordi-

nation) coherence of a SoS is related to its ability to

reach the goal state or to accomplish the given task.

Generally, the goal state is given as a set of predi-

cates on some of the SoS entities - the tactical entities.

For example, if a SoS contains agents, obstacles and

landmarks the speciﬁcations related to the target for-

mation are given only with reference to agents and

landmarks but not to obstacles. It is intuitive that

in a general sense the formation can be stated as a

collection of relations, R

, among the tactical enti-

ties. Thus, the ultimate goal the SoS should achieve

could be coded into the SoS structure through a tar-

get formation, that imposes the desired relations, R

among the tactical entities.

Beside the ultimate goal of a problem it might be

necessary to impose partial goals or strategies for each

agent. From a mathematical point of view these are

restrictions imposed to the agents behaviour. Partial

goals are often dealing with some order relations and

generally they require sub-plans in order to be ac-

complished. For example, the ultimate goal for the

RoboCup game may be coded as a zero distance be-

tween the ball and the adversary’s gate. The partial

goals are related to the game strategy. Note, for ex-

ample, that if the ball is surrounded by many players

from both teams, a strategy should be deﬁned so that

the ball is made attractive only for two adverse play-

ers and repulsive for all the others.

The interactions among entities may be described

mathematically by relations among them. If the ulti-

mate goal is coded into the SoS structure as a target

formation the decision making system should always

minimize the misﬁt measure (e

, q

) = R

³ R

between the desired, R

, and actual, R

relations;

³ stands for a comparison operator. Stated in this

manner, the SoS problem of goal achieving resembles

the well-known regulator (tracking) problem from the

control systems theory if R

are time invariant (time-

varying). Moreover, the coherence concept deﬁned

for SoSs becomes equivalent to the stability concept

from control systems theory. Thus, if the dynamical

system of the misﬁt measure is stable and converges

towards the origin, the SoS behaves coherently and

consequently, the goal is achieved. When the com-

munication among agents is only partial, the ”partial

information control” can be stated as given by the ro-

AGENTS COORDINATION IN FLAT HIERARCHICAL SOCIETY-ORIENTED SYSTEMS

bust control framework that is formalized by the con-

trol theory.

3 THE MODEL OF THE AGENT

From a philosophical point of view, in our approach

the agent consists of three functional components:

Eye, Mind and Body.

The Eye has the functionality of some generalized

sensors, that is the uniﬁcation of physical and logi-

cal sensors. Therefore, the Eye stands for the ability

to detect, perceive or ”see” the neighboring entities.

Like in (Cistelecan, 2004a) the agents are allowed to

”see” each other when some conditions are met and

the pooling of the required conditions is the task of

the generalized sensors. The conditions are given in

relation to the misﬁt measure (e

, q

) between the

entities k and j:

, q

) = (g

, q

, x

, q

, w

, q

, x

, q

, w

, q

))

(1)

where g

and g

are polynomials and the target for-

mation is described through R

= (w

, q

We can notice that the scheduling system - dynam-

ical and distributed - is intrinsic to our control frame-

work: as soon as a generalized sensor ﬁres, the syn-

chronization (communication) with the perceived tac-

tical entity (agent, respectively), may be initiated.

The Ag

’s Eye may be as follows:

= Z

, e, q

, δ) p

(t) = P

(e, q

, z

, δ)

(2)

where (e ∈ R

, q

∈ (Z/

)

) is the misﬁt measure

gathered from all SoS entities and q

is a vector of

variables in the Galois ﬁeld with p elements {0, ..., p−

1}, with the usual multiplication and addition modulo

p, where p is prime, (Gunnarsson and Plantin, 1998).

It is possible to introduce temporal relations

through ”clock” variables. Two temporal relations of-

ten used in distributed systems are ”agent k commu-

nicates with agent j within δ

time units and ”agent k

does some activity after agent j completes some ac-

tivity”. For the ﬁrst example, if the communication

is controlled through the ”clock” variable q

clk

, the

logical error variable in the form q

= δ

− q

clk

= min{δ

, q

clk

} should inﬂuence any other vari-

able involved in the communication process. For the

second example, a possibility is through the following

variables:

= o

; z

= Z

, q

); p

= z

where o

is a binary signal that informs Ag

about the

state of activity performed by Ag

, and Z

should be

interpolated from the table that describes the possible

histories of the variables o

and z

The Mind implements the interaction among enti-

ties and also the planning activity. This component

is mathematically modeled as a sliding manifold to-

gether with a method for adjusting the structure of the

sliding manifold. The planning capability is due to

embedding of the logic components into the sliding

manifold. The partial goals, strategies and sub-plans

are coded into the sliding manifold.

As deﬁned in the control systems theory the slid-

ing mode control is implemented for a continuous dy-

namical system ˙x = f(x, u), x ∈ R

, u ∈ R

the following discontinuous control law: u

= u

(x)

if s

(x) > 0 and u

= u

−

(x) if s

(x) < 0, where

(x) = 0 is the sliding manifold for the control law

and i = 1, .., m. The stability of the dynamical

system is guaranteed by either the reaching condition

˙s

< 0, i = 1, .., m or the Lyapunov type condition

˙s

< 0, d

> 0. The sliding mode regime is

described by s

(x) = 0.

In the proposed framework the decision is taken, as

is customary in the conventional sliding mode control,

so that the system state approaches the sliding mani-

fold and through it the goal state. The Mind adapts

the structure of the sliding manifold and the control

law according to the environment, as will be detailed

in Section 8. The Ag

’s Mind is modeled as a sliding

manifold that aggregates the two vectors that consist

of e

and q

. We assume the sliding manifold on

the following form:

(t) =

j=1

, q

) =

j=1

= 0

(3)

where q

∈ Z

...q

...] and N

is the number

of entities perceived by Ag

. The coefﬁcients c

are

estimated in the off-line design stage so that the whole

dynamics is stable. The mappings h

: (Z/

)

→

Z are also estimated off-line so that the sliding man-

ifolds implement the required strategies or sub-plans.

The mappings h

should make the entities j attrac-

tive/ repulsive to some degree for Ag

The Body stands for the action the agent has to

take. Actions may be motion related like in robotics

but they may also be in the form of message commu-

nications or data-bases interrogation, e.t.c. The Ag

’s

Body aggregates a continuous part and a logical part,

as follows:

˙x

= F

, q

, u

, v

) q

= F

, q

, u

, v

)

(4)

where F

and F

are vectors of polynomials, u

and v

are the control variables for the continuous and

discrete dynamics, respectively, x

∈ R

, and q

∈

(Z/

)

is a vector of variables in the Galois ﬁeld.

The passive entities dynamics is as given in (4), with

(t) = 0 and v

(t) = 0.

ICINCO 2004 - INTELLIGENT CONTROL SYSTEMS AND OPTIMIZATION

4 NEIGHBORHOODS,

ALLIANCES AND COALITIONS

Since in the technological world ”intelligence” is a

synonym for ”complexity solver”, when referring to

SoSs the equivalence between intelligent behaviors

and DES / hybrid systems is not exactly true. The

so called ”understanding of a new situation” or ”re-

ﬁning the knowledge” are not implicitly modeled

into the DES / hybrid system framework unless the

agents are enriched with some further abilities. These

abilities should imply robustness and consequently

they should require that the SoS behave appropriately

when faced with tasks and situations that were not ex-

plicitly considered during the off-line design. There-

fore, the agents have to ”adjust” their behaviours ac-

cording to the real, actual situation. For example,

when the task of some agents is to collect objects /

consume resources the number of objects / resources

is not exactly known, in advance, during the off-line

design stage. Siltt, it is obvious that the appearance /

disappearance of entities (objects, agents, processes,

tasks) is strongly related to coherence. For these

cases, a wise SoS should behave coherently while

adapting on-line some of its parameters. We refer to

this problem in Section 8.

We introduce some concepts that help us explain-

ing the manner the agents self-adjust their behaviour

as a function of the environment. Using the Eye

component the δ-neighborhood, of an agent Ag

{Ag

}, is deﬁned as the set of the entities that

are perceived by the agent Ag

at the time t. The

vector δ contains the ﬁring thresholds of the gener-

alized sensors. Since both the number and type of

entities that an agent perceives are non-deterministic,

it is hard to imagine at the design time all the possi-

ble conﬁgurations of a neighborhood. On the other

hand, there are some special situations, related to the

required strategies or special orders of actions that

should be taken into account at the off-line design

stage. This special situations are described through

special neighborhoods that we call either alliances or

coalitions depending on the team concept. Inside a

team the agents collaborate for solving a problem.

Therefore, for single-team-like SoSs, such as robots

collecting objects, resource allocation problems, dis-

tributed sensor networks, fault-tolerant software sys-

tems, our framework works with alliances. Given for

a minimal or strategic neighborhood, the δ-alliance

of an agent Ag

, A

{Ag

} is deﬁned as the set of

the tactical entities that are perceived by the agent

. For multi-adverse-team-like SoSs such as the

RoboCup or CaptureTheFlag games, multi-team mili-

tary operations, immune systems, beside alliances our

framework works with coalitions. A coalition is de-

ﬁned as the set of tactical entities coming from all

teams. The concepts of alliance and coalition are in-

tended to be used in the off-line design stage in or-

der to get a nominal control system from the problem

speciﬁcations. The nominal control system should be

seen here as a ”minimal” (or abstracted) control sys-

tem derived off-line for alliances and coalitions, that

may be extended on-line in order to deal with neigh-

borhoods. The Mind component is responsible for ex-

panding the control given for an alliance / coalition to

the control required by a neighborhood.

Similar to (Cistelecan, 2004a), in the proposed con-

trol framework, the agents are coordinated through

sliding manifolds. Since the agents functionality is

limited to a limited horizon by the Eye component,

the SoS dynamics emerges as an autonomous switch-

ing system. The switching depends on the envi-

ronment and for this reason no switching ordering

should be imposed by the off-line design. Note that

the sub-plans interleaving is equivalent to switching

through neighborhoods, changing alliances or tradi-

tionally said, mode switching. Moreover, the slid-

ing manifolds are tools for the ”negotiations” ac-

complishment. Since each agent is allowed to inter-

act only with its neighbors, the negotiation of each

agent is based on its local perspective like in AI-

based approaches. The importance of a temporary

alliance and the difﬁculty to regulating threads of ma-

chine intermingled agents as they form and disolve

is observed also by (Klavins and Koditschek, 2000).

Concepts like ”overlapping coalitions of agents” and

”each agent negotiates based on its local perspective”

are also deﬁned in (Sims et al., 2003).

5 THE CONTROL LAW DESIGN

In the proposed SoS, the decision an agent should

take is given by a sliding mode control law. The con-

trol system is transformed into a dynamical polyno-

mial system. Using tools from differential algebra the

speciﬁcations given, both for the continuous dynam-

ics and logical components, are aggregated. Sliding

mode for hybrid systems was already formalized in

(Dogruel et al., 1996). Polynomial dynamical sys-

tems have already been used to model and / or con-

trol hybrid systems, (Forsman, 1994), (Marchand and

LeBorgne, 1998), (Gunnarsson and Plantin, 1998).

Sliding mode control design using tools from differ-

ential algebra is given in (Fortell, 1995).

The decision making system uses a sliding mode

control law, as follows:

= K

, q

)tanh(s

)

= K

, q

)tanh(s

)

(5)

The dynamical system of the misﬁt measure for the

SoS is as follows:

˙e = F (e, q

, u, v) q

= µ(e, q

, u, v)

(6)

AGENTS COORDINATION IN FLAT HIERARCHICAL SOCIETY-ORIENTED SYSTEMS

where u = [u

]

k=1..N

, v = [v

]

k=1..N

and N is

the number of agents. The system (6) was obtained

by appending the dynamics of the misﬁt measure

[

˙e

] for all N agents and also the algebraic

constraint given by the overlapping variables.

The nominal SoS structure (see Section 4) is devel-

oped off-line. The decentralized control laws and the

sliding manifolds are estimated off-line so that when

they work together on-line the coherence of the SoS is

guaranteed. The algebraic framework has been cho-

sen in order to get a parametric control solution. The

concepts of alliance and coalition are used in the off-

line design stage in order to get the nominal control

system from the problem speciﬁcations. The Mind

component adds, on-line, extra-terms both to the slid-

ing manifolds and control laws while coherence is

preserved.

Note that the error vector in (6) contains both con-

tinuous and logical variables. Since we are interested

in a possible practical implementation of the proposed

control system using available computational tools,

we choose to transform the non-homogeneous error

vector into a homogeneous one containing only con-

tinuous time variables. This is possible by a transfor-

mation of the error vector as follows:

˙e q

−→ ˙ω

= [

˙e ˙q

˙p

]

= [

e q

] =

= [

[ω

] [ω

]

k=1,..,N

(7)

where a logical variable z

given as z

= Z(z, e)

is replaced by the continuous time derivative ˙z =

(Z(z, e) − z)p, p = σ((Z(z, e) − z)

), and σ(x) is

the sigmoid function. Thus, the off-line design works

not with (6) but with the polynomial system:

˙ω = f (ω, u, v) (8)

For the nominal SoS structure the Lyapunov function

is chosen in the following form:

(ω, η) = (s

)

P (η)s

+ K

(9)

where:

= [K

(ω

)]

k=1,N

, i = 1, 2,

)

= [s

(ω

), ..., s

(ω

)]

(10)

N is the number of agents, and s

, as given by (3), are

constructed for alliances (and coalitions). The super-

script A stands for alliance in a generic sense. More

precisely, the nominal SoS structure should work with

many different alliances: A

,..., A

. Each alliance A

induces a different manifold s

. We assume that for

the set {s

,..., s

} there exists a generic s

where

polynomials S

are gathered from all s

,..., s

Then, for each alliance the sliding manifold is ob-

tained by deleting some terms from the generic slid-

ing manifold, s

6 EXAMPLE: AGENTS

COLLECTING OBJECTS

In this section we show through an example that the

sliding manifolds can be used as tools for accomplish-

ing cooperation and collaboration in SoSs. We mean

by collaboration an explicit exchange of information.

For a group of agents collecting objects, the ulti-

mate goal could be coded as a zero distance between

any agent and any object (no object on the ﬁeld) and,

in order to assure the coverage of the whole area, a

given distance among agents. The ultimate goal is

given through R

, as follows:

(BT

= F

= F )

(BK

= E

= E)

(¬CO

¬CO

)

(ρ

= ρ

= w

)

(11)

where k, j = 1, .., N, k 6= j, N is the number of

agents, BT , BK stand for the battery and basket, re-

spectively, ρ

and w

are the actual and desired dis-

tances between agents k and j, and ¬CO stands for

’no object on the ﬁeld’. As an example, we assume

that the battery and basket may be full (F/2), empty

(E/0) or nonempty (NE/1).

Since each agent has to collect many objects, the

agent should reason about the order of approaching

the objects. It is intuitive that this order should be

based on the distances between agents and objects,

, but sometimes some other important considera-

tions change this order. For example, note that the ul-

timate goal can be accomplished only if some passive

entities like reloading-stations (RS) and storehouses

(SH) are introduced into the SoS. Thus, the battery

can be loaded if it is empty and the basket can be

cleared when it is full. When the battery is empty

or the basket is full the agent must go to RS or SH

respectively, ignoring the objects that are on its way.

On the other hand, a well designed SoS should allow

the agent, in non-critical situations, to charge its bat-

tery or clear the basket, while collecting the objects

encountered on its way. Moreover, the collaboration

among agents should allow a change of the order the

objects are collected.

If the actions required by agents are: collect ob-

jects (CO), keep distance from agents (KD), approach

agent (AA), approach reloading-station (AR), ap-

proach storehouse (AS), look for a RS (?RS), look for

a SH (?SH), some of the rules that have to be wired

into the sliding manifolds are the following:

(BT = F, BK = E) −→ (CO, KD)

(BT = NE, BK = E) −→ (CO, KD, ?RS)

(BT = NE, BK = NE) −→

−→ (CO, KD, ?RS, ?SH)

(BT = E) −→ (?RS, AR, AA/AR)

(BK = F ) −→ (?SH, AS, AA/AS

(12)

ICINCO 2004 - INTELLIGENT CONTROL SYSTEMS AND OPTIMIZATION

where AA/AR and AA/AS have the meaning: ap-

proach the RS or SH that is in the neighborhood

of another agent. When q

RS/SH

req

= 1 Ag

asks the

neighboring agents (if any) if there is a RS/SH in

their neighborhood. If the answer is positive, q

ack

1 and e

ack

is given the smallest distance to an agent

that has the RS/HS in its neighborhood. Otherwise

ack

= 0 and Ag

starts to wander based on a given

constant $.

If the logical state of an agent implies (BT =

NE, BK = N E) and both RS and SH are in

the N

{Ag

} some logical variables q

RS/SH

(BT, BK, v

) ∈ (Z/

), decide an order of ap-

proaching the entities. We give a simpler, but not so

ﬂexible, solution to this case in the next section.

The agents may implement a collaboration if they

are willing to help each other in collecting objects.

In order to implement collaboration each agent Ag

should estimate a logical error variable q

coll

max(q

, q

), where the value of q

should be com-

municated by Ag

, as it is not visible. The variable q

that should be appended to the Ag

’s Body:

˙x

= Ax

+ Bu

, A ∈ R

2×2

, B ∈ R

2×1

(13)

is as follows:

: (Z/

)

−→ (Z/

)

= Q

(W E, BT, BK)

(14)

where W E is related to the number of objects in

{Ag

} and stands for the working effort (0 - no

object to collect, 1 - a reasonable number of objects,

2 - many objects to collect). A possible assignment

for q

is as follows: Q

(0, 2, 0) = 0, Q

(1, 2, 0) = 1,

(0, 2, 1) = 1, Q

(2, 2, 3) = 2, Q

(2, 2, 0) = 2,

where

= 0 means Ready&ReadyT oHelp

= 1 means

NotReady&

DoNotNeedHelp&

CouldHelp

= 2 means NotReady&N eedHelp

If the sliding variable s

is made dependent on q

coll

the agents may agree to collaborate.

The sliding manifold implementing these rules for

the N

{Ag

} is as follows:

(t) =

coll

RS/SH

ack

RS/SH

ack

+ h

ack

$ = 0

(15)

where: h

: (Z/

)

−→ Z, e

is the error be-

tween Ag

and the entity j that belongs to N

{Ag

are the sliding mode coefﬁcients, the superscripts

”o”, ”RS/SH” and ”a” stand for objects, RS/SH

and agents, respectively. The order of approaching the

neighboring entities is controlled through the map-

pings h

that depend on some logical variables q

Moreover, if an agent is willing to give some help

only after ”sleeping” for δ

minutes then a logical er-

ror variable depending on the ”clock” variable should

inﬂuence the mappings h

coll

7 AGENTS MAKING SUB-PLANS

The ordering relations, strategies and sub-plans are

wired into the sliding manifolds during the off-line

design stage. The mappings h

are estimated for al-

liances during the off-line stage. Let us assume that

is in a logical state characterized by (BT =

NE, BK = NE) and its neighborhood consists of a

RS, a SH and an object: A

{Ag

} = {RS, SH, o}.

A coherent behaviour of the agent would imply a se-

quence of plans so that if, for example, at the begin-

ning it is closer to RS than to SH it heads to the RS

while collecting the objects it encounters on its way.

In order to explain how sub-plans are constructed we

assume a set of sub-plans {P 1, P 2, P 3} that stands

for the following conﬁgurations (partitions):

| < |e

| → P 1

| ≥ |e

| ∧ |e

| < |e

| → P 2

| > |e

| ∧ e

= 0 → P 3

(16)

where e

, e

are the errors between Ag

and

RS, SH and object, respectively. In order to track the

change of conﬁguration some logical variables should

be appended to the Ag

’s Body, as follows:

` |e

| ∼ |e

|; q

` |e

| ∼ |e

` |e

| ∼ |e

|; [BK = 2]; [BT = 2]

(17)

where ∼ stands for {>, <, =} and q

∈ Z/

. For

the conﬁguration C

the sub-plan P 1 implements an

aggregated movement towards the RS and object. If

this movement does not disrupt the movement closer

to the RS then the object is collected. After the object

was collected, if the evaluation of the ﬁeld shows that

the distance to RS is higher than the distance to SH

then, a switch from P 1 to P 3 is done. On the other

hand, if the movement closer to the object implies at a

moment that the agent gets farther from the RS then

the sub-plan P 2 should replace P 1. The switching

among sub-plans is required when a divergent move-

ment of the sliding variable is encountered. For ex-

ample, when the agent starts to go away from the RS

while approaching the object the sliding mode reach-

ing condition s

˙s

< 0 should not be fulﬁlled any

longer. Consequently, a new sub-plan is required in

order to correct the agent dynamics by implement-

ing a repulsive ﬁeld around the object. Notice that

for each sub-plan the stable sliding mode dynamics

AGENTS COORDINATION IN FLAT HIERARCHICAL SOCIETY-ORIENTED SYSTEMS

should implement a movement of the system closer

to the goal. For the conﬁgurations given above the

sliding manifolds should be implemented on the fol-

lowing form:

P 1 :: s

= X

+ X

= 0

P 2 :: s

= X

+ X

= 0

P 3 :: s

= X

+ X

= 0

(18)

where X

, X

stand for the variables that induce an

either attractive or repulsive movement, respectively.

Note that X

, X

are generic coefﬁcients that repre-

sent multiplications of the mappings h

by the slid-

ing mode coefﬁcients c

A quantiﬁed formula written in the theory of reals

(Jirstrand and Glad, 1997) can describe the switching

of sub-plans. For example, the transition P 1 7→ P 2

can be described, as follows:

∃e

, e

∈ [−δ, δ],

∃e

, e

∈ [−δ, δ], ∃e

, e

∈ [−δ, δ],

ϕ |=

((s

> 0 ∧ C

) ⇒ (s

> 0 ∧ C

∧ s

> s

))

∨((s

< 0 ∧ C

) ⇒ (s

< 0 ∧ C

∧ s

< s

))

⇒ (s

> 0 ∧ s

> s

∨

∨s

< 0 ∧ s

< s

∨ s

< 0)

(19)

where

) = s

= X

+ X

) = s

= X

+ X

) = s

= X

+ X

) = s

= X

+ X

(20)

where the variables X

= c

(C)

and X

= c

(C), ζ

are the sliding mode coefﬁcients

that should be estimated in the off-line design stage

so that the dynamics of the whole distributed system

is stable. The variables k are estimated from the re-

lation that results after the quantiﬁer elimination and

cylindrical algebraic decomposition act on (19).

If a neighborhood contains two objects instead

of one object as was wired into the sliding mani-

fold using the alliance given above, N

{Ag

} =

{RS, SH, o1, o2}, the variables X

and X

change

their values only because of the new computed slid-

ing mode coefﬁcients, bc, as will be shown in Section

8. The new sliding manifold used by the agent Ag

order to accomplish its goal is as follows:

(t) = bc

(C)e

+ bc

(C)e

+bc

(C)e

+ bc

(C)e

(21)

When the problem speciﬁcations require that every al-

liance contain at least two agents, the alliance given

above has to be changed so that the most stringent

condition is that the two agents perceive each other.

Therefore, the sliding manifold for the agent Ag

must contain a term referring the distance between

the two agents. The peer agent may be attractive or

repulsive like any other entity in the neighborhood

of Ag

. From the above given analysis it is obvi-

ous that the mappings ζ(C) are very important. Still,

their computation is not trivial and we have to in-

vestigate this topic thoroughly. An important fea-

ture of the mappings ζ(C) is that they have to cap-

ture a given order of the logical variables. For exam-

ple, if we study an alliance of the form A

{Ag

} =

{RS, SH, o1} the mappings should be computed us-

ing the following ordering for the logical variables:

BT Â BK Â {q

, q

} The importance de-

gree of {q

, q

} depends on the conﬁguration

and therefore it should be properly revealed by the

mappings h

8 ON-LINE SELF ADAPTATION

When dealing with neighborhoods the following Lya-

punov function should be considered:

(ω, η) = V

+ V

= (s

)

P (η)s

+ (K

)

+ (K

)

(22)

where: s

= s

+ s

and s

= [s

(ω

)]

k=1,..,N

a vector of polynomials. It is to be noticed that there

exist a difference in the energy given by the Lyapunov

functions in the cases of alliances and neighborhoods.

This is due to the entities the neighborhoods con-

tain but alliances do not. Therefore, the off-line de-

sign provides a control law that can be extended on-

line so that to take into account all the entities of a

neighborhood. The extension is possible because of

a reserve of energy that is foreseen for a neighbor-

hood in relation to the generic alliance. The control

laws for an alliance and a neighborhood are: u

(ω

)tanh(s

), v

= K

(ω

)tanh(s

), u

(ω

)tanh(s

), v

= K

(ω

)tanh(s

= K

(ω

) + k

(ω

), K

= K

(ω

) +

(ω

), with K

(0) = 0, i=1,2, k

(0) = 0,

(0) = 0, where k

= [k

]

k=1,..,N

]

k=1,..,N

should be estimated.

We proceed similar to (Forsman, 1998). We re-

fer in the following to four regions, Ω

(η), Ω

(η)

and B

(η), B

(η), deﬁned as follows: Ω

(η) =

{ω|Q

(ω, η) ≤ 0}, Ω

(η) = {ω|Q

(ω, η) ≤ 0},

(η) = {ω|V

(ω, η) ≤ V

(ω, η)}, B

(η) =

{ω|V

(ω, η) ≤ d}

∂V

∂s

˙s

+ K

∂K

∂ω

˙ω + K

∂K

∂ω

˙ω

∂V

∂s

˙s

)

∂K

∂ω

+ (K

)

∂K

∂ω

˙ω|

(23)

ICINCO 2004 - INTELLIGENT CONTROL SYSTEMS AND OPTIMIZATION

Stability implies: B

(η) ⊆ Ω

(η), B

(η) ⊆

Ω

(η), B

(η) ⊆ B

(η), Ω

(η) ⊆ Ω

(η). The

design has as a primary objective to ﬁnd polynomi-

als relating c

, d and η by examining the following

ideals for all switching systems that are derived from

alliances A

=< V

− V

, Q

, g

, ..., g

=< V

− d, Q

, g

, ..., g

=< (ω

)

− ω

, ..., (ω

)

− ω

(24)

where g

∂V

∂ω

− λ

∂Q

∂ω

, g

∂V

∂ω

− λ

∂Q

∂ω

and ω

= [

... ω

] is as in (7). The Grobner

basis for the ideal

ϑ = ϑ

+ ϑ

, with the alge-

braic restrictions related to the overlapping variables,

is computed w.r.t. the following ranking:

d ≺ {η} ≺ {c

} ≺ ω

≺ {s

} ≺ {K

} ≺ {k

} ≺ {λ}

(25)

and the restrictions s

∈ R[ω

, η, c

, d], K

, K

∈

R[ω

, η, c

, d, s

], k

, k

∈ R[ω

, η, c

, d, s

The self-adaptation is performed by extending al-

liances to neighborhoods, using the variables k

and

. This extension means to implement a stable dy-

namics in the state space ω

using the stable dynam-

ics in the state space ω, ω 7→ ω

= [

ω ˆω

]. The

Mind component has to ﬁnd the components ˆc

, s

) and the control functions

(ω

) =

(ω

), k

, ω

), i=1,2. The sliding manifolds

, with off-line computed c

are extended on-line

to the sliding manifolds s

where:

j=1

(ω) = 0

j=1

ˆc

, s

)

(ω

) = 0

(26)

The control gain (K

(ω

), K

(ω

)) should be ex-

tended on-line to depend on ω

, as follows:

(ω

), K

(ω

)) 7→

7→ (

(ω

, k

(ω

, k

))

(27)

We will approach this extension into a future paper.

9 CONCLUSIONS

The proposed framework is intended for developing

control systems for distributed systems. The control

law is designed in order to implement a coherent co-

ordination for a SoS while the dynamics is constraint

by sub-plans. The agents have the ability of self-

adaptation because they are able to change the sliding

manifolds that are parts of the decision making pro-

cess.

REFERENCES

Antsaklis, P. I. and Passino, K. M. (1993). An Introduc-

tion To Intelligent and Autonomous Control. Kluver

Academic Publisher.

Cistelecan, R. (2004a). Dynamic networks: Decentralized

control and feasibility. In Sci. Bull. of Politehnica

Univ. of Timisoara, Romania, Vol.49(63),121-127.

Cistelecan, R. (2004b). Flat hierarchical society-

oriented systems and agents coordination. In Sci.

Bull. of Politehnica Univ. of Timisoara, Romania,

Vol.49(63),115-121.

Dogruel, M., Ozguner, U., and Drakunov, S. (1996).

Sliding-mode control in discrete-state and hybrid sys-

tems. In IEEE Trans. on AC, vol. 41, no. 3 , 414-419.

Fisher, M. and Wooldridge, M. (2003). A logical approach

to simulating societies,1-14. In G.N. Gilbert and R.

Conte (eds), Artiﬁcial Societies: the computer simula-

tion of social life, UCL Press, London.

Forsman, K. (1994). Hybrid control systems and compre-

hensive grobner bases. In Tech. Report, Automatic

Control, Linkoping.

Forsman, K. (1998). Constructive algebraic geometry in

nonlinear control. In 29th IEEE CDC.

Fortell, H. (1995). Algebraic approaches to normal forms

and zero dynamics. In Ph. D. Thesis, Linkoping Univ.

Gunnarsson, J. and Plantin, J. (1998). Synthesis of a dis-

crete system using algebraic methods. In WODES96.

Jirstrand, M. and Glad, T. (1997). Computationl questions

of equilibrium calculation with application to nonlin-

ear aircraft dynamics. In Tech. Rep., Linkoping Univ.

Klavins, E. and Koditschek (2000). A formalism for the

composition of concurent robot behaviors. In Proc. of

ICRA,3395-3402.

Marchand, H. and LeBorgne, M. (1998). Partial order con-

trol of discrete event systems modeled as polinomial

dynamical systems. In IEEE Int. Conf. on Systems,

Man and Cybernetics, 770-775.

Ortiz, C., Hsu, E., and DesJardins, M. (2001). Incremen-

tal negotiation and coalition formation for resource-

bounded agents. In Proc of AAAI Fall Symposium on

Negotiation Methods for Autonomous Coop. Syst.

Sen, S. and Dutta, P. (2002). The evolution and stability of

cooperative traits. In Proc. of the First International

Joint Conference on autonomous agents and Multi-

agents systems, Italy,1114-1120.

Sims, M., Goldman, C., and Lesser, V. (2003). Self-

organization through bottom-up coalition formation,

867-874. In AAMAS’03, Australia.

Yan, P., Gao, Y., and Ozbay, H. (2003). A variable structure

control approach to active queue management for tcp

with ecn. In IEEE Trans. on Control Systems Technol.

AGENTS COORDINATION IN FLAT HIERARCHICAL SOCIETY-ORIENTED SYSTEMS