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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
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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” ?
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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
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Large-scale& Complex
Idea of “Glocal Control” ?
Power NWsMeteorologicalPhenomena
Bio-system
TransportationNWs
PredictionGlobal Behaviors By Locally
Real World
ControlLocally
Measurement
Locally
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Fans Temperature, Wind Sensors
GlocalControl
Measurement
Prediction
Control
Urban Heat Island Problem
Where (space resolution) When (time resolution)How (level resolution)
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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
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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
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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
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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
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OUTLINEOOUTLINEUTLINE
1. Glocal Control2. Unified Framework with Stability
Conditions 3. Cooperative Stabilization4. Robust Stability Analysis5. Hierarchical Consensus 6. Conclusion
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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)
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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
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Cyclic Pursuit
Control LawInformationNetwork
observe &pursuit
An Example : Cyclic Pursuit
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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 ?
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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
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(Tanaka et al., ASCC, 2009)
Algebraic condition
LMI feasibility problem
Extended Routh-Hurwitz
Criterion [Frank,1946]
Generalized Lyapunov inequality
Key lemma
Stability Conditions
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Extended Routh-Hurwitz CriterionExtended Routh-Hurwitz Criterion
Stability regionStability region
Numerical Example : 2nd order (1/2)
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Generalized Lyapunov inequalityGeneralized Lyapunov inequality
Numerical Example : 2nd order (2/2)
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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
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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
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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
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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
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• 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)
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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
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OUTLINEOOUTLINEUTLINE
1. Glocal Control2. Unified Framework with Stability
Conditions 3. Cooperative Stabilization4. Robust Stability Analysis5. Hierarchical Consensus 6. Conclusion
(Hara et al.: CDC-CCC2009)
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RemarksRemarks : No physical interactions: No physical interactionsmemorymemory--less feedbackless feedback
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Not stabilizable !
Cooperatively stabilizable ?
k
An Application:An Application: Inverted Pendulum
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Cooperatively stabilizable:
Solely stabilizable:
N : odd:Coop. Stab. = Solely Stab.
N : even:Coop Stab. (N=2) any N=2m
Re
Im
Property 1
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Property 2
Solely Stabilizable:Stabilizable by a real gain output feedback
Cooperatively Stablizable:Stabilizable by a complex gain output feedback
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2nd order systems:
Theorem: Coop. Stabliz. = Soley Stabiliz.
Higher order systems:
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Example :Example : Inverted Pendulum
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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
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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)
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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)
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① Cooperation realizes complex gain feedback virtually.
② Different roles of two agents are required for getting an advantage for stabilization.
Message : Cooperative Stabilization
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OUTLINEOOUTLINEUTLINE
1. Glocal Control2. Unified Framework with Stability
Conditions 3. Cooperative Stabilization4. Robust Stability Analysis5. Hierarchical Consensus 6. Conclusion
(Hara et al.: CDC2010)
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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
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Nominal system: homogeneous
Multiplicative PerturbationsMultiplicative Perturbations
Independent perturbations
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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)
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・・・
Each gene’s dynamics
Interaction structure
Linearized Gene Network Model
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Robust Stability TestRobust Stability Test
The smaller values of N, Q, and/or Rmore robust for maintaining stability
(Osawa et. al., ASCC2011) tomorrow morning
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Theorem (LMI)Theorem (LMI)
Stability for Dissipative Agents Stability for Dissipative Agents
Agent DynamicsAgent Dynamics
(Hirsch, Hara: IFAC2008)
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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
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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)
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# total agents : n1×n2×n3
Layer 3
Layer 2
Layer 1
Subsystem: Agent
Larg
er S
cale
n1
n2
n3
Hierarchical Consensus Problem
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: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
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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
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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”
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New Framework for System Theory
2D System Singular Perturbed System
Multi-resolved Systems
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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
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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
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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 ?
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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”
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