1

Glocal ControlA Realization of Global Functions

by Local Measurement and Control

GlocalGlocal ControlControlA Realization of Global Functions A Realization of Global Functions

by Local Measurement and Control by Local Measurement and Control

Shinji HARAShinji HARAThe University of Tokyo, Japan

The 8th Asian Control ConferenceKaohsiung, Taiwan

May 16, 2011

2

Recently, systems to be treated in various fields of engineering including control have became large and complex, and more high level control such as adaptation against changes of environments for open systems is required. Typical example includes meteorological phenomena and bio systems, where our available actions of measurement and control are restricted locally although our main purpose is to achieve the desired global behaviors.

This motivates us to develop a new research areanew research area so called "GlocalGlocal ControlControl," which means that the desired global behaviors is achieved by only local actions.

Why “Glocal Control” ?

3

Advanced Science & Technology Driven by Control

GlocalGlocalControlControl

ClassicalControl

ModernControl

RobustControl

HybridControl

Stabilization

Automation

High Performance

Multiple Functions

Steel process

Engine control

Linear motor car

Robotics

Bio-system

MeteorologicalPhenomena

Transportation

Power NWs

Realization of High Quality Products Solving Social Problems such as

Energy, Environments, and Medicine

4

Large-scale& Complex

Idea of “Glocal Control” ?

Power NWsMeteorologicalPhenomena

Bio-system

TransportationNWs

PredictionGlobal Behaviors By Locally

Real World

ControlLocally

Measurement

Locally

5

Fans Temperature, Wind Sensors

GlocalControl

Measurement

Prediction

Control

Urban Heat Island Problem

Where (space resolution) When (time resolution)How (level resolution)

6

Proteins GeneNetwork

Cell Communityof Cells

HGMIS(www.ornl.gov)

Layer 2: CELL SignalingNetworks

Layer 3: METABOLICNetworks

Hierarchical Bio-Network Systems

Larg

er S

cale

Layer 1: GENE RegulatoryNetworks

Cyclic Structure

: Subsystem 1 : Subsystem 2: Subsystem 3 : Subsystem 4

Hierarchical Bio-Network Systems

7

Power NWsMeteorologicalPhenomena

Bio-system

Prediction

Real World

ControlLocally

Measurement

Locally

TransportationNWs

Framework for Glocal Control

Physical Network

Dynamical Logical Network

(Measurement, Prediction, Control)

① Dual Network

② Hierarchical Dynamical Systems with Multi-resolution

8

① Paradigm Shift in ControlRealization of High Quality Products Solving Social Problems

② New Unified Framework

③ New Control Theory for Systematic Waysof Analysis and SynthesisFundamental Results & New Notions/PrinciplesPractical Applications

Three Key Issues

LTI System with Generalized Frequency VariableHierarchical Dynamical System with Multiple Resolutions

GlocalControl

9

① Paradigm Shift in ControlRealization of High Quality Products Solving Social Problems

② New Unified Framework

③ New Control Theory for Systematic Waysof Analysis and SynthesisFundamental Results & New Notions/PrinciplesPractical Applications

LTI System with Generalized Frequency VariableHierarchical Dynamical System with Multiple Resolutions

Idea of GlocalControl

Toward GlocalControl

Three Key Issues

10

OUTLINEOOUTLINEUTLINE

1. Glocal Control2. Unified Framework with Stability

Conditions 3. Cooperative Stabilization4. Robust Stability Analysis5. Hierarchical Consensus 6. Conclusion

11

OUTLINEOOUTLINEUTLINE

1. Glocal Control2. Unified Framework with Stability

Conditions 3. Cooperative Stabilization4. Robust Stability Analysis5. Hierarchical Consensus 6. Conclusion

(Hara et al.: CDC2007, Tanaka et al.: ASCC2009)

12

LTI System with Generalized Frequency Variable

A unified representation for multi-agent dynamical systems

…Group RobotGroup Robot Gene Reg. NetworksGene Reg. Networks

1/s 1/s h(sh(s))

ΦΦ(s(s) = 1/h(s)) = 1/h(s)

Generalized Freq. Variable

Dynamics +

Information Structure

13

Cyclic Pursuit

Control LawInformationNetwork

observe &pursuit

An Example : Cyclic Pursuit

14

Define: Domains

Re

Im

Re

Im

Re

Im

(Hara et al. IEEE CDC, 2007)

All eigenvaluesbelong to…

Stability Region for LTISwGFV

How to characterize the region ? How to characterize the region ? How to check the condition ?How to check the condition ?

15

Graphical Algebraic Numeric（LMI）

Nyquist – type Hurwitz – type Lyapunov – type

Polyak & Tsypkin (1996)Fax & Murray (2004)

Hara et al. (2007)

Tanaka, Hara,Iwasaki (ASCC2009)

Tanaka, Hara,Iwasaki (ASCC2009)

Hurwitz test for complex

coefficients

GeneralizedLyapunovInequality

Stability Tests for LTISwGFV

16

(Tanaka et al., ASCC, 2009)

Algebraic condition

LMI feasibility problem

Extended Routh-Hurwitz

Criterion [Frank,1946]

Generalized Lyapunov inequality

Key lemma

Stability Conditions

17

Extended Routh-Hurwitz CriterionExtended Routh-Hurwitz Criterion

Stability regionStability region

Numerical Example : 2nd order (1/2)

18

Generalized Lyapunov inequalityGeneralized Lyapunov inequality

Numerical Example : 2nd order (2/2)

19

Algorithm

Given Data: all coefficients of numerator and denominator of h(s)

Systematic methods

for stability analysis

Hurwitz-type & LMI

Systematic methods

for stability analysis

Hurwitz-type & LMI

Result:

20Δ3 > 0

Numerical Example : 4th order

Δ1 > 0

Δ2 > 0 Δ4 > 0

Unstable & NMP

Stability Region

21

An Application : Biological rhythms

• Biological rhythms– 24h-cycle, heart beat, sleep cycle etc.– caused by periodic oscillations of protein

concentrations in Gene Regulatory Networks

• Medical and engineering applications– Artificially engineered biological oscillators

(e.g.) Repressilator [Elowitz & Leibler, Nature, 2000]

PB PC2 PC1PE

PD PG

PH1 PH2 PH3

PF

PJ

PIGene B Gene C

Gene D

Gene E

Gene F

Gene G

Gene H

Gene I

Gene J

protein A

protein C

protein F

protein D

protein GPAGene A

Motivation

22

What are analytic conditions for convergence and the existence of oscillations ?

Time [s] Time [s]C

once

ntra

tion

Con

cent

ratio

n

Cyclic Gene Regulatory Networks

・・・DNA

mRNA

protein

convergence Oscillation

23

The cyclic GRN has periodic oscillations if at least one of eigenvalue of lies inside where

TimeP

rote

in c

once

ntra

tion

Locallyunstable

Condition for Existence of Oscillations

24

• Assumptions: All interactions are repressive

The cyclic GRN has periodic oscillations, if an analytic

condition in terms of is satisfied.

Theorem [Hori et al., CDC2009]

Analytic Criteria

(Hori, Hara: CDC2010)

25

① LTI system with generalized freq. variablea proper class of homogeneous multi-agent dynamical systems

② Three types of stability tests, namely graphical, algebraic, and numeric (LMI)

③ Parametric stability analysis for gene regulatory networks

new biological insight

Message : Framework and Stability

powerful tools for analysis

26

OUTLINEOOUTLINEUTLINE

1. Glocal Control2. Unified Framework with Stability

Conditions 3. Cooperative Stabilization4. Robust Stability Analysis5. Hierarchical Consensus 6. Conclusion

(Hara et al.: CDC-CCC2009)

27

RemarksRemarks : No physical interactions: No physical interactionsmemorymemory--less feedbackless feedback

AA

Not stabilizable !

Cooperatively stabilizable ?

k

An Application:An Application: Inverted Pendulum

28

Cooperatively stabilizable：

Solely stabilizable：

N : odd：Coop. Stab. = Solely Stab.

N : even：Coop Stab. (N=2) any N=2m

Re

Im

Property 1

29

Property 2

Solely Stabilizable：Stabilizable by a real gain output feedback

Cooperatively Stablizable：Stabilizable by a complex gain output feedback

30

2nd order systems:

Theorem: Coop. Stabliz. = Soley Stabiliz.

Higher order systems:

31

Example :Example : Inverted Pendulum

32

An example：Cope. Stab. ＝ Soley Stab.

Stability Region

0 0.5 1 1.5 2 2.5 3-15

-10

-5

0

5

10

15

Time

Control Law

Leader

Follower

33

We can prove by a symbolic computation (QE) that the system can not be stabilized alone no matter how we choose T>0 .

Inverted Pendulum : PD control (1/2)

34

Stability Region

A =

T=1/2 :

0 20 40 60 80 100−0.04

−0.03

−0.02

−0.01

0

0.01

0.02

0.03

Time[s]

thet

a[ra

d]

pendulum1pendulum2

Inverted Pendulum : PD control (2/2)

35

① Cooperation realizes complex gain feedback virtually.

② Different roles of two agents are required for getting an advantage for stabilization.

Message : Cooperative Stabilization

36

OUTLINEOOUTLINEUTLINE

1. Glocal Control2. Unified Framework with Stability

Conditions 3. Cooperative Stabilization4. Robust Stability Analysis5. Hierarchical Consensus 6. Conclusion

(Hara et al.: CDC2010)

37

FromFrom StabilityStability to

・Robust StabilityRobust Stability ?* homogeneous heterogeneous* physical inter-agent interactions

・Control PerformanceControl Performance ?H∞-norm Computation ?

Fundamental Questions in ControlFundamental Questions in Control

Robust Stability for LTI Systems with GFV

38

Nominal system: homogeneous

Multiplicative PerturbationsMultiplicative Perturbations

Independent perturbations

39

AssumptionAssumption

Robust Stability Condition for Robust Stability Condition for Heterogeneous PerturbationsHeterogeneous Perturbations

TheoremTheorem:: The following conditions are equivalent. The following conditions are equivalent.

(Hara et al.: CDC2010)

40

・・・

Each gene’s dynamics

Interaction structure

Linearized Gene Network Model

41

Robust Stability TestRobust Stability Test

The smaller values of N, Q, and/or Rmore robust for maintaining stability

(Osawa et. al., ASCC2011) tomorrow morning

42

Theorem (LMI)Theorem (LMI)

Stability for Dissipative Agents Stability for Dissipative Agents

Agent DynamicsAgent Dynamics

(Hirsch, Hara: IFAC2008)

43

① Methods of robust stability analysis for standard systems such as D-scaling work well for stability analysis for heterogeneous multi-agent dynamical systems.

② Although the results are not complete, there are many potential practical application fields to which we can apply them.

Message : Robust Stability

44

OUTLINEOOUTLINEUTLINE

1. Glocal Control2. Unified Framework with Stability

Conditions 3. Cooperative Stabilization4. Robust Stability Analysis5. Hierarchical Consensus 6. Conclusion

(Shimizu, Hara: SICE2008, Hara et al.: ACC2009)

45

# total agents : n1×n2×n3

Layer 3

Layer 2

Layer 1

Subsystem: Agent

Larg

er S

cale

n1

n2

n3

Hierarchical Consensus Problem

46

：Cyclic Pursuit inside sub-group

Homogeneousstructure

Fractalstructure

Property on Interactions

weak interactionweak interaction:Sparse

Small gain

Share an aggregated informationControl uniformly

Low Rank Interaction:

Hierarchical Structure

47-5 -4 -3 -2 -1 0 1

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

Re

Im

Δ= 1⋅ ζT

Δ= I

○：rank 1

×：Identity

n1=25> n2=4

⊿: Rank 1

Eigenvalue Distributions

480 1 2 3 4 5 6

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Time

x-P

osi

tion

0 1 2 3 4 5 60

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Time

x-P

osi

tion

⊿ = I

⊿: Rank 1n1 > n2Rapid

Consensus

Time Responses (n1=25, n2=4)

Aggregation Distribution

49

① Proper ways of aggregation and distribution are important to achieve rapid consensus.

② Low rankness of interlayer connection captures them properly.

Message : Hierarchical Consensus

50

A Unified Framework for Decentralized Cooperative Control

of Large Scale Networked Dynamical Systems

Dynamical System with Generalized Frequency Variable

Key Idea :Key Idea :

Stability & Robust Stability AnalysisCooperative Stabilization

Hierarchical caseHierarchical caseNonNon--linear caselinear caseControl Performances, Synthesis Control Performances, Synthesis ??

ExtensionsExtensions ::

Toward “Glocal Control”

51

New Framework for System Theory

2D System Singular Perturbed System

Multi-resolved Systems

52

MR model

HR model

LR model

Local input

LocalControl

input

Local measurement (HR)

Global measurement (LR)

Glocal Adaptor

集約予測器（低分解能）

集約予測器（中分解能）

分散予測器（高分解能）

Glocal Predictor

Desired Global

Behavior

Large scale complex system

Inversemodel

Generator for local reference

commands

Image of Glocal Control System

HierarchicalDistributedController

53

Smart Energy NW and Energy Saving

http://tinycomb.com/wp-content/uploads/2009/05/smart-grid.jpg

Electric power network + Gas energy network

Smart Energy Network

Hierarchical Air Conditioning System

Group of buildingsSet of floorsSet of rooms

54

Evacuation Guidance for Tsunami

Wave measurement system by GPS

GPS Wave

Sensor

How to set up GPS wave sensors to predict the time and height of “tsunami” properly for effective evacuation guidance ?

Optimal time-, space-, level- resolution ?

55

① Glocal ControlJun-ichi Imura (Tokyo Tech.)

Koji Tsumura (U. Tokyo)

Koichiro Deguchi (Tohoku U.)

② LTI Systems with Generalized Freq. Vars.Tetsuya Iwasaki (UCLA)

Hideaki Tanaka (U. Tokyo)

Masaaki Kanno (Niigata U.)

③ Gene Regulatory NetworksYutaka Hori (U. Tokyo)

Tae-Hyoung Kim (Chung-Ang U.)

Acknowledgements

56AA

Thank you Thank you very much !very much !

Please join us to develop “Glocal Control Theory”

and to solve social problems through “Glocal Control”