Projection based model reduction of multi-agent …trentelman/seminars/Tokyo.pdfProjection based...

Projection based model reduction of multi-agent systemsusing graph partitions

Harry L. Trentelman, N. Monshizadeh and M.K. Camlibel

Johann Bernoulli Institute for Mathematics and Computer Science,University of Groningen,

Groningen, The Netherlands.

The University of Tokyo, 2014

April 12, 2014 1 / 38

Outline

1 Introduction

2 Leader-Follower Multi-Agent Systems

3 Network Approximation by Graph Partitions

4 Preservation of Consensus

5 H2 Analysis of the Model Reduction Error and Almost EquitablePartitions

6 Directions for Future Research

April 12, 2014 2 / 38

Introduction

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General problem: given a large scale networked multi-agent systemwith leaders and followers, approximate it by a lower order network,while preserving (some of) the network structure.

Does the approximated network preserve important properties such asconsensus?

How good is the approximation? Can one compute (upper and lowerbounds on) the error?

April 12, 2014 3 / 38

Outline

1 Introduction






April 12, 2014 4 / 38

Graph theory

Weighted directed graph G “ pV ,E ,Aq, V “ t1, 2, . . . , nu, E set ofarcs, adjacency matrix A.

Laplacian of G : L “ D ´ A, D “ diagpd1, d2, . . . , dnq, di “ř

j aij .

Incidence matrix of G : R “ rrij s with

rij “

$

’

&

’

%

1 if vertex i is the head of arc j

´1 if vertex i is the tail of arc j

0 otherwise

for i “ 1, 2, . . . , n and j “ 1, 2, . . . , k (k is the total number of arcs).

Incidence matrix undirected graph: assign arbitrary orientation to theedges and take the incidence matrix of the corresponding directedgraph.

Edge weight matrix: W “ diagpw1,w2, . . . ,wkq with wj is the weightassociated to the edge (arc) j , j “ 1, 2, . . . , k .

For undirected graphs Laplacian L then L “ RWRJ.

April 12, 2014 5 / 38

Leader follower multi-agent systems

G “ pV ,E ,Aq weighted undirected graph, A “ raij s, Assume G connectedgraph.VL “ tv1, v2, . . . , vmu Ă V leaders, VF :“ V zVL followers.Leader-follower multi-agent system:

9xi “

#

řnj“1 aijpxj ´ xi q if i P VF

řnj“1 aijpxj ´ xi q ` u` if i P VL

xi P R state of agent i , u` P R external input applied to agent i “ v`. Incompact form

9x “ ´Lx `Mu,

with L the Laplacian of G , M is given by

Mi` “

"

1 if i “ v`0 otherwise.

April 12, 2014 6 / 38

Example

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9x “ ´Lx `Mu

L “

»

—

—

—

—

—

—

—

—

–

5 0 0 0 0 ´5 0 0 0 00 5 0 0 ´3 ´2 0 0 0 00 0 6 ´1 ´2 ´3 0 0 0 00 0 ´1 6 ´5 0 0 0 0 00 ´3 ´2 ´5 25 ´2 ´6 ´7 0 0´5 ´2 ´3 0 ´2 25 ´6 ´7 0 00 0 0 0 ´6 ´6 15 ´1 ´1 ´10 0 0 0 ´7 ´7 ´1 15 0 00 0 0 0 0 0 ´1 0 1 00 0 0 0 0 0 ´1 0 0 1

fi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

fl

, M “

»

—

—

—

—

—

—

—

—

–

1 00 00 00 00 00 00 10 00 00 0

fi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

fl

.

April 12, 2014 7 / 38

Problem Statement

Problem

Approximate the leader-follower multi-agent system

9x “ ´Lx `Mu

by a reduced order multi-agent system

9x “ ´Lx ` Mu

such that the spatial structure of the network is preserved.

Idea

We want to do this by reducing the size of the network graph throughclustering of the agents/vertices

April 12, 2014 8 / 38

Outline

1 Introduction






April 12, 2014 9 / 38

Petrov-Galerkin Projections

Consider an arbitrary system

9x “ Ax ` Bu, y “ Cx

with state space Rn. Let W ,V P Rnˆr (with r ă n) such that WJV “ I .Then VWJ is a projection.A reduced order model (projected model) is then given by

9x “ WJAV x `WJBu, y “ CV x

with reduced order state space Rr .General reduction framework: Krylov based, truncation methods, momentmatching use this projection with appropriate choice of matrices V andW .

Idea

Put idea of clustering of network graph into the Petrov-Galerkin framework

April 12, 2014 10 / 38

Graph partitions

V “ t1, 2, . . . , nu vertex set of a graph G .

Any nonempty subset of V is called a cell of V .

Collection of cells, given by π “ tC1,C2, . . . ,Cru, called a partition ofV if YiCi “ V and Ci X Cj “ ∅ whenever i ‰ j .

For a cell C Ă V , the characteristic vector of C is the n-dimensionalcolumn vector ppC q with

pi pC q “

#

1 if i P C ,

0 otherwise.

For a partition π “ tC1,C2, . . . ,Cru, the characteristic matrix of π is

Ppπq ““

ppC1q ppC2q ¨ ¨ ¨ ppCr q‰

.

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Graph partitions

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Vertex set: V “ t1, 2, 3, 4, 5, 6, 7, 8, 9, 10u.Cell: any nonempty subset of VPartition: π “ tt1, 2, 3, 4u, t5, 6u, t7u, t8u, t9, 10uuCharacteristic matrix:

Ppπq “

»

—

–

1 1 1 1 0 0 0 0 0 00 0 0 0 1 1 0 0 0 00 0 0 0 0 0 1 0 0 00 0 0 0 0 0 0 1 0 00 0 0 0 0 0 0 0 1 1

fi

ffi

fl

J

.

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Projection by graph partitions

A direct application of Petrov-Galerkin projections will destroy thespatial structure of the network (e.g. balanced truncation, momentmatching)

Instead: choose a graph partition, characteristic matrix Ppπq.

W :“ PpπqpPJpπqPpπqq´1

V :“ Ppπq.

Corresponding reduced order system:

9x “ ´Lx ` Mu,

L “ pPJPq´1PJLP

M “ pPJPq´1PJM

April 12, 2014 13 / 38

L is the Laplacian of a weighted directed graph, G “ pV , E , Aq.

Each cell of π in G becomes a vertex in G .

There is an arc from vertex p to vertex q in G if and only if thereexist i P Cp and j P Cq with p ‰ q such that ti , ju P E .

Therefore, G is a symmetric directed graph, i.e.pi , jq P E ô pj , iq P E .

Relationship between A and A “ rapqs given by

apq “1

|Cp|

ÿ

iPCp ,jPCq

aij ,

for p ‰ q.

The input weights in M depend on the cardinality of the leader cells.

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Relationship between the original and the reduced ordermodel

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The reduced order model: 9x “ ´Lx ` Mu

L “

»

—

—

—

–

5 ´5 0 0 0´10 23 ´6 ´7 00 ´12 15 ´1 ´20 ´14 ´1 15 00 0 ´1 0 1

fi

ffi

ffi

ffi

fl

, M “

»

—

—

—

–

0.25 00 00 10 00 0

fi

ffi

ffi

ffi

fl

.

April 12, 2014 15 / 38

Relationship between the original and the reduced ordermodel

Original graph:

Reduced graph:

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11 21 31

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April 12, 2014 16 / 38

Outline

1 Introduction






April 12, 2014 17 / 38

Preservation of consensus

We have assumed that our weighted undirected graph G is connected.Hence the multi-agent system 9x “ ´Lx reaches consensus, i.e. for alli , j P V we have xi ptq ´ xjptq Ñ 0 as t Ñ8.

Since L “ pPJPq´1PJLP, we see that

pPJPq12 LpPJPq´

12 “ pPJPq´

12 PJLPpPJPq´

12 . Thus L is similar to

a symmetric matrix, and hence has only real eigenvalues.

Moreover: the eigenvalues λi of L interlace the eigenvalues λi of L:λi ď λi ď λn´r`i for i “ 1, 2 . . . , r .

In particular λ2 ď λ2 so the reduced order system 9x “ ´Lx reachesconsensus with rate of convergence at least as fast as the originalnetwork.

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Outline

1 Introduction






April 12, 2014 19 / 38

Input-Output approximation of networks

Recall G “ pV ,E ,Aq undirected weighted graph. Incidence matrix R,edge weighting matrix W .

Original model: 9x “ ´Lx `Mu

We add output variables as: y “ W12 RJx

}y}2 “ xJLx

Example:

y “

„

2 00 5

12„

1 ´1 00 1 ´1

»

–

x1x2x3

fi

fl

1 2 32 5

Output variables after projection: y “ W12 RJPx

In this example: y “

„

2 00 5

12„

0 01 ´1

„

x1x2

April 12, 2014 20 / 38

Model reduction error

Original model:

9x “ ´Lx `Mu

y “ W12 RJx

Reduced model:

9x “ ´Lx ` Mu

y “ W12 RJx

Transfer matrices Spsq “ W12 RJpsI ` Lq´1M and

Spsq “ W12 RJpsI ` Lq´1M

Approximation error }S ´ S}2 (H2-norm). Note S and S are in H2.

Problem: for a given partition, find an expression for this error. Finda priori upper or lower bounds.

First for almost equitable partitions.

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Almost equitable partitions

Definition

G “ pV ,E q unweighted undirected graph. For a given cell C Ď V :Npi ,C q “ tj P C | ti , ju P Eu.Partition π “ tC1,C2, . . . ,Cru called an almost equitable partition (AEP)of G if for each p, q P t1, 2, ¨ ¨ ¨ , ru with p ‰ q there exists an integer dpq

such that |Npi ,Cqq| “ dpq for all i P Cp.

For weighted graphs, the notion of almost equitability is extended asfollows:

Definition

Let G “ pV ,E ,Aq be a weighted undirected graph. Recall aij indicates theweight associated to the edge ti , ju. Partition π “ tC1,C2, . . . ,Cru calledan almost equitable partition (AEP) of G if for each p, q P t1, 2, ¨ ¨ ¨ , ruwith p ‰ q there exists dpq P R such that

ř

jPNpi ,Cqqaij “ dpq for all

i P Cp.

April 12, 2014 22 / 38

Almost equitable partitions

An almost equitable partition can be best described by an example:

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π “ tt1, 2, 3, 4u, t5, 6u, t7u, t8u, t9, 10uu is an an almost equitablepartition

April 12, 2014 23 / 38

Model reduction error

Let π “ tC1,C2, . . . ,Cru be an AEP of G , VL “ tv1, v2, . . . , vmuleader vertices

Let ki be s.t. vi P Cki (i “ 1, 2 . . . ,m)

Theorem

The (normalized) model reduction error is given by

E pπq “}S ´ S}22}S}22

“

řmi“1p1´

1|Cki

|q

mp1´ 1n q

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Example

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Let VL “ t1, 7u

Then,

E pπq “

řmi“1p1´

1|Cki

|q

mp1´ 1n q

“p1´ 1

4q ` p1´11q

2p1´ 110q

“5

12

April 12, 2014 25 / 38

Outline of the proof

Spsq “ W12 RJpsI ` Lq´1M and Spsq “ W

12 RJpsI ` Lq´1M.

∆psq :“ Spsq ´ Spsq. Recall E pπq “ }∆}22{}S}22

π almost equitable partition ô impPq is L-invariant ñ∆Jp´sqSpsq “ 0

Using this: }S}22 “ }S}22 ` }∆}

22

}S}22 “ tracepMJX1Mq and }S}22 “ tracepMJY1Mq, where X1 and

Y1 are the Grammians of pW12 RJ, Lq and pW

12 RJ, Lq, respectively

X1 “12 In ´

12n1n1Jn , Y1 “ PTX1P

ñ }S}22 “m2 p1´

1n q and }S}22 “

12

řmi“1

1|Cki

|´ m

2n

}∆}22 “m2 p1´

1n q ´

12

řmi“1

1|Cki

|` m

2n “12

řmi“1p1´

1|Cki

|q

April 12, 2014 26 / 38

Remarks

0 ď E pπq ď 1

E pπq “ 0 for π “ tt1u, t2u, . . . , tnuu.

E pπq “ 1 for π “ tV u.

E pπq determined by the cardinalities of those cells in π that containthe leaders

Corollary

If π is an almost equitable partition with the propery that tviu P π for alli “ 1, 2 . . . ,m (i.e. each leader is the unique element in its cell) thenE pπq “ 0.

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Arbitrary partitions

What if we start with an arbitrary (not necessarily AEP) partition? Inthat case the closed form expression fails. But: can we make anestimate of E pπq?

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π “ tt1, 2, 3, 4, 5, 6u, t7, 8u, t9, 10uu is not an an almost equitablepartition.

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Partial ordering of partitions

Let G “ pV ,E ,Aq be a weighted undirected graph. Partial orderingon all partitions of G . Let π1 and π2 be two partitions. We sayπ1 ď π2 if every cel in π1 is a subset of a cel in π2: π1 is finer thenπ2, π2 is coarser than π1.

In terms of the characteristic matrices: π1 ď π2 if and only ifimpPpπ2qq Ă impPpπ1qq.

Using the fact that for any AEP π0, impPpπ0qq is L-invariant weobtain:

Theorem

Let π0 be an AEP of G . Then for every partition π that is coarser than π0we have E pπ0q ď E pπq.

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Estimate of the error for arbitrary partitions

Let π be an arbitrary partition of G . Idea: find a finer AEP of Gwhich is ”as close as possible” to π.

Define ΠAEPpπq :“ tπ0 | π0 AEP and π0 ď πu.

Since ”ď” is a partial ordering, this set contains a unique maximalelement π˚AEPpπq, the maximal almost equitable partition finer thanπ. Thus we get:

Corollary

Let π be a partition of G and let π˚AEPpπq be the maximal AEP finer thanπ. Then E pπ˚AEPpπqq ď E pπq.

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Computation of π˚AEPpπq

Let G be a weighted undirected graph with vertex setV “ t1, 2, . . . , nu and π a partition. How to compute π˚AEPpπq?

Let Π be the set of all partitions of G . Define a map Ψ : Rnˆ‚ Ñ Πas follows: for X P Rnˆ‚, i , j P V are in the cell ΨpX q if and only ifthe the ith and jth row of X are the same.

Example:

Ψp

»

–

2 0 4 12 0 4 10 5 2 0

fi

flq “ 1 2 32 5

Ψp

»

–

2 0 4 1 10 5 2 0 92 0 4 1 1

fi

flq “ 1 2 32 5

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Computation of π˚AEPpπq

Theorem

Let π be a partition of G . Define a sequence of partitions tπku by

π0 “ π

πk`1 “ Ψp“

Ppπkq LPpπkq‰

q

pk “ 1, 2, . . .q. Then

1. π0 ě π1 ě π2, . . ..

2. There exists q with 0 ď q ď n such that πq “ πq``for all ` ą 0.

3. For any such q we have πq “ π˚AEPpπq.

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Example

π “ tt1, 2, 3, 4, 5, 6u, t7, 8u, t9, 10uu (not an an almost equitablepartition).

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Example

Ppπq “

»

—

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—

—

—

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—

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—

—

–

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, LpPpπqq “

»

—

—

—

—

—

—

—

—

—

—

—

—

—

—

–

0 0 00 0 00 0 00 0 0

13 ´13 013 ´13 0´12 14 ´2´14 14 0

0 0 10 0 1

fi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

fl

By inspection we see thatπ1 “ Ψp

“

Ppπq LPpπq‰

q “ tt1, 2, 3, 4u, t5, 6u, t7u, t8u, t9, 10uu,which is the partition we studied before:

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Example

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This partition is almost equitable, and therefore equal to π˚AEPpπq.

Recall that with VL “ t1, 7u we had computed E pπ˚AEPpπqq “512 . Let

E pπq be the error associated with π. Conclusion: E pπq ě 512 .

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Outline

1 Introduction






April 12, 2014 36 / 38

Directions for future research

Extension to directed weighted graphs.

Instead of a lower bound on the error associated with an arbitrarypartition we would like to find upper bounds.

Extension to the H8 norm of the error transfer matrix.

Extension to the discrete time case.

Extension to multi-agent systems with arbitrary homogeneous(heterogeneous) linear dynamics at the vertices.

Combination of our clustering method with model reduction of thedynamics of the vertices.

April 12, 2014 37 / 38

THANKS FOR YOUR HOSPITALITY!

April 12, 2014 38 / 38

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