hn-models in control

SOME MODERN TRENDSIN CONTROL THEORY

Dmitry A. [email protected], www.ipu.ru, www.mtas.ru

Institute of Control Sciences RASMoscow

Institute of Physics and Technology Moscow

1. HISTORY AND TRENDS OF CONTROL THEORY2. HIERARCHICAL AND NETWORKED MODELS3. INTERDISCIPLINARY CONTROL MODELS4. EMPHASIS OF RECENT CONFERENCES5. MULTIAGENT SYSTEMS: SPECIFICITY AND ARCHITECTURE OF AN

AGENT6. HIERARCHICAL MODELS (EXAMLES):

- Diffusive bomb- Social networks- Combat modeling

7. SOME TRENDS IN NETWORKED AND HIERARCHICAL MODELING8. CONTROL THEORY: ANTI-INTUITIVE RESULTS (EXAMPLES)

- Optimization of hierarchies- Labor supply- Ecology VS Economy- Is truthtelling profitable?- Hierarchical automatization

9. PAST, PRESENT AND FUTURE OF CONTROL THEORY

PLAN

STRUCTURE OF CONTROL SYSTEM

…

Control is “the process of checking to make certain that rules or standards being applied” (Macmillan Dictionary).

Control is “the act or activity of looking after and making decisions about something” (Merriam-Webster Dictionary).

Control is “an influence on a controlled system with the aim of providing the required behavior of the latter” (Theory of Organizations Control)

CONTROL SYSTEM

CONTROLLED OBJECT

Stateof

controlledobject

Control

External disturbances

OBJECTS, METHODS AND MEANS OF CONTROL

CONTROL

OBJECTS METHODS

MEANS

Measuring, transformative,

actuating Informational, computational

…

Control is “the process of checking to make certain that rules or standards being applied” (Macmillan Dictionary).

Control is “the act or activity of looking after and making decisions about something” (Merriam-Webster Dictionary).

Control is “an influence on a controlled system with the aim of providing the required behavior of the latter” (Theory of Organizations Control)

Scientific knowledge

about the object of control

Control theory

Applications

Technologies of control …

…

t

Typical time of development

CYCLE OF THEORY DEVELOPMENT

t

1860-е 1900 1930 1940 1950 1960 1970 1980 1990 2000 2010КонецXVIII века

150 YEARS OF CONTROL THEORY (Technics)

Scientific knowledge

about the object of control

Control theory

Applications

Technologies of control …

…

t

Typical time of development

Institute of Control Sciences (ICS RAS, Moscow):- System Theory and General Control Theory;- Techniques of Control in Complicated Engineering and Man-Machine Systems;- Theory of Control in Inter-Disciplinary Models of Organizational, Social, Economic, Medical and Biological and Environment Protection Systems;- Theory and Techniques in Development of Software-and-Hardware and Engineering Tools of Control and Complicated Data Processing and Control Systems;- Scientific Fundamentals of Technologies in Vehicle Control and Navigation;- Scientific Fundamentals of Integrated Control Systems and Automation of Technological Industrial Processes.

USA: CalTech, Harvard Univ., MIT, Stanford Univ., Univ. of California, etc.UK: Imperial College, Oxford Univ., Cambridge Univ., etc.Germany: Univ. of Stutgard, Techn. Univ. ofDarmstadt , etc.Italy: University of Rome, Politecnico di Torino, Politecnico di Milano, etc.

France: SUPELEC Paris, Univ. of Grenoble, LAAS-SNRS Toulose,etc.Australia: Univ. of New-castle, Australian Nat. Univ., etc.Sweden: University of Linkoping, Lund University.Japan: Japan Advanced Inst. of Science and Techn., Kyoto Univ., etc.

CONTROL THEORY IN RUSSIA

… and in the World

…

SCIENTIFIC RESEARCHESAND APPLIED PROJECTS OF ICS RAS. I

Classical linear control theory

Control of mechanical

systems

Nonlinear systems

Control in quantum systems

Control of systems with distributed parameters

Theory of stability and stabilization

Control of moving objects

Robust and adaptive control

General control theory

)(),,(.

twtuxfx +=

Study of control problems for a multi-mode submarine in the case of

emergency. The intelligent control level is based on production rules for current and predicted situations

Full-scale multi-mode computer training complex for Russian Navy

General view of the training complex

SCIENTIFIC RESEARCHESAND APPLIED PROJECTS OF ICS RAS. II

Integrated solutions for upper level control systems in nuclear power engineering with application in Russia and abroadØProgramming languageØOperating systemØEngineering equipment (in cooperation with the Nizhny Novgorod Scientific Research Institute for Engineering Systems)

ØApplied software

SCIENTIFIC RESEARCHESAND APPLIED PROJECTS OF ICS RAS. III

Equipment for automation devices and computers

New effects in nanostructures

Modern jet devices(operable up to 500˚ C, in vibration and radiation environment)

Multi-channel sensors

SCIENTIFIC RESEARCHESAND APPLIED PROJECTS OF ICS RAS. IV

Aviation maps

Navigation maps

Digital topographic mapsLarge-scale plans

Digital relief matrix

3D-modeling

3D-modeling

Map making by field surveyMap updating by air survey

Map updating by space survey

Geographic Information Systems

SCIENTIFIC RESEARCHESAND APPLIED PROJECTS OF ICS RAS. V

3D-Modeling in control

SCIENTIFIC RESEARCHESAND APPLIED PROJECTS OF ICS RAS. VI

APPLICATIONS OF CONTROL THEORY







PLAN

USA (NSF) priorities:Group control. Combat control. Control in financial and economic systems. Control in biological and ecological systems. Man and team in a control loop.Unified theory of control, computation and communication (С3). …

European priorities:Man-machine symbiosis (modeling a man in a control loop and as a controlled subject). Distributed and networked systems. Production, safety and strategies of heterogeneous control. New principles of interdisciplinary coordination and control. …

Russian Academy of Sciences:Methods and means of communicational and networkedcontrol of multi-level and distributed dynamic systems under uncertainty; intellectual control.

HETEROGENEOUS = DIVERSIVE(NETWORKED)

+ DISTRIBUTED+ HIERARCHICAL

(controlled object, control system and communications).

HETEROGENEOUS CONTROL MODELS

HIERARCHIES AND NETWORKES:CONTROLLED OBLECT, CONTROL SYSTEM

AND COMMUNICATIONS(example: Project Management)

FROM NETWORKES AND HIERARCHIES

TO HIERARCHIES OF NETWORKESAND NETWORKES OF HIERARCHIES

Пакет работ

УровеньWBS

УровеньWBS

УровеньWBS

Пакетработ






. . . . . .

Центр

Центр

Центр

OBS

Руководство проекта


Планy * A'∈

Результатдеятельности

z A0∈

Действие)y А'∈

ЦентрЦентр Центр

АЭ АЭ АЭ АЭ АЭ АЭ

RBS

Функциональная структура предприятия

Стимулирующее воздействие

Группапроектов

ПроектПроектПроектУровень проектов

NETWORK SHEDULE

WBS OBS

RBS

Responsibility allocation

Resources allocation

Authority allocation

Степень

достижения

поставленны

х це

лей

Время реализации

Исходное состояние

Целевое состояние

Оценка состояния

Планируемое состояние

Реальное состояние

Плановая траектория

Реализуемая траектория

Прогноз реализуемой траектория

Возможная траектория в рамках

существующей стратегии (работа над

ошибками)

Оценка затрат для возврата на плановую траекторию развития







PLAN

2000 1950 1940 1960 1970 1980 1990

Theory of automatic control

Game theory, operations research

Cybernetics

Decision-making, ;public choice theory

Mechanism design (MD)

Multiagent systems(DAI)

Discrete optimization, optimal control

2010

Network structures (С3)

Organizational systems (MAN)

Social systems

(SOCIETY)

Ecological systems

(NATURE)

Technico-organizational and man-machine

systems Ecologo- economical

systems

Economical systems

PRODUCTION

Regulatory-axiological

systems

Noospheric systems

Socio-ecological systems

Technical systems

Socio- economical

systems

INTERDISCIPLINARY SYSTEMS

Optimization of hierarchicaland network structures

Territory

Government

Industry

Enterprise

Coordination of interaction and decision-making in multiagent systems

Interests concordance in ecologo-economical systems

Reflexive control

Mechanisms of organizations control

Temporal network organization

А

Б В

Г

2 12 121 А В ВА ВАВ

EXAMPLES OF INTERDISCIPLINARY SYSTEMS

Confrontation Hierarchies Collective Decision-making Cooperative Control

Interaction. Distributed Optimization (e.g. Task Assignment) Mission Planning

“Implementation”. Formation Control

Stabilization. Consensus Problem

Operational level

Action

Tactical level

Strategic level (decision-making,

adaptation, learning, reflexion)

Level of goal-setting and choice of functioning

mechanisms

Exte

rnal

inf

orm

atio

n

Dyn

amic

syst

ems

Art

ifici

al

Inte

llige

nce

Mod

els o

f C

olle

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e B

ehav

ior

Gam

e T

heor

y







PLAN

GENERAL TOPICS

ACC-2011***

*CDC – Conference on Decision and ControlECC – European Control Conference12-15 December 2011

Organized by IEEE, Orlando, Florida> 1500 papers

**IFAC World Congress28 Aug – 2 Sept., 2011> 1500 papers

***ACC – American Control Conference29-30 June 2011Organized by IEEE, San Francisco, USA > 900 papers

CDC-ECC-2011*

Математи-ческая теория

9%

Приложения19%

"Cредства"4%

"Классика"49%

"Сетевизм"19%

Applications 19%

Mathematical theory

9%

Technical means 4% “Networks”

19%

Classic 49%



Cредства5%



9%

IFAC-2011**

Applications 44%

“Networks” 11%

Classic 32%

Mathematical theory

9%

Technical means 5%


5%


Cредства5%



Technical means 5%

Applications 33%

Mathematical theory

5%

Classic 42%

“Networks” 14%

МАС и консенсус

35%

Кооператив-ное

управление21%

Коммуникации в МАС35%

Верхние уровни

управления4%

Другое4%Highest

control levels 4%

Communi- cations in MAS

35%

Cooperative control

21%

Consensus problems

35%

Others 4%

Другое8%Верхние

уровни управления

10%


Кооператив-ное

управление10%


46%

Highest control levels

10%

Communi- cations in MAS

26%

Cooperative control

10%

Consensus problems

46%

Others 8%


33%

Кооперативное управление

15%


Верхние уровни управления

13%

Другое8%

Highest control levels

13%

Communications in MAS

31% Cooperative

control 15%

Consensus problems

31%

Others 8%

«NETWORKS»

II

I III

IV

ACC-2011

СDС-ECCC-2011 IFAC-2011

II

IIII

IV

I

II

III

IV

Другие19%

Морские подвижные

объекты2%

Автомобили и автотрафик

11%

Биология и медицина

13%Мехатроника и

роботы11%

Производство2%

Авиация и космос

9%

Энергетика33%

Others 19%

Energetics 33%

Aero-Space 9%

Production 2% Mechatronics

and robots 11%

Maritime mobile objects

2%

Avto-vehicles and traffic

11%

Bio-Med 13%



7%


Мехатроника и роботы

13%


13%


13%

Морские подвижные

объекты3%

Другие16%

Others 16%

Energetics 17%

Aero-Space 7%

Production 18%

Mechatronics and robots

13%

Bio-Med 13%


13%


3%

Другие8%Морские

подвижные объекты

11%


8%


17%Мехатроника и

роботы18%



8%


Others 8% Energetics

22%

Aero-Space 8%

Production 8%

Mechatronics and robots

18%

Bio-Med 17%


8%


11%

APPLICATIONS

ACC-2011

СDС-ECCC-2011 IFAC-2011







PLAN

MULTIAGENT SYSTEMS (MAS)

«… a rat or individual locusts are not too clever, and almost harmless. However, flocks of rats or swarms of locusts can have a devastating impact».

Modern trends:- decentralization- miniaturization- intellectualization

Centralized control

Decentralized (group) control

NETWORK

MAS

Material MAS

Virtual MAS (softbots)

Wheeled UAVs AUVs …

Specificity of MAS:üMultiple components;üDistributed, networked

communications;üHierarchy;ü Intelligence (autonomy):• rationality (decision-making under

uncertainty and cognitive restrictions);• autonomous goal-setting, goal-

oriented behavior;• reflection;• cooperative and/or competitive

interactions (the formation of coalitions, information, and other confrontation).

MULTIAGENT SYSTEMS: SPECIFICITY

MULTIAGENT SYSTEMS: ARCHITECTURE OF AN AGENT





Operational level

Action

Tactical level




mechanisms

Exte

rnal

inf

orm

atio

n

Dyn

amic

syst

ems

Art

ifici

al

Inte

llige

nce

Mod

els o

f C

olle

ctiv

e Be

havi

or

Gam

e T

heor

y





Operational level

Action

Tactical level




mechanisms Ex

tern

al

i

nfor

mat

ion

Dyn

amic

syst

ems

Art

ifici

al

Inte

llige

nce

Mod

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f C

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ehav

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Gam

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eory

MULTIAGENT SYSTEMS: CONSENSUS

Multi-agent system

( ) nitxtxatxn

jjiiji ..., ,1,)()()(

1=−−= ∑

=& – characteristic of agent i,)(tx i,

),()( txLtx −=& ( ) ,)(..., ),()( T1 txtxtx n=

[ ] ,ij n nL ×= l

=≠−

= ∑≠

.,,,

)( ijaija

tik

ik

ij

ijl

Theorem*Consensus is reachable iff the communication graph is “connected”.

CONSENSUS PROBLEM

*Agaev R., Chebotarev P. (A&RC. 9. 2000)





Operational level

Action

Tactical level




mechanisms Ex

tern

al

i

nfor

mat

ion

Dyn

amic

syst

ems

Art

ifici

al

Inte

llige

nce

Mod

els o

f C

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ehav

ior

Gam

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heor

y

MULTIAGENT SYSTEMS: FORMATION CONTROL

С3 & FORMATIONS CONTROL

Basic consensus problem

( )ix t& = 1

( ( ) ( ))n

ij j ij

a x t x t=

−∑

Communications & computations

( )ix t& = 1

( )( ( ) ( ))n

cij j ij i ij

ja t x t x tτ τ

=

− − −∑

Model of the controlled object

Autonomous Underwater Vehicles. Edited by N. Cruz. – Rijeka: InTech, 2011.

+ nonlinearity + observability + adaptivity + switching communication matrix

…

Formation control ( )ix t& = vi(t),

( )iv t& = 1

( ( ) ( ))n

ij j ij

b x t x t=

−∑ +

1( ( ) ( ))

n

ij j ij

b v t v t=

−∑ + ci(V(t) – vi(t))

x(t), v(t) V(t)





Operational level

Action

Tactical level




mechanisms Ex

tern

al

i

nfor

mat

ion

Dyn

amic

syst

ems

Art

ifici

al

Inte

llige

nce

Mod

els o

f C

olle

ctiv

e B

ehav

ior

Gam

e Th

eory

MULTIAGENT SYSTEMS: PLANNING





Operational level

Action

Tactical level




mechanisms Ex

tern

al

i

nfor

mat

ion

Dyn

amic

syst

ems

Art

ifici

al

Inte

llige

nce

Mod

els o

f C

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or

Gam

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heor

y

MULTIAGENT SYSTEMS: COOPERATIVE CONTROL







PLAN

).,...,,,,...,,(maxArg *,

*1,

*1,

*1

*niiiiiiiiiXxi xxxxxfx

iiσσσσσσ θ +−

∈∈

Informational equilibrium

Information control problem.max),(min

)(→Φ ℑ∈Ψ∈ IIx

IxX

ΨX (I) ⊆ X' – the set of real agents actions, which are stable under information structure I; Φ(x, I) – control efficiency criterion;ℑ – set of feasible information structures.

Taking into account the nontrivial mutual beliefs of agents allows:

1. (normative point of view) to enlarge the set of game’s outcomes, which in its turn increases the efficiency of information control;

2. (descriptive point of view) to describe many practically observed situations, which can not be interpreted as a Nash equilibrium under common knowledge, as informational equilibriums under proper information structure.

ГI = N, (Xi)i ∈ N, (fi(⋅))i ∈ N, Ω, I – reflexive game; Ω – set of feasible states of nature; I – information structure; N – set of players (agents); fi: Ω × X’ → ℜ1 – goal function of the i-th agent.

Information (beliefs) structure

θ1 θi θn

θi1 θij θin

… …

… …

I1

Ii1

Real agent

Phantom agent

Reflexion

. . . . . .

. . . . . .

INFORMATIONAL REFLEXION AND CONTROL

Reflexive model of group aircraft battle

Strategic behavior in collective decision-making. Suppose, that the principal is interested in the result x0∈[d;D] of expert examination. Let the opinions of n experts ri∈[d;D]i∈N, be known by the principal only, who is able to form second-order beliefs of the experts. Collective decision x = π(s), where s ∈ [d;D]n is the vector of opinions, revealed by experts. Denote: x0i(a,ri) is the solution of the following equation:π(a, …, x0, …, a) = ri , di(ri) = max d; x0i(D, ri), Di(ri) = min D; x0i(d, ri),d(r) = (d1(r1), d2(r2), …, dn(rn)), D(r) = (D1(r1), D2(r2), …, Dn(rn)).Statement 15. Any result x0∈[π(d(r));π(D(r))] may be implemented as the collective decision by means of information control with the second rank of reflexion.

SOME APPLICATIONS OF INFORMATIONAL CONTROL

INFORMATIONAL AND STRATEGIC REFLEXION

«Optimizational models of collective behavior

Models of games

Game theory

Models of collective behavior

Informational structures

Concept of equilibrium Informational equilibrium

Nash equilibrium

Control problems

Informational control

MODELS OF STRATEGIC REFLEXION

Reflexive structures k-level models;

cognitive hierarchies models, etc

Models of reflexion in

bimatrix games

Informational reflexion

Strategic reflexion

Prognostical

Reflexive equilibrium

Reflexive control

“Reflexive” models

Level

Phenomenological (descriptive)

Normative

Models of reflexive decision-making

Game theory

(super-intelligent players)

Theory of collective behavior

(rational agents)

LEVEL OF “INTELLIGENCE”

REFLEXION

«Probability» of target destruction:ü Direct attack of 40 “simple” agents: 0,125ü 8 «investigators» + 32 «reflexive» agents: 0,985ü 40 «investigators»: 0,999

Level of hierarchy

Modelled processes Modelling tools

6 Choice of agents and their characteristics

Discrete and multicriteriaoptimization

5 Choice of agents’trajectories and velocities

Optimal control

4 Forecasting by the agent opponents’behavior

Reflexive games

3 Minimization of the probabilityof detection

Optimization and heuristics of choosing the direction of motion

2 Collision and obstacle avoidance

Algorithms of “local”trajectories’ choice

1 Motion toward the target

Equations of dynamics

DIFFUSIVE BOMB MODEL: STRATEGIC REFLEXION







PLAN





Operational level

Action

Tactical level




mechanisms Ex

tern

al

i

nfor

mat

ion

Dyn

amic

syst

ems

Art

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Inte

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MULTIAGENT SYSTEMS: CONFRONTATION

MODELS OF SOCIAL NETWORKS

The set M of principals

The set of agentsN

…

Problem – find a control vector u, s.t. Φ(X, u) = H(X) – c(u) →

Uu∈max ,

H(⋅) – “gain”, с(⋅) – control costs.

Each principal (from set M) is able to influence on the initial opinions of the agents uij and is interested in forming the final opinions XM. Problem – find equilibrium of the game:

Г = (M, Ujj ∈ M, Gj(⋅)j ∈ M).

A set N of agents form social network G = (N, E). x – vector of initial opinions, X – final opinions.

aij ≥0 – the trust of i-th agent to j-th, k-th agent indirectly influences on i-th agent: xk+1 = A [xk + B uk]. Problem – find total influence of one agents on another, find the agents, that form the final opinions

3. Informational interaction (dynamics)

4. Informational control

5. Information warfare

Facebook

Twitter

Newsvine

Habr

2. Structural analysis

1. «Statistical» analysis

ANALYSIS AND MODELINGOF INFORMATIONAL PROCESSES IN SOCIAL MEDIA

11.11.11 06.12 .11 31.12.11 25.01.12 19.02 .12 15.03 .12 09.04 .12 04.05 .12 29.05.12 23.06 .12 18.07 .12

0

5000

10000

15000

Суточное количество сообщений : Политики A и B

0 10 20 30 40 50 60 70?0.4

?0.2

0.0

0.2

0.4

0.6

0.8

1.0Автокорреляционная функция по сообщениям : Политики A и B

0 20 40 60 80 100 120 140

?0.2

?0.1

0.0

0.1

0.2

0.3

0.4

0.5Парная корреляционная функция по сообщениям

Политик A Политик B

11.11 .11 06.12 .11 31.12 .11 25.01 .12 19.02 .12 15.03 .12 09.04 .12 04.05 .12 29.05 .12 23.06 .12

11

10

9

8

7

6

5

4

3

2

1Вейвлет скалограмма ?объём сообщений ?: Политик A

11.11 .11 06.12 .11 31.12 .11 25.01 .12 19.02 .12 15.03 .12 09.04 .12 04.05 .12 29.05 .12 23.06 .12

11

10

9

8

7

6

5

4

3

2

1Вейвлет скалограмма ?объём сообщений ?: Политик B

Политическая жизнь России

Политик A Политик B

Общество / Блогосфера / СМИ

Активностьблоггеров

Активностьблоггеров

11.11 .11 06.12 .11 31.12 .11 25.01 .12 19.02 .12 15.03 .12 09.04 .12 04.05 .12 29.05 .12 23.06 .12

0

20

40

60

80

100Точки статистической разладки: Политик A

11.11 .11 06.12 .11 31.12 .11 25.01 .12 19.02 .12 15.03 .12 09.04 .12 04.05 .12 29.05 .12 23.06 .12

0

100

200

300

400Точки статистической разладки: Политик B

Political activity Person A Person B

Society / Blogoshere / Media

Blogers’ activity Blogers’ activity

Daily mentioned: persons A and B

Autocorrelation by persons

Autocorrelation by messages

Person B Person A

Statistical discord: person A

Statistical discord: person B

Scalogramma: person B

Scalogramma: person A

Characteristics of maximal component• Number of agents – 362.000, • Number of “connections”– 802.000 (12.000.000 comments)Structure of components1 – Discussing component (25%). 2 - Un-popular component (45%).3 – Popular component (8%).4 - Others (22%).

The structure is time-stable:• May 2011 • November 2011• December 2011

45

The rate of information (average) spread - 5.3 steps to receive information FROM any agent;- 5.3 steps to transmit information TO any agent.

THE STRUCTURE OF MAXIMALWEAKLY CONNECTED COMPONENT

1

2

3

4

Number of removed nodes Si

ze o

f max

imal

clu

ster

(%)

Number of removed nodes

Num

ber o

f cla

ster

s

CONTROL IN SOCIAL NETWORKSCONTROL:- OPINIONS,- BELIEFS (TRUST),- REPUTATION,- …- MEMBERSHIP and STRUCTURE.

The removal of small number of the most significant nodes leads to the appearance of a great number of unconnected groups (А). But this groups are small, and the largest group is still connected (B).

Число

авторов

Число

авторов

64.000 – cumulative estimate of politically active blogers in Russian LiveJournal

A

B

BACKGROUND: CONSENSUS PROBLEM

French J.R. A Formal Theory of Social Power // Psychological Review. 1956. 63. P. 181 – 194.

Harary F. A Criterion for Unanimity in French’s Theory of Social Power / Studies in Social Power. – Michigan: Institute of Sociological Research. 1959. P. 168 – 182.

De Groot M.H. Reaching a Consensus // Journal of American Statistical Association. 1974. 69. P. 118 – 121.

Roberts F. Discrete Mathematical Models with Applications to Social, Biological, and Environmental Problems. – Prentice: Prentice Hall, 1976.

Jackson M. Social and Economic Networks. – Princeton: Princeton University Press, 2008.

BACKGROUND: GAME THEORY

Network games

Network formation games Network-based games

Networking games

Cognitive games

Social networks games

Games over the project network-schedules

Network – is a result of game-theoret ical interaction

Network is f ixed and determines the dependence of players gains from their

actions

SOCIAL NETWORKS AND LINEAR DYNAMIC SYSTEMS

xk+1 = A [xk + B uk], k = 0, 1, …

MULTIAGENT SYSTEMS

SOCIAL NETWORKS COGNITIVE MAPS

PageRank Problem

ui

xi

Φ(X, u)







PLAN

Level of hierarchy

Modeled phenomena and processes

Technique of modeling

5 Allocation of forces and means in space

Game theory(Colonel Blotto game, etc.)

4 Allocation of forces and means in time

Optimal control, repeated games, etc.

3 Number of troops dynamics

Lanchester’s equations and its modifications

2 «Local»interactionsof squads

Markovian and other stochastic models

1 Interactionof separatebattle units

Dynamic systems.Finite state automata.Simulation.

HIERARCHICAL COMBAT MODELS

20 40 60 80 100Номер ИПБ

500

1000

1500

2000МЦРMRRR

REFLEXIVE COLONEL BLOTTO GAME

Average maximal rational rank of reflexion (MRRR) ∼ 230 (nonsense!)

Denote by N = [1, …, n] the set of objects, x = (x1, …, xn) – first player’s action, y = (y1, …, yn) – second player’s action, where xi ≥ 0 (yi ≥ 0) – amount of resources, allocated by the first (second) player to i-th object, i = 1,n . Resources are limited (1) i

i Nx

∈∑ ≤ Rx, i

i Ny

∈∑ ≤ Ry.

In probabilistic Colonel Blotto Model (CBM) the probability px(xi, yi) of the first player win on the i-th objectdoes not depend on other objects and is “proportional” to the amount or resources, allocated on this object.:

(2) px(xi, yi) = ( )

( ) ( )

i

i i

ri i

r ri i i

xx yα

α +, py(xi, yi) = 1 – px(xi, yi), где ri ∈ (0; 1], αi > 0, px(xi = 0, yi = 0) =

1i

i

αα +

.

Let BRx(y) = (u1 y1 + ε, …, un yn + ε) denote the best response of first player, where n-dimentional vector u = (u1,…, un) is a solution of the

following knapsack problem: (3)0;11

1

max ,

,

i

n

i i uin

i i xi

uV

u y R

∈=

=

→

≤

∑

∑, ε =

1

1 ( )n

x i ii

R u yn =

− ∑ , i.e. let’s assume that the player tries to win on the most valuable set of

objects, and the rest of resources are equally divided among other objects. Let BRy(x) = (v1 x1 + δ, …, vn xn + δ) denote the best response of first player, where n-dimentional vector v = (v1,…, vn) is a solution of the

following knapsack problem: (4) 0;11

1

max,

,

i

n

i i vin

i i yi

vV

v x R

∈=

=

→ ≤

∑

∑, δ =

1

1 ( )n

y i ii

R u xn =

− ∑ .

Rank gameexploration

MRRR





- Optimization of hierarchies- Labor supply- Ecology VS Economy- Is truthtelling profitable?- Hierarchical automation


PLAN

MODERN TRENDS IN MAS

1) Strategicdecision-making

2) Increasing roleof game-theory

3) Benchmark scenarios and problems





Operational level

Action

Tactical level




mechanisms

Exte

rnal

inf

orm

atio

n

Dyn

amic

syst

ems

Art

ifici

al

Inte

llige

nce

Mod

els o

f C

olle

ctiv

e B

ehav

ior

Gam

e T

heor

y

MAS & STRATEGIC BEHAVIOR: As Is

Game theory

MAS, group control

Distributed optimization

Game theory

MAS, group control

Algorithmic game theory


Models of collective behavior, bounded

rationality

MAS & STRATEGIC BEHAVIOR: As Is

Game theory, mechanism design

MAS, group control

Algorithmic game theory


Models of collective behavior, bounded

rationality

Experimental economics, Behavioral

Game Theory

MAS & STRATEGIC BEHAVIOR: To Be

«MAXIMAL INTELLECTUALIZATION»

Intention to maximize the “intellectualization” is limited not only by “costs”(computational, cognitive, economical, etc), but by the inefficient “over-complexity”.

«Intellectualization»

«Costs»

«Result»

«Effect»

MAS: SOME QUALITATIVE RESULTS

1) The level of MAS’ “intellectualization” should be adequate to the problem in hand (taking into account various “costs”).

2) The intention to maximize the “intellectualization” at the higher levels of agent’s architecture leads to a centralized scheme.

3) Trend to the integration of networked MAS, game theory and artificial intelligence.

GENERAL TRENDS

1) Inter-disciplinary: control objects, methods and means of control.2) Network/hierarchical structure of controlled object, control system and

communications.3) Intra-paradigmal problems: «linearity» of development, desire to reduce the

problem to well-known, i.e. «internal» problems of any subject field. Self-isolation of different braches of control science. The demand for the creation of new adequate mathematical technique.

4) «Heuristical» applications: the concept of bounded rationality (under the lack of time, ability or necessity) – instead of optimal pseudo-optimal solutions are heuristically searched.

5) Unification:5C = Control + Computation + Communication + Cost + Cycle.

6) Heterogeneous (hierarchical, complex, integrated) modeling. Problems of models «coupling», search for common language. Generating and replicating typical solutions of control problems.







PLAN

K(ρm) = )(

maxmEy ρ∈

f0(y) → ℘∈mρ

max .

Вертикальные связи

Горизонтальные связи

А

Б В

Г

Временная сетевая

организацияC Network

organization

Network organization -a structure, where temporal connections between the elements are actualized for the period of solving certain problem.

The problem of structural synthesis: find the number of levels of hierarchy m and distribution of agents among the levels (feasible structure ρm), which maximize the efficiency f0(y), given that the agents in corresponding hierarchical game choose their equilibrium actions: Examles:

- distributed production;- IT-projects;- group control, etc.

HIERARCHICAL AND NETWORK ORGANIZATIONS

Statement. For any technological structure and any equilibrium x ∈ E1

N(ρm) there exist the set of decision-making strategies

in a game Г2(ρm0), which guarantees to all the

decision-makers the same levels of utilities, as in the initial game.

OPTIMIZATION OF HIERARCHIES

Development of the organization management structure

Data collection and processing algorithms

Design of assembly plant Task management in network structures

…

Optimization of control hierarchy







PLAN

LABOR SUPPLY (theory)

α α0

τ(α)

0

I = 1

α α0

τ(α)

0

I = 2

α α0

τ(α)

0

I = 3

α

α0

τ(α)

0 αmax

I = 4

α α0

τ(α)

0 αmax

I = 5

τ(α) – desired working time (hours/day), α – wage rate

Индекс (1999)

I25%

II53%

III6%

IV16%

Индекс (2009)

I25%

II41%

III16%

IV8%

V10%

Sample volume> 500

Sample volume> 5000

LABOR SUPPLY (“practice”)

α α0

τ(α)

0

I = 1

α α0

τ(α)

0

I = 2

α α0

τ(α)

0

I = 3

α α0

τ(α)

0 αmax

I = 4

α α0

τ(α)

0 αmax

I = 5

τ(α) – desired working time (hours/day), α – wage rate







PLAN

Economic agents (enterprises)

Environment (NATURE)

Control Center (principal)

Interaction with environment

Results of

activity

Control

State of nature

ECOLOGO-ECONOMICAL SYSTEM

Model: u ≥ 0 – production level; y ≥ 0 – security level; z(u), ϕ(y) – enterprise’s costs; H i(u, y) – principal’s income;

σi(u′, u, x, y) =

≠≠==

'

'

или,0,,

uuxyuuxyV i ;

K = 1, 2, …, k – the set of principals. Goal function of i-th principal: Φ i(σi(⋅), u, y) = H i(u, y) – σi(u, y), i ∈ K, Goal function of the enterprise: f(σi(⋅), u, y) = c u + ∑

∈σ

Kii yu ),( – z(u) – ϕ(y).

u* = arg 0

max≥u

[c u – z(u)];

Φ*i =

0,max

≥yu [Hi(u, y) – c (u* – u) + [z(u*) – z(u)] – ϕ(y)], i ∈ K;

S = u ≥ 0, y ≥ 0 | ∃ V ∈ k+ℜ : Hi(u, y) – Vi ≥ Φ*

i, i ∈ K,

∑∈Ki

iV = c (u* – u) – [z(u*) – z(u)] + ϕ(y);

Λ = u ≥ 0, y ≥ 0, V ∈ k+ℜ | Hi(u, y) – Vi ≥ Φ*

i, i ∈ K,

∑∈Ki

iV = c (u* – u) – [z(u*) – z(u)] + ϕ(y).

Φ*0 =

0,max

≥yu [ ∑

∈Kii yuH ),( – c (u* – u) + [z(u*) – z(u)] – ϕ(y)].

Theorem. The compromise set is not empty iff Φ*

0 ≥ ∑∈

ΦKi

i* .

ECOLOGY VS ECONOMY

Territory

Government

Industry

Enterprise







PLAN

The problem of project’s duration reductionModel: ∆ – the required reduction of project duration; yi – the required reduction of i-th operation duration; xi – plan of reduction of i- th operation duration; ri – «efficiency» of i-th agent; N – set of agents (operations); ci (yi, ri) = ri ϕ (yi / ri) – costs function of i-th agent; λ – rate of payment for the reduction of project duration; fi (yi, ri) = λ yi – ci (yi, ri) – agent’s goal function; s = (s1, s2, ..., sn) – agents’ message;

Theorem. The procedure of decision-making xi (∆, s) = Vsi ∆,

λ (∆, s) = ϕ' (∆ / V), where V = ∑∈Ni

is , is straightforward, minimizes

total costs and allows any decentralization.

Procedures of decision-making (examples):manipulable non-manipulable

IS TRUTH-TELLING PROFITABLE?







PLAN

HIERARCHICAL AUTOMATIZATION

INFORMATIONAL SYSTEMS

OLAP, BSC, DSS ERP MES SCM, CRM, PMS MRP2 MRP, CRP SCADA, DCS PLC, MicroPC

Finances,material

resources,personnel

MRP2

MRP, CRP

SCADA,DCS

PLC, MicroPC

INFORMATIONALSYSTEMS

Control ofTS «Optimization » Control of OS

OLAP, BSC, DSS

ERP

MES

SCM, CRM, PMS

? ?…







PLAN

PAST, PRESENT AND FUTURE OF CONTROL THEORY (objects of control)

1860-е 1900 1930 1940 1950 1960 1970 1980 1990 2000 2010 2020 …

Mechanical systems

Technical systems Organizational and informational systems

Decentralized intellectual systems

t

- Technical systems

- Economical systems

- Ecological systems

- Live systems

- Social systems

???Степень

достижения поставленных целей

«Relevancy»

“Application field”

«UNCERTAINTY» PRINCIPLE

(«Application field») x («Relevancy») ≤ Const

Mathematics

Psyghology, Sociology, Pedagogy

Economics

Biology

Chemesrty Physics

“Weak” sciences

“Strong” sciences

CONTROL THEORY

LIMITS OF SCIENCE

Thank you !

hn-models in control

Education