Lecture Series, TU Munich October 22, 29 & November 5, 2013 Glocal Control for Hierarchical Dynamical Systems Theoretical Foundations with Applications in Energy Networks Shinji HARA The University of Tokyo, Japan
OUTLINE 1. Glocal Control & Energy Networks 2. A Unified Framework for Networked Dynamical Systems with Stability Analysis 3. From Homogeneous to Heterogeneous 4. From Frat to Hierarchical 5. Decentralized Hierarchical Control Synthesis 6. Applications in Energy Networks 2
Framework for Glocal Control Realization of Global Functions by Local Measurement and Control Real World Glocal Control System Local Control Meteorological Phenomena Power NWs Bio-Systems LR model Transportation MR model HR model Local Measurement 3
Messages : A New Framework 1 LTI system with generalized freq. variable a proper class of homogeneous multi-agent dynamical systems 2 Three types of stability tests, namely graphical, algebraic, and numeric (LMI) powerful tools for analysis Q3: from Homogeneous to Heterogeneous? Q4: from Flat Structure to Hierarchical Structure? 4
From Frat to Hierarchical Structures Larger Scale : Agent Layer 3 Layer 2 Layer 1 Subsystem n3 n2 n1 5
OUTLINE : Part 4 4. From Frat to Hierarchical Low-rank Interlayer Connections Hierarchical Consensus for Heterogeneous Networks Hierarchical Adaptive Consensus 6
OUTLINE : Part 4 4. From Frat to Hierarchical Low-rank Interlayer Connections Hierarchical Consensus for Heterogeneous Networks Hierarchical Adaptive Consensus (Shimizu, Hara: SICE2008) Q9: What are Key Properties in Hierarchical Systems? 7
Hierarchical NW Dynamical Systems Larger Scale : Agent Layer 3 Layer 2 Layer 1 Subsystem n3 n2 n1 # total agents : n1 n2 n3 8
Hierarchical Structure Homogeneous structure Upper-layer structure Property on Interactions Low Rank Interaction: Share an aggregated information Control uniformly weak interaction: Sparse Small gain 9
Eigenvalue Distributions : Rank 1 2 1.5 1 0.5 = 1 T = I n1 = 25 > n2 = 4 Im 0-0.5-1 -1.5 :rank 1 :Identity -2-5 -4-3 -2-1 0 1 Re 10
Time Responses (n1=25, n2=4) x-position 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 : Rank 1 Rapid n1 > n2 Consensus 0.2 0.1 0 0 1 2 3 4 5 6 Time x-position 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 = I 0 0 1 2 3 4 5 6 Time Distribution Aggregation Low-rank Interlayer Interactions Multiple resolution 11
General Case with h(s) = I Stability Boundary : Rank 1 : = I *: = Rank 1 12
OUTLINE : Part 4 4. From Frat to Hierarchical Low-rank Interlayer Connections Hierarchical Consensus for Heterogeneous Networks Hierarchical Adaptive Consensus (Fujimori et. al., CDC2011) Q10: What is a General Framework for Heterogeneous Network Synthesis? 13
Hierarchical Structure Design, and Local connection Distribution of information Aggregation of information 14
Hierarchical Structure Design, and Hierarchical structure is compressed into matrix 15
Homogeneous vs Heterogeneous Homogeneous Heterogeneous can be written using Kronecker product. We need Khatori-Rao product. 16
Eigen-connection Matrix is right eigen-connection matrix of associated with eigenvalue Def ( is the right eigenvector associated with ) Analogously, left eigen-connection matrix can also be defined by using left eigenvector. 17
Assumption Theorem : Rank 1 Case has at least one simple eigenvalue is a right eigen-connection matrix of associated with eigenvalue Theorem: Rank 1 For any, the set of all the eigenvalues of is given by The eigenvalues of local interconnection The eigenvalues determined by hierarchical structure An analogous result is obtained for left eigen-connection matrices. 18
Numerical Examples (1/2) Without Control y 7 6 5 4 3 2 1 0 7 0 2 4 6 8 10 Time[s] y^1 y^2 6 5 Rank 1 y 4 3 2 1 0 y^1 y^2 0 2 4 6 8 10 Time[s] : eigenvalue of 19
Numerical Examples (2/2) 12 10 Without Control 8 6 y 4 2 0-2 y^1 y^2 0 2 4 6 8 10 Time[s] 8 6 Rank 1 10 8 Rank 2 y 4 y 6 2 0 y 1 y 2 0 5 10 15 20 Time[s] 4 2 0 y^1 y^2 0 2 4 6 8 10 12 14 Time[s] 20
Assumption Theorem : Rank 2 Case has at least two simple eigenvalues is a right eigen-connection matrix of associated with eigenvalue Theorem: Rank 2 For any, the set of all the eigenvalues of is given by An analogous result is obtained for left eigen-connection matrices. 21
OUTLINE : Part 4 4. From Frat to Hierarchical Low-rank Interlayer Connections Hierarchical Consensus for Heterogeneous Networks Hierarchical Adaptive Consensus (Fujimori et.al., SICE2011) 22
Stability for Dissipative Agents Agent Dynamics Theorem (LMI) (Hirsch, Hara: IFAC2008) 23
Passive Systems : Non-hierarchical Case Nonlinear Dynamics Goal of Consensus Conformation of Outputs Assumption Passivity Graph -L is Strongly Connected with positive edges Theorem All agents achieve consensus robustly for unknown positive weights. 24
Passive Systems : Non-hierarchical Case Nonlinear Dynamics Goal of Consensus Conformation of Outputs Assumption Passivity Graph -L is Strongly Connected with positive edges Theorem All agents achieve consensus robustly for unknown positive weights. 25
Passive Systems : Hierarchical Case Regard S. C. multiple pendulums as a subsystem Subsystems exchange the weighted sum of information :input weight :output weight 26
An Example of Hierarchical Structure Whole Graph Laplacian aggregation There are Negative Edge! We need another criterion Q: How can we decide the weights so that the subsystem is passive? 27
Consensus of Hierarchical Structure Assumption Passivity subsystem: S. C. satisfy Proposition All the subsystems are Passive The positive left eigenvector of the graph Laplacian representing the connection inside the subsystem associated with the eigenvalue 0. Outputs of subsystems achieve consensus Agents in each subsystem achieve consensus When sums of output weights are coincident, all the agents achieve consensus. 28
Numerical Simulation (1/2) Case1) The Assumption is satisfied aggregation Sum of output weight is equivalent Angular velocity[rad/s] 1.5 1 0.5 0-0.5-1 -1.5-2 -2.5 0 2 4 6 8 10 12 14 29
Numerical Simulation (2/2) Case2) The Assumption is not satisfied aggregation Angular velocity[rad/s] 5 4 3 2 1 0-1 -2-3 -4-5 86 88 90 92 94 96 98 100 Time[s] 30
Three Messages 1 Low rank interlayer connections are quite helpful for rapid consensus. aggregation 2 Heterogeneous agents: Khatri-Rao Product hierarchical network synthesis based on left eigenvectors 3 Nonlinear agents: left eigenvector strongly connected graph in the upper layer + subsystems which can be passive 31
Messages : A New Framework 1 LTI system with generalized freq. variable a proper class of homogeneous multi-agent systems 2 Three types of stability tests, namely graphical, algebraic, and numeric (LMI) powerful tools for analysis 3 From Homogeneous to Heterogeneous robust stability analysis (Hinf norm condition) 4 From Flat to Hierarchical Structure low-rank interlayer connection (aggregation & distribution) How to Design Decentralized Control Systems Systematically? 32