Clustering-based State Aggregation of Dynamical Networks

Transcription

1 Clustering-based State Aggregation of Dynamical Networks Takayuki Ishizaki Ph.D. from Tokyo Institute of Technology (March 2012) Research Fellow of the Japan Society for the Promotion of Science More than 10 hours From Tokyo to Stockholm I am (was?) tennis man

2 Model Reduction via Projection Given stable system Find Stable reduced model input-to-state map state-to-output map Dim. of state: Find such that is small enough 2/22

3 Typical System Norms : Laplace transform : Frobenius norm Stable system Solution Impulse response Transfer function -norm Maximum gain -norm Energy of -norm of 3/22

4 Contents Clustering-based State Aggregation in terms of -norm How to reduce systems while preserving network topology? Use of positive tri-diagonalization Application to diffusion process over complex network -aggregation of Positive Networks Preservation of network topology as well as positivity Use of controllability gramian Application to Chemical Master Equation 4/22

5 System Description [Definition] Bidirectional Network with and is said to be bidirectional network if is symmetric and stable. Including reaction-diffusion systems: : reaction of : diffusion between and 5/22

6 Traditional Model Reduction Traditional model reduction methods Balanced truncation, Krylov projection, Hankel norm approximation No specific structure in transformation matrix Drawback: Network structure (spatial information) is destroyed Given : Sparse Reduced model : Dense : Dense Need to impose suitable sparse structure on 6/22

7 Clustering-based State Aggregation Aggregation of disjoint sets of states (clusters) Block-diagonally structured aggregation matrix Interconnection topology among clusters is preserved Coarse Cluster Fine How to find reducible clusters? For simplicity, Aggregation = Averaging: 7/22

8 Key Observation to Construct Reducible Clusters Given 50 th order [State trajectory under random input] Clustering accordingly to behavior About trajectories can be aggregated into dimensional variables?? [Definition] Reducible Cluster A cluster is said to be reducible if Cluster under where any input signal.

9 Positive Tri-diagonalization [Lemma] For every bidirectional network, there exists a unitary such that has the structure below. Bidirectional network Positive tri-diagonal realization (not necessarily positive) Metzler for all

10 Reducibility Characterization Bidirectional network : positive tri-diagonal realization : transformation matrix Index matrix reducible reducible : DC-gain Maximal gain due to positivity identical identical Cluster reducibility is characterized by rows of 10/22

11 Reducibility: Reducible Cluster Aggregation [Theorem] A cluster is reducible iff Furthermore, if all clusters are reducible, then holds. Coarse Aggregated model with Dynamical network Fine Relaxation to?? 11/22

12 Reducibility Relaxation [Definition] -reducible Cluster for row vector A cluster is said to be -reducible if : coarseness parameter [Theorem] If all clusters are -reducible, then holds where linear dependence on Preservation: Stability and Interconnection topology among clusters In addition, represents average of original state 12/22

13 Cluster Set Construction Give, Initialize While Choose, Set For all, if satisfies, then - reducibility : where Cluster set to be obtained is not necessarily unique 13/22

14 Numerical Example Diffusion process over the Holme-Kim model (3000 th dim.) (less than 0.5% error) if Dim. of reduced model Error bound The value of 276 clusters (276 th dimensional model) 14/22

15 Contents Clustering-based State Aggregation in terms of -norm How to reduce systems while preserving network topology? Use of positive tri-diagonalization Application to diffusion process over complex network -aggregation of Positive Networks Preservation of network topology as well as positivity Use of controllability gramian Application to Chemical Master Equation 15/22

16 System Description [Definition] Positive Network with and is said to be positive network if is Metzler and (marginally) stable, and. Metzler matrix : having non-negative off-diagonal entries : non-negative non-negative property e.g., Heat diffusion systems, Electric circuit systems, Markovian processes Model reduction while preserving positivity, stability and network 16/22

17 Reducibility Characterization ( -case) Given with stable Controllability gramian Lyapunov equation reducible reducible Cholesky factorization identical identical Cluster reducibility is characterized by rows of 17/22

18 18/22 -State Aggregation [Definition] -reducible Cluster The cluster is said to be -weakly reducible if : coarseness parameter [Theorem] If all clusters are -weakly reducible, then holds where linearly bounded by Preservation: Stability, Positivity, Interconnection topology among clusters

19 Generalization to Marginally Stable Positive Networks Gramian is not defined if Projected gramian has zero-eigenvalue where is orthogonal complement of such that Controllability gramian of stable projected system Unique positive semi-definite matrix for Cholesky factorization identical identical : Graph Laplacian 19/22

20 Application to Chemical Master Equation(CME) Michaelis-Menten system ex) Initial number of are both [Realizable distributions] : reaction rate constant Realization probability State with CME expression: Continuous-time Markovian process off-diagonal entries of are non-negative column sums of are zero (zero-eigenvalue) th dimensional 20/22

21 21/22 Numerical Example th order if th order [Reduced order versus ] [Trajectories of and ] Validates as coarseness parameter : solid lines : dot lines Relative error of in -norm:

22 Summary Clustering-based State Aggregation positive tri-diagonalization leads to -aggregation controllability graman leads to -aggregation Preserving interconnection topology as well as stability, positivity Application to diffusion process over complex networks and CMEs Coarse Aggregated model with Dynamical network Fine 22/22