research article algorithm for identifying minimum driver...

Research ArticleAlgorithm for Identifying Minimum Driver Nodes Based onStructural Controllability

Reza Haghighi1 and HamidReza Namazi2

1Department of Electrical and Computer Engineering National University of Singapore Singapore 1190772Department of Mechanical Engineering Faculty of Engineering Universiti Malaysia Sarawak 94300 Kuching Malaysia

Correspondence should be addressed to HamidReza Namazi m080012entuedusg

Received 15 May 2015 Accepted 30 July 2015

Academic Editor Hou-Sheng Su

Copyright copy 2015 R Haghighi and H Namazi This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

Existing methods on structural controllability of networked systems are based on critical assumptions such as nodal dynamics withinfinite time constants and availability of input signals to all nodes In this paper we relax these assumptions and examine the struc-tural controllability for practical model of networked systems We explore the relationship between structural controllability andgraph reachability Consequently a simple graph-based algorithm is presented to obtain the minimum driver nodes Finally simu-lation results are presented to illustrate the performance of the proposed algorithm in dealing with large-scale networked systems

1 Introduction

Advances in communications technology have opened upnew challenges in the area of networked systems Control-lability of multiagent networked systems as a fundamentalconcept in this field has received considerable attentionThe pioneer work in analysing controllability of multiagentsystems with leader-follower architecture had been carriedout by Tanner [1] where controllability conditions wereprovided for multiagent systems with undirected graphtopology based on eigenvectors of the Laplacian matrix Infurther development some algebraic conditions for con-trollability of multiagent systems are presented in [2 3] Jiand Egerstedt [4] introduced network equitable partitions topresent a necessary condition for the controllability of leader-follower multiagent systems Inspired by [4] Rahmani et al[5] proposed the controllability of multiagent systems withmultiple leaders Liu et al [6] derived a simple controllabilitycondition for discrete-time single-leader switching networkswhich was further extended to continuous-time single-leaderswitching networks [7] Ji et al [8] derived a necessaryand sufficient condition for the controllability of leader-follower multiagent systems by dividing the overall systeminto several connected components The other related topics

in this area are leader-follower consensus [9 10] leader-follower formation control [11ndash13] containment control [1415] and pinning-controllability of networked systems [16 17]

The concept of structural controllability has been studiedextensively since the classical work by Lin [18] In [18]structural controllability of SISO linear systems was exploredby introducing a notion of structured matrix whose ele-ments are either fixed zeros or independent free parametersShields and Pearson [19] extended the results of [18] tostructural controllability of multiinput linear systems Sincethen various works have been carried out on the structuralcontrollability of linear systems [20ndash22] Recently structuralcontrollability of networked systems has emerged as a majorinterest in the network sciences A notable work in thisarea is carried out by Liu et al [23] which addressed thestructural controllability of complex networks Jafari et al[24] studied structural controllability of a leader-followermultiagent systemwithmultiple leaders Sundaram andHad-jicostis [25] developed a graph-theoretic characterization ofcontrollability and observability of linear systems over finitefields Haghighi and Cheah [26] employed the concept ofstructural observability to examine the weight-balanceabilityof networked systems

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2015 Article ID 192307 9 pageshttpdxdoiorg1011552015192307

2 Mathematical Problems in Engineering

For large-scale networked systems it is infeasible to applyinput signals to all network nodes due to the high controlcost and the difficulty of practical implementations In thiscase a fundamental problem is to identify a certain amountof nodes to be driven externally to bring the whole networkunder control This problem was addressed in [23] wherea theoretical framework was developed to solve the mini-mum input problem based on Linrsquos structural controllabilitytheorem [18] As pointed out in [27] the results in [23] arebased on the assumption that each node has an infinite timeconstant which do not generally represent the dynamics ofthe physical and biological systems

Despite the model in [23] Cowan et al [27] consideredinternal dynamics for all nodes of the network Cowanrsquosresult states that structural controllability does not depend ondegree distribution Hence the structural controllability canalways be conferred with a single independent control inputHowever the result in [27] suffers from a drawback that eachindependent input is connected to all nodes in the networkwhich is practically infeasible

In this paper we examined the structural controllabilityin networked systems by relaxing the critical assumptions inprevious results Consequently we provide a graph-theoreticmethod to identify driver nodes We present an algorithmto determine minimum driver nodes in networked systemsThe contribution of this paper is twofold (i) we relaxedthe assumptions in existing methods such as infinite timeconstant for each node and having direct access to inputsignals by all nodes on structural controllability of networkedsystems (ii) We provide a simple algorithm to obtain mini-mum driver nodes in networked systems

Thepaper is organized as follows Section 2 presents somepreliminaries in graph theory and controllability Section 3presents the model of the networked systems Section 4addresses the structural controllability in networked systemsand presents the linkage between the structural controlla-bility and the graph reachability in networked systems Analgorithm for identifying minimum driver nodes in net-worked systems is proposed in Section 5 Section 6 presentsthe simulation results and Section 7 concludes this paper

2 Preliminaries

The communication between nodes can be expressed bya weighted directed graph G(VEA) such that V =

V1 V2 V

119873 represents the set of nodes E sube V times V is

the edge set andA = [119886119894119895

] is the weighted adjacency matrixwhere 119886

119894119895gt 0 if (119894 119895) isin E and 119886

119894119895= 0 otherwise A graphH

is said to be a subgraph of a graphG ifV(H) sube V(G) andE(H) sube E(G)

A directed path in a digraph is an ordered sequence ofnodes so that any two consecutive nodes in the sequence arean edge of the digraph An undirected graph is a tree if andonly if for any two nodes there is a unique path connectingthem A directed spanning tree or arborescence is a digraphsuch that there is a unique directed path from a designatedroot node to every other node

Definition 1 A digraphG is called an arborescence divergingfrom node 119906 if there is only one directed path between root119906 and any other node ofG IfG is an arborescence divergingfrom 119906 then its reverse digraph (ie all edges of G arereversed) is called an arborescence converging to 119906 [28]

If there is an arborescence subdigraph diverging from anarbitrary node 119906 then 119906 is called a globally reachable node

Definition 2 Driver nodes are nodes in a network that haveto be controlled in order to completely control the entirenetwork

Definition 3 A matrix 119860119904is said to be a structured matrix

if its elements are either fixed zeros or independent freeparameters [29]

Definition 4 Two dynamical systems are called structurallyequivalent if their interconnection structures are identicalHence we can say (119860 119861) has the same structure as (1198601 1198611)if for every fixed zero entry of the matrix [119860 119861] thecorresponding entry of the matrix [1198601 1198611] is fixed zero andvice versa

Definition 5 The structural rank (srank) of a matrix is themaximum rank of all structurally equivalent matrices [30]

Theorem 6 (controllability test [31]) (119860119899times119899

119861119899times119898

) is control-lable if and only if there is no left eigenvector of 119860 that isorthogonal to 119861 that is

forall119908 119908ℎ119890119903119890 119908119879

119860 = 120582119908119879

997904rArr 119908119879

119861 = 0 (1)

Theorem 7 (Popov-Belevitch-Hautus controllability test[32]) (119860

119899times119899 119861119899times119898

) is controllable if and only if

Rank [120582119868 minus 119860 119861] = 119899 (2)

where 120582 is an eigenvalue of 119860

3 Model of Interconnected Networks

We consider each node in the network corresponding to adynamical system governed by the following equation

119894

= minus119886119894119894

119909119894

+

119873

sum119895=1

119895 =119894

119886119894119895

119909119895

+ 119887119894119906119894 (3)

where119909119894

isin 119877 denotes the state of node 119894119873 is the total numberof nodes and 119906

119894denotes an external input

Interconnected system (3) can be represented in matrixform

= 119860119909 + 119861119906 (4)

where 119909 = [1199091 1199092 119909

119873]119879 119906 = [119906

1 1199062 119906

119898]119879 119861 isin 119877119873times119898

is inputmatrix and 119860 isin 119877119899times119899 is defined as 119860 = AminusD whereA is the adjacency matrix andD = diag119886

11 11988622

119886119873119873

Mathematical Problems in Engineering 3

4 Controllability and Graph Reachability

According to classical control theory a dynamical system iscontrollable if for any initial state there exists an input thatcan drive the system to any final state in a finite time It iswell known that the system (119860 119861) is controllable if and onlyif the following controllability matrix

C = [119861 119860119861 119860119873minus1

119861] (5)

has full rank Even though a system with a pair of (119860 119861)

might be uncontrollable it can be controllable for anotherstructurally equivalent pair (119860lowast 119861lowast) [18]

Definition 8 A dynamical system (119860 119861) is structurallycontrollable if there exists a structurally equivalent system(119860lowast 119861lowast) that is controllable [33]

In what follows we first consider network with singledriver node and present the relation between graph reacha-bility and controllability

41 Controllability of Networks with Single Driver NodeConsider a network of nodes with single driver node 119894 whichis expressed as follows

= 119860119909 + 119861119906 (6)

where 119861 = [0 0 119887119894 0 0⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119894th element]119879 To examine the controlla-

bility we form the following zero-state response

119909 (119905) = int119905

0

119890119860(119905minus120591)

119861119906 (120591) 119889120591 (7)

The term 119890119860(119905minus120591)119861 refers to the 119894th column of thematrix 119890119860(119905minus120591)

multiplied by 119887119894 Using Cayley-Hamilton theorem 119890119860(119905minus120591) can

be expanded as follows

119890119860(119905minus120591)

=

119873minus1

sum119894=0

120572119894

(119905 minus 120591) 119860119894 (8)

where 120572119894(sdot) are scalar functions We state the following

theorem

Theorem 9 Consider the network expressed by (6) Thenetwork is structurally controllable if and only if there is anarborescence subdigraph diverging from driver node 119894

Proof (necessity condition) According to Lemma A1 in theAppendix the (119895 119894)th element of matrix series (8) is zeroif there is no path from node 119894 to node 119895 In this case the119895th element of the 119890

119860(119905minus120591)119861 is zero and remains zero for allvalues of the network link weights therefore at least 119909lowast =

[0 0 120572 0 0⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟119895th element

]119879 is an uncontrollable state of the system

Sufficiency Condition We show that a network which con-tains an arborescence subdigraph diverging from its drivernode is structurally controllable Without loss of generality

we assume that node 1 is the driver node In structuralcontrollability independent nonzero parameters can take anyvalues including zero Hence we zero out the weights forredundant links in such a way that the digraph associatedwith the network becomes an arborescence diverging fromthe driver node

Matrices 119860 and 119861 in (4) for an arborescence divergingfrom driver node can be expressed as follows

119860 =

[[[[[[[[

[

minus11988611

0 sdot sdot sdot 0

11988621

minus11988622

d 0

1198861198731

1198861198732

sdot sdot sdot minus119886119873119873

]]]]]]]]

]

119861 = [1198871

0 sdot sdot sdot 0]119879

(9)

Let 119908119894

= [1199081198941 119908119894

119873]119879 be the left eigenvector associated with

eigenvalue 120582119894 UsingTheorem 6 we have

119908119879

119894119861 = 0 lArrrArr 119908

119894

11198871

= 0 lArrrArr 119908119894

1= 0 (10)

Therefore to show that the network is structurally control-lable we need to prove the existence of weights such that1199081198941

= 0 for 119894 = 1 119873 To do so let 119886119894119894be a strictly

monotonic sequence for 119894 = 1 119873 Since 119860 is triangularmatrix with distinct diagonal entries eigenvalues of 119860 are itsdiagonal entries that is 120582

119894= minus119886119894119894for 119894 = 1 119873 Therefore

we obtain 1199081198941by solving 119908119879

119894119860 = 120582

119894119908119894as follows

1205821

= minus11988611

997904rArr 1199081

1= 1

1205822

= minus11988622

997904rArr 1199082

1=

11988621

11988611

minus 11988622

1205823

= minus11988633

997904rArr 1199083

1=

11988631

11988611

minus 11988633

+11988632

11988621

(11988622

minus 11988633

) (11988611

minus 11988633

)

120582119873

= minus119886119873119873

997904rArr 119908119873

1=

1198861198731

11988611

minus 119886119873119873

+

119873minus1

sum1198971=2

1198861198731198971

11988611989711

(11988611989711198971

minus 119886119873119873

) (11988611

minus 119886119873119873

)

+

119873minus1

sum1198971=3

1198971minus1

sum1198972=2

1198861198731198971

11988611989711198972

11988611989721

(11988611989711198971

minus 119886119873119873

) (11988611989721198972

minus 119886119873119873

) (11988611

minus 119886119873119873

)

+

119873minus1

sum1198971=119873minus1

sdot sdot sdot

119897119873minus2minus1

sum119897119873minus1=2

1198861198731198971

11988611989711198972

sdot sdot sdot 119886119897119873minus11

(11988611989711198971

minus 119886119873119873

) sdot sdot sdot (119886119897119873minus1119897119873minus1

minus 119886119873119873

) (11988611

minus 119886119873119873

)

(11)


Since all the denominators of (11) have the same sign and 1199081198941

for 119894 = 1 119873 can be expressed as

119908119894

1= 1205721

11198861198941

+

119894minus1

sum1198971=1

1205721198971

21198861198941198971

11988611989711⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

path of length2 from 1 to 119894

+

119894minus1

sum1198971=1

1198971minus1

sum1198972=1

12057211989711198972

31198861198941198971

11988611989711198972

11988611989721⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟


+ sdot sdot sdot

+

119894minus1

sum1198971=119894minus1

sdot sdot sdot


sum119897119894minus1=2

1205721

119894minus11198861198941198971

11988611989711198972

sdot sdot sdot 119886119897119894minus11⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

path of length119894minus1 from 1 to 119894

(12)

where 120572119897119896are same sign scalars for different values of 119896 and 119897

Existence of arborescence diverging from node 1 guaranteesthat 119908119894

1= 0 Hence 119908119894119861 = 0 for 119894 = 1 119873

Since an arborescence diverging from the driver node isstructurally controllable we can conclude that any networkedsystemwhich contains an arborescence subdigraph divergingfrom the driver node is structurally controllable

Corollary 10 Anetwork with a globally reachable driver nodeis structurally controllable

In the above we examine networks with single drivernode In what follows we generalize the result for networkswith multiple driver nodes

42 Controllability of Networks with Multiple Driver NodesConsider a network of nodes with multiple driver nodeswhich is expressed as follows

= 119860119909 + 119861119906 (13)

where 119861 = 119903 diag1198871 1198872 119887

119898 isin 119877

119873times119898 where 119903 diagsdot

refers to the rectangular diagonal matrix and 119887119903are positive

scalars The following theorem expresses the controllabilitycondition in networks with multiple driver nodes

Theorem 11 Consider the network expressed by (13) whichconsists of multiple driver nodes The network is structurallycontrollable if and only if there is a path from at least one drivernode to any arbitrary node

Proof For simplicity we assume that nodes 119894 = 1 119898 aredriver nodes of the networkHencematrix119861 can be expressedas follows

119861 =

[[[[[[[[[[[

[

1198871

0 sdot sdot sdot 0

0 1198872

d 0

0 sdot sdot sdot 0 119887119898

0(119873minus119898)times119898

]]]]]]]]]]]

]

(14)

Necessity Condition We assume that there is node 119895 andthat there is no path from any input node to that nodeAccording to Lemma A1 in the Appendix matrix series (8)has zero elements in columns 1 to 119898 of row 119895 Therefore119890119860(119905minus120591)119861 has zero row 119895 which yields existence of 119909lowast =

[0 0 120572 0 0⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟119895th element

]119879 as an uncontrollable state of the system

SufficiencyConditionWeassume that the network contains119898

driver nodes By zeroing out the weights of redundant linkswe decompose the network into 119898 components such thatdriver node 119896 controls over nodes of component 119896 Hencematrices 119860 and 119861 can be expressed as follows

119860 =[[

[

1198601

0d

0 119860119898

]]

]

119861 =[[

[

1198611

0d

0 119861119898

]]

]

(15)

If 119897th node in 119896th component is driver node we have 119861119896

=

[0 0 119887119896 0 0⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119897th element]119879 for 119896 = 1 119873 Using Theorem 7 we

obtain

Rank [120582119868 minus 119860 119861]

= Rank [[

[

1205821198681

minus 1198601

1198611

0d

0 120582119868119898

minus 119860119898

119861119898

]]

]

(16)

where 119868119896are identity matrices with the same size as 119860

119896for

119896 = 1 119873 Using Theorem 9 for an arbitrary component119896 [120582119868

119896minus 119860119896

119861119896] is full row-rank Since [120582119868 minus 119860 119861] is block

diagonal matrix with full row-rank block matrices therefore

Rank [120582119868 minus 119860 119861] = 119873 (17)

For better underdressing in what follows we comparethe proposed structural controllability condition and Liursquosstructural controllability condition An example of Liursquosstructural controllability is presented in Figure 1 It is shownthat controlling node 1 is not sufficient for full control (seeFigure 1(a)) To gain full control we must simultaneouslycontrol node 1 and any node among 119909

2 1199093(see Figure 1(b))

In contrast in the proposed structural controllability (seeFigure 2) controlling node 1 is sufficient for full control overthe networked system

5 Algorithm for IdentifyingMinimum Driver Nodes

We have shown the relationship between the structuralcontrollability and graph reachability Thus the problem of


A =

0 0 0

a21 0 0

a31 0 0

lfloorlceil

rfloorrceil

B =

b1

0

0lfloorlceilrfloorrceil

u1

b1x1

x2

a31a21

x3

b1 0 0 0

0 b1a21 0 0

0 b1a31 0 0lfloorlceil

rfloorrceil

C =

n = 3 m = 1 srank(119966) = 2

(a)

A =

0 0 0

a21 0 0

a31 0 0

lfloorlceil

rfloorrceil

B =

b1 0

0 b2

0 0

lfloorlceilrfloorrceil

u1

u2

b1

b2

x1

x2

a31a21

x3

b1 0 0 0 0 0

0 b2 b1a21 0 0 0

0 0 b1a31 0 0 0lfloorlceil

rfloorrceil

C =

n = 3 m = 2 srank(119966) = 3

(b)Figure 1 An example of Liursquos structural controllability

B =

b1

0


u1

b1x1

x2

a31a21

x3

A =

a11 0 0

a21 a22 0

a31 0 a33

lfloorlceil

rfloorrceil

N = 3 m = 1 srank(119966) = 3

b1 b1a11 b1a211

0 b1a21 b1a21(a11 + a22)0 b1a31 b1a31(a11 + a33)lfloorlceil

rfloorC =

rceil

Figure 2 An example of the proposed structural controllability

examining the structural controllability of the networkedsystems described by (5) can be converted into graph reach-ability problem Here we are interested in determining theminimum number of driver nodes in a directed networkdenoted by 119873

119863 to obtain controllability over the networked

systems However difficulties in identifying minimum num-ber of driver nodes in large-scale networks lead to therequirement for a simple systematic method In what followswe propose a simple algorithm to determine the minimumnumber of driver nodes using graph reachability approach

To check the graph reachability between each two arbi-trary nodes we present the following theorem

Theorem 12 Consider a network of nodes with an associatedstructured adjacency matrixA

119904 For any two arbitrary nodes 119894

and 119895 if (119895 119894)th element of the matrix (I minus A119904119905)minus1 is zero then

there is no path from node 119894 to node 119895 where I is an identitymatrix and 119905 is a positive constant such that the spectral radiusofA119904119905 is less than 1

Proof To prove this theorem we first use the Taylor seriesexpansion of the matrix inverse (see Lemma A3 in theAppendix) Consider

(I minus A119904119905)minus1

=

infin

sum119894=0

119905119894A119894

119904 (18)

Using Lemma A2 (A119896119904)119895119894for 119896 = 1 2 infin is zero if and

only if there is no path from node 119894 to node 119895 Hence thezeroness of the (119895 119894)th entry of A119896

119904for 119896 = 1 2 infin leads

to the zeroness of the (119895 119894)th entry of (I minus A119904119905)minus1

Remark 13 Using Gershgorinrsquos theorem [34] the suitable 119905

which satisfies the condition in Theorem 12 is obtained asfollows

119905 = (max119894

119873

sum119895=1

119895 =119894

119886119894119895

)

minus1

minus 120576 (19)

where 120576 is a small number

To illustrate the result in Theorem 12 let us consider thenetwork in Figure 3

The associated structured matrix can be defined asBoolean matrix as follows

A119904

=

[[[[[

[

0 1 1 1

0 0 0 1

1 1 0 1

0 1 0 0

]]]]]

]

(20)


1

2

3

4

a14 = 03 a12 = 02a13 = 08

a24 = 05

a42 = 07

a34 = 04 a32 = 01a31 = 06

Figure 3 An example of a network with 4 nodes

From (19) we obtain 119905 = 01 Therefore matrix (I minus A119904119905)minus1 is

obtained in structured format as follows

[[[[[

[

lowast lowast lowast lowast

0 lowast 0 lowast


0 lowast 0 lowast

]]]]]

]

(21)

where lowast represents nonzero parameters such that in matrix(21) for example entry (4 1) is zero which means that thereis no path from node 1 to node 4 Since the network is smalldriver nodes in Figure 3 can be easily identified which areeither node 2 or node 4 The same result can be obtained byexamining (IminusA

119904119905)minus1 Inmatrix (21) columns full of nonzero

elements represent globally reachable nodes For columnswhich contain zero elements we define graph reachabilityindex as follows

Definition 14 Node 119906 is said to have graph reachability index119903 if there are paths from 119906 to maximum 119903 other nodes of thenetwork

Therefore we can express the following corollary

Corollary 15 In matrix (I minus A119905)minus1 columns with higher

nonzero elements represent nodes with higher graph reachabil-ity index

We can deduce that nodes with higher graph reachabilityindex are suitable to be assigned as driver nodes

Remark 16 To find the minimum driver nodes to obtain astructurally controllable network we start by assigning thenode with the highest graph reachability index as the drivernode Then we remove all the nodes that are in the pathrooted for the assigned driver node We repeat the aboveprocedure for the remaining network till the condition inTheorem 11 is satisfied

Using the above mentioned results we present a system-atic algorithm to identify the minimum driver nodes in anetworked system such that the structural controllability ofthe network is guaranteedThe algorithm for determining theminimumdriver nodes of the network is described as follows

1

2

3

4

5

6

7

8

910

11

12

13

14

15

1617

18

19 20

21

22

23

24

25

26

27 28

29

30

Figure 4 An example of a network consisting of 30 nodes

Consider graph G with the associated structured adja-cency matrixA

119904

Step 1 Compute graph reachability matrixS = (I119873

minusA119904119905)minus1

Step 2 Identify the node with the highest graph reachabilityindex by finding the columns of matrix S with the largestnonzero elements If there is more than one node with thehighest graph reachability index we can randomly chooseone of them

Step 3 Assign that node as the driver node and zero out allthe rows with the nonzero elements in the column associatedwith that driver node

Step 4 Go back to Step 2 and repeat the procedure till allelements of matrixS are zero

The above procedure is expressed in Algorithm 1

Remark 17 It should be noted that the set of minimumdriver nodes is usually not unique depending on the networkconfigurations and one can determine other sets with thesame number of driver nodes

6 Simulations

In this section we present simulation results to illustratethe performance of the proposed method for networkedsystems of various sizes and topologies For the numericalcalculations and simulations we used MATLAB softwareFor illustration purpose we first consider a network with 30nodeswhich are distributed randomly as depicted in Figure 4The weights of links are randomly selected from [0 1] Wecompute (I minus A119905)

minus1 where A is the associated Laplacianmatrix The sparsity pattern of matrix (I minus A119905)minus1 is plottedin Figure 5 where the blue solid circles represent nonzeroelements of thematrix Applying the proposed algorithm thedriver nodes of the network are identified by magenta circlesin Figure 6 The result of the first simulation is summarizedin Table 1 where 119873 is the number of nodes 119871 is the number


Input A119904

Method(1) Compute 119905 from (19)(2) Compute graph reachability matrixS = (I

119873minus A119904119905)minus1

(3) 119896 = 0(4) while max(any(S)) = 0 do(5) 119896 = 119896 + 1 119896 represents the number of driver nodes(6) V = sum(S = 0 1) V represents the vector of the number of nonzero elements in each column(7) [value ind] = max(V) ind represents the column with the largest graph reachability index(8) Dnode(119896) = ind Dnode represents the array of driver nodes(9) 119908 = find(S( ind)) 119908 represents the rows with nonzero elements in the driver node column(10) S(119908 ) = 0(11) end while

Algorithm 1 Finding driver nodes in each connected component

Table 1 The characteristics of the network represented in Figure 6

119873 119871 119873119863

119899119889

30 41 6 02

[1 29 30]272523211917151311975 6 8 10 12 14 16 18 20 22 24 26 283 42

[30

1]2

34

56

78

910

1112

1314

1516

1718

1920

2122

2324

2526

2728

29

Figure 5 The sparsity pattern of matrix (I minus A119905)minus1

of links 119873119863is the computed number of driver nodes and

119899119889is the computed density of driver nodes obtained by 119899

119889=

119873119889

119873To illustrate the capability of the purposed algorithm in

dealing with large-scale networks we consider a network of1000 nodes which are distributed randomly within a squareregion as shown in Figure 7 The communication links aregenerated between neighboring nodes with the probability of05 The weights of links are randomly selected from [0 1]The sparsity pattern of matrix (I minus A119905)

minus1 is plotted forthe network in Figure 8 Applying the proposed algorithmthe driver nodes of the network are identified by magentacircles in Figure 9 The result of the second simulation issummarized in Table 2

1

2

3

4

5

6

7

8

910

11

12

13

14

15

1617

18

19 20

21

22

23

24

25

26

27 28

29

30

Figure 6 Driver nodes of the network identified bymagenta circles


119873 119871 119873119863

119899119889

1000 1361 96 0096

Table 3 The characteristics of some randomly generated networks

119873 119871 119873119863

119899119889

2000 2929 128 006405000 6931 408 0081610000 13951 748 00748

We applied the proposed algorithm on some randomlygenerated networks and the results are illustrated in Table 3

7 Conclusion

In this paper we have addressed the structural controlla-bility problem for networked systems Despite the existingmethods governed by some impractical assumptions onnodal dynamics and availability of input signals we haveexamined structural controllability for networked systems in



[1 1000]800600400200[100

01]

200

400

600

800



practical framework Using controllability analysis we havepresented the connection between networks driver nodesand graph reachability Consequently based on results ongraph reachability we have put forward a simple algorithmto determine minimum driver nodes in networked systemsFinally simulation results have been presented to illustratethe performance of the proposed methods

Appendix

LemmaA1 LetL = AminusDwhereA is the adjacencymatrixand D = diag119886

11 11988622

119886119873119873

Consider the followingmatrix

P =

119873minus1

sum119894=1

120573119894L119894 (A1)

where 120573119894are scalars (P)

119894119895is zero for any arbitrary values of 120573

119894

if there is no path of any length from node 119895 to node 119894

Proof To prove the lemma we show that the (119894 119895)th elementof all matricesL119894 where 119894 = 1 2 119873 is zero if there is nopath of any length from node 119895 to node 119894 Since there is noadjacent path from node 119895 to 119894 then 119886

119894119895= 0 Therefore the

(119894 119895)th element of theL2 can be expressed as follows

(L2)119894119895

=

119873

sum1198961=1

1198961=119895

1198861198941198961

1198861198961119895

= (A2)119894119895

(A2)

Using Lemma A2 we obtain (L2)119894119895

= 0 Therefore the(119894 119895)th element of theL3 can be expressed as follows

(L3)119894119895

=

119873

sum1198961=1

1198961=119894119895

119873

sum1198962=1

1198962=119894119895

1198861198941198961

11988611989611198962

1198861198962119895

= (A3)119894119895

(A3)


= 0 Similarly we canproceed forL4L5 L119873minus1 and show that (L119896)

119894119895= 0 for

119896 = 1 2 119873 minus 1

Lemma A2 (see [35]) Let A be the adjacency matrix of adigraphG then (A119896)

119894119895is greater than zero if and only if there

is a path of length 119896 from node 119895 to node 119894

Lemma A3 For two arbitrary matrices 119860 and 119861 the Taylorseries expansion of the matrix inverse is expressed as follows

(119860 + 119861)minus1

= 119860minus1

infin

sum119894=0

(minus1)119894

(119861119860minus1

)119894

(A4)

where the spectral radius of 119861119860minus1 is less than 1

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper


References

[1] H G Tanner ldquoOn the controllability of nearest neighborinterconnectionsrdquo in Proceedings of the 43rd IEEE Conferenceon Decision and Control (CDC rsquo04) vol 3 pp 2467ndash2472December 2004

[2] M Ji A Muhammad and M Egerstedt ldquoLeader-based multi-agent coordination controllability and optimal controlrdquo inProceedings of the American Control Conference pp 1358ndash1363June 2006

[3] A Rahmani and M Mesbahi ldquoOn the controlled agreementproblemrdquo inProceedings of theAmericanControl Conference pp1376ndash1381 IEEE Minneapolis Minn USA June 2006

[4] M Ji and M Egerstedt ldquoA graph-theoretic characterization ofcontrollability for multi-agent systemsrdquo in Proceedings of theAmerican Control Conference (ACC rsquo07) pp 4588ndash4593 IEEENew York NY USA July 2007

[5] A RahmaniM JiMMesbahi andMEgerstedt ldquoControllabil-ity of multi-agent systems from a graph-theoretic perspectiverdquoSIAM Journal on Control and Optimization vol 48 no 1 pp162ndash186 2009

[6] B Liu T Chu L Wang and G Xie ldquoControllability of aleader-follower dynamic network with switching topologyrdquoIEEETransactions onAutomatic Control vol 53 no 4 pp 1009ndash1013 2008

[7] B Liu T Chu L Wang Z Zuo G Chen and H SuldquoControllability of switching networks of multi-agent systemsrdquoInternational Journal of Robust and Nonlinear Control vol 22no 6 pp 630ndash644 2012

[8] Z J Ji Z D Wang H Lin and Z Wang ldquoInterconnectiontopologies for multi-agent coordination under leaderndashfollowerframeworkrdquo Automatica vol 45 no 12 pp 2857ndash2863 2009

[9] K Peng andY Yang ldquoLeader-following consensus problemwitha varying-velocity leader and time-varying delaysrdquo Physica Avol 388 no 2-3 pp 193ndash208 2009

[10] W Ni and D Cheng ldquoLeader-following consensus of multi-agent systems under fixed and switching topologiesrdquo Systems ampControl Letters vol 59 no 3-4 pp 209ndash217 2010

[11] L Consolini F Morbidi D Prattichizzo and M TosquesldquoLeader-follower formation control of nonholonomic mobilerobots with input constraintsrdquo Automatica vol 44 no 5 pp1343ndash1349 2008

[12] R Haghighi and C C Cheah ldquoOn leader-based shape coordi-nationrdquo in Proceedings of the 11th International Conference onControl Automation Robotics amp Vision (ICARCV rsquo10) pp 404ndash409 IEEE Singapore December 2010

[13] R Haghighi and C C Cheah ldquoMulti-group coordinationcontrol for robot swarmsrdquoAutomatica vol 48 no 10 pp 2526ndash2534 2012

[14] H Su G Jia and M Z Q Chen ldquoSemi-global containmentcontrol of multi-agent systems with input saturationrdquo IETControl Theory amp Applications vol 8 no 18 pp 2229ndash22372014

[15] H Su and M Z Q Chen ldquoMulti-agent containment controlwith input saturation on switching topologiesrdquo IET ControlTheory amp Applications vol 9 no 3 pp 399ndash409 2015

[16] M Porfiri and M di Bernardo ldquoCriteria for global pinning-controllability of complex networksrdquo Automatica vol 44 no12 pp 3100ndash3106 2008

[17] Q Song and J Cao ldquoOn pinning synchronization of directedand undirected complex dynamical networksrdquo IEEE Transac-tions on Circuits and Systems I Regular Papers vol 57 no 3 pp672ndash680 2010

[18] C-T Lin ldquoStructural controllabilityrdquo IEEE Transactions onAutomatic Control vol 19 no 3 pp 201ndash208 1974

[19] R W Shields and J B Pearson ldquoStructural controllability ofmulti-input linear systemsrdquo IEEE Transactions on AutomaticControl vol AC-21 no 2 pp 203ndash212 1976

[20] S Hosoe and K Matsumoto ldquoOn the irreducibility conditionin the structural controllability theoremrdquo IEEE Transactions onAutomatic Control vol 24 no 6 pp 963ndash966 1979

[21] J-M Dion C Commault and J van der Woude ldquoGenericproperties and control of linear structured systems a surveyrdquoAutomatica vol 39 no 7 pp 1125ndash1144 2003

[22] H Mayeda ldquoOn structural controllability theoremrdquo IEEETransactions on Automatic Control vol 26 no 3 pp 795ndash7981981

[23] Y-Y Liu J-J Slotine and A-L Barabasi ldquoControllability ofcomplex networksrdquoNature vol 473 no 7346 pp 167ndash173 2011

[24] S Jafari A Ajorlou and A G Aghdam ldquoLeader localizationin multi-agent systems subject to failure a graph-theoreticapproachrdquo Automatica vol 47 no 8 pp 1744ndash1750 2011

[25] S Sundaram and C N Hadjicostis ldquoStructural controllabilityand observability of linear systems over finite fields withapplications to multi-agent systemsrdquo IEEE Transactions onAutomatic Control vol 58 no 1 pp 60ndash73 2013

[26] R Haghighi and C C Cheah ldquoDistributed average consensusbased on structural weight-balanceabilityrdquo IET Control Theoryamp Applications vol 9 no 2 pp 176ndash183 2015

[27] N J Cowan E J Chastain D A Vilhena J S Freudenberg andC T Bergstrom ldquoNodal dynamics not degree distributionsdetermine the structural controllability of complex networksrdquoPLoS ONE vol 7 no 6 Article ID e38398 2012

[28] W Tutte GraphTheory Addison-Wesley 1984[29] D D Siljak Decentralized Control of Complex Systems Aca-

demic Press New York NY USA 1991[30] K J Reinschke Multivariable Control A Graph-Theoretic

Approach Springer 1988[31] P R Belanger Control Engineering A Modern Approach Saun-

ders College Publishing 1995[32] R L Williams and D A Lawrence Linear State-Space Control

Systems John Wiley amp Sons 2007[33] C Sueur and G Dauphin-Tanguy ldquoBond-graph approach for

structural analysis of MIMO linear systemsrdquo Journal of theFranklin Institute vol 328 no 1 pp 55ndash70 1991

[34] R S Varga Gershgorin and His Circles Springer Berlin Ger-many 2004

[35] GWilliamsLinearAlgebrawithApplications Jones andBartlettPublishers 2012

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


Operations ResearchAdvances in

Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


For large-scale networked systems it is infeasible to applyinput signals to all network nodes due to the high controlcost and the difficulty of practical implementations In thiscase a fundamental problem is to identify a certain amountof nodes to be driven externally to bring the whole networkunder control This problem was addressed in [23] wherea theoretical framework was developed to solve the mini-mum input problem based on Linrsquos structural controllabilitytheorem [18] As pointed out in [27] the results in [23] arebased on the assumption that each node has an infinite timeconstant which do not generally represent the dynamics ofthe physical and biological systems

Despite the model in [23] Cowan et al [27] consideredinternal dynamics for all nodes of the network Cowanrsquosresult states that structural controllability does not depend ondegree distribution Hence the structural controllability canalways be conferred with a single independent control inputHowever the result in [27] suffers from a drawback that eachindependent input is connected to all nodes in the networkwhich is practically infeasible

In this paper we examined the structural controllabilityin networked systems by relaxing the critical assumptions inprevious results Consequently we provide a graph-theoreticmethod to identify driver nodes We present an algorithmto determine minimum driver nodes in networked systemsThe contribution of this paper is twofold (i) we relaxedthe assumptions in existing methods such as infinite timeconstant for each node and having direct access to inputsignals by all nodes on structural controllability of networkedsystems (ii) We provide a simple algorithm to obtain mini-mum driver nodes in networked systems

Thepaper is organized as follows Section 2 presents somepreliminaries in graph theory and controllability Section 3presents the model of the networked systems Section 4addresses the structural controllability in networked systemsand presents the linkage between the structural controlla-bility and the graph reachability in networked systems Analgorithm for identifying minimum driver nodes in net-worked systems is proposed in Section 5 Section 6 presentsthe simulation results and Section 7 concludes this paper

2 Preliminaries

The communication between nodes can be expressed bya weighted directed graph G(VEA) such that V =

V1 V2 V

119873 represents the set of nodes E sube V times V is

the edge set andA = [119886119894119895

] is the weighted adjacency matrixwhere 119886

119894119895gt 0 if (119894 119895) isin E and 119886

119894119895= 0 otherwise A graphH

is said to be a subgraph of a graphG ifV(H) sube V(G) andE(H) sube E(G)

A directed path in a digraph is an ordered sequence ofnodes so that any two consecutive nodes in the sequence arean edge of the digraph An undirected graph is a tree if andonly if for any two nodes there is a unique path connectingthem A directed spanning tree or arborescence is a digraphsuch that there is a unique directed path from a designatedroot node to every other node

Definition 1 A digraphG is called an arborescence divergingfrom node 119906 if there is only one directed path between root119906 and any other node ofG IfG is an arborescence divergingfrom 119906 then its reverse digraph (ie all edges of G arereversed) is called an arborescence converging to 119906 [28]

If there is an arborescence subdigraph diverging from anarbitrary node 119906 then 119906 is called a globally reachable node

Definition 2 Driver nodes are nodes in a network that haveto be controlled in order to completely control the entirenetwork

Definition 3 A matrix 119860119904is said to be a structured matrix

if its elements are either fixed zeros or independent freeparameters [29]

Definition 4 Two dynamical systems are called structurallyequivalent if their interconnection structures are identicalHence we can say (119860 119861) has the same structure as (1198601 1198611)if for every fixed zero entry of the matrix [119860 119861] thecorresponding entry of the matrix [1198601 1198611] is fixed zero andvice versa

Definition 5 The structural rank (srank) of a matrix is themaximum rank of all structurally equivalent matrices [30]

Theorem 6 (controllability test [31]) (119860119899times119899

119861119899times119898

) is control-lable if and only if there is no left eigenvector of 119860 that isorthogonal to 119861 that is

forall119908 119908ℎ119890119903119890 119908119879

119860 = 120582119908119879

997904rArr 119908119879

119861 = 0 (1)

Theorem 7 (Popov-Belevitch-Hautus controllability test[32]) (119860

119899times119899 119861119899times119898

) is controllable if and only if

Rank [120582119868 minus 119860 119861] = 119899 (2)

where 120582 is an eigenvalue of 119860

3 Model of Interconnected Networks

We consider each node in the network corresponding to adynamical system governed by the following equation

119894

= minus119886119894119894

119909119894

+

119873

sum119895=1

119895 =119894

119886119894119895

119909119895

+ 119887119894119906119894 (3)

where119909119894

isin 119877 denotes the state of node 119894119873 is the total numberof nodes and 119906

119894denotes an external input

Interconnected system (3) can be represented in matrixform

= 119860119909 + 119861119906 (4)

where 119909 = [1199091 1199092 119909

119873]119879 119906 = [119906

1 1199062 119906

119898]119879 119861 isin 119877119873times119898

is inputmatrix and 119860 isin 119877119899times119899 is defined as 119860 = AminusD whereA is the adjacency matrix andD = diag119886

11 11988622

119886119873119873




C = [119861 119860119861 119860119873minus1

119861] (5)






= 119860119909 + 119861119906 (6)

where 119861 = [0 0 119887119894 0 0⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟



119909 (119905) = int119905

0

119890119860(119905minus120591)

119861119906 (120591) 119889120591 (7)




119890119860(119905minus120591)

=

119873minus1

sum119894=0

120572119894

(119905 minus 120591) 119860119894 (8)


theorem




[0 0 120572 0 0⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟119895th element





119860 =

[[[[[[[[

[

minus11988611

0 sdot sdot sdot 0

11988621

minus11988622

d 0

1198861198731

1198861198732


]]]]]]]]

]

119861 = [1198871


(9)

Let 119908119894

= [1199081198941 119908119894



119908119879

119894119861 = 0 lArrrArr 119908

119894

11198871

= 0 lArrrArr 119908119894

1= 0 (10)






119894119860 = 120582

119894119908119894as follows

1205821

= minus11988611

997904rArr 1199081

1= 1

1205822

= minus11988622

997904rArr 1199082

1=

11988621

11988611

minus 11988622

1205823

= minus11988633

997904rArr 1199083

1=

11988631

11988611

minus 11988633

+11988632

11988621

(11988622

minus 11988633

) (11988611

minus 11988633

)

120582119873

= minus119886119873119873

997904rArr 119908119873

1=

1198861198731

11988611

minus 119886119873119873

+

119873minus1

sum1198971=2

1198861198731198971

11988611989711

(11988611989711198971

minus 119886119873119873

) (11988611

minus 119886119873119873

)

+

119873minus1

sum1198971=3

1198971minus1

sum1198972=2

1198861198731198971

11988611989711198972

11988611989721

(11988611989711198971

minus 119886119873119873

) (11988611989721198972

minus 119886119873119873

) (11988611

minus 119886119873119873

)

+

119873minus1

sum1198971=119873minus1

sdot sdot sdot


sum119897119873minus1=2

1198861198731198971

11988611989711198972


(11988611989711198971

minus 119886119873119873


minus 119886119873119873

) (11988611

minus 119886119873119873

)

(11)




119908119894

1= 1205721

11198861198941

+

119894minus1

sum1198971=1

1205721198971

21198861198941198971

11988611989711⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟


+

119894minus1

sum1198971=1

1198971minus1

sum1198972=1

12057211989711198972

31198861198941198971

11988611989711198972

11988611989721⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟


+ sdot sdot sdot

+

119894minus1

sum1198971=119894minus1

sdot sdot sdot


sum119897119894minus1=2

1205721

119894minus11198861198941198971

11988611989711198972



(12)



1= 0 Hence 119908119894119861 = 0 for 119894 = 1 119873





= 119860119909 + 119861119906 (13)

where 119861 = 119903 diag1198871 1198872 119887

119898 isin 119877






119861 =

[[[[[[[[[[[

[

1198871

0 sdot sdot sdot 0

0 1198872

d 0

0 sdot sdot sdot 0 119887119898

0(119873minus119898)times119898

]]]]]]]]]]]

]

(14)


[0 0 120572 0 0⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟119895th element




119860 =[[

[

1198601

0d

0 119860119898

]]

]

119861 =[[

[

1198611

0d

0 119861119898

]]

]

(15)


=

[0 0 119887119896 0 0⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟


obtain

Rank [120582119868 minus 119860 119861]

= Rank [[

[

1205821198681

minus 1198601

1198611

0d

0 120582119868119898

minus 119860119898

119861119898

]]

]

(16)


119896for


119896minus 119860119896



Rank [120582119868 minus 119860 119861] = 119873 (17)


2 1199093(see Figure 1(b))





A =

0 0 0

a21 0 0

a31 0 0

lfloorlceil

rfloorrceil

B =

b1

0


u1

b1x1

x2

a31a21

x3

b1 0 0 0

0 b1a21 0 0


rfloorrceil

C =

n = 3 m = 1 srank(119966) = 2

(a)

A =

0 0 0

a21 0 0

a31 0 0

lfloorlceil

rfloorrceil

B =

b1 0

0 b2

0 0


u1

u2

b1

b2

x1

x2

a31a21

x3

b1 0 0 0 0 0

0 b2 b1a21 0 0 0


rfloorrceil

C =

n = 3 m = 2 srank(119966) = 3


B =

b1

0


u1

b1x1

x2

a31a21

x3

A =

a11 0 0

a21 a22 0

a31 0 a33

lfloorlceil

rfloorrceil

N = 3 m = 1 srank(119966) = 3

b1 b1a11 b1a211


rfloorC =

rceil












=

infin

sum119894=0

119905119894A119894

119904 (18)



119904for 119896 = 1 2 infin leads




119905 = (max119894

119873

sum119895=1

119895 =119894

119886119894119895

)

minus1

minus 120576 (19)




A119904

=

[[[[[

[

0 1 1 1

0 0 0 1

1 1 0 1

0 1 0 0

]]]]]

]

(20)


1

2

3

4

a14 = 03 a12 = 02a13 = 08

a24 = 05

a42 = 07

a34 = 04 a32 = 01a31 = 06




[[[[[

[


0 lowast 0 lowast


0 lowast 0 lowast

]]]]]

]

(21)











1

2

3

4

5

6

7

8

910

11

12

13

14

15

1617

18

19 20

21

22

23

24

25

26

27 28

29

30



119904








6 Simulations




Input A119904


119873minus A119904119905)minus1




119873 119871 119873119863

119899119889

30 41 6 02

[1 29 30]272523211917151311975 6 8 10 12 14 16 18 20 22 24 26 283 42

[30

1]2

34

56

78

910

1112

1314

1516

1718

1920

2122

2324

2526

2728

29




119889=

119873119889




1

2

3

4

5

6

7

8

910

11

12

13

14

15

1617

18

19 20

21

22

23

24

25

26

27 28

29

30



119873 119871 119873119863

119899119889

1000 1361 96 0096


119873 119871 119873119863

119899119889

2000 2929 128 006405000 6931 408 0081610000 13951 748 00748


7 Conclusion




[1 1000]800600400200[100

01]

200

400

600

800




Appendix


11 11988622

119886119873119873


P =

119873minus1

sum119894=1

120573119894L119894 (A1)



119894





(L2)119894119895

=

119873

sum1198961=1

1198961=119895

1198861198941198961

1198861198961119895

= (A2)119894119895

(A2)



(L3)119894119895

=

119873

sum1198961=1

1198961=119894119895

119873

sum1198962=1

1198962=119894119895

1198861198941198961

11988611989611198962

1198861198962119895

= (A3)119894119895

(A3)



119894119895= 0 for

119896 = 1 2 119873 minus 1





(119860 + 119861)minus1

= 119860minus1

infin

sum119894=0

(minus1)119894

(119861119860minus1

)119894

(A4)





References











































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












C = [119861 119860119861 119860119873minus1

119861] (5)






= 119860119909 + 119861119906 (6)

where 119861 = [0 0 119887119894 0 0⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟



119909 (119905) = int119905

0

119890119860(119905minus120591)

119861119906 (120591) 119889120591 (7)




119890119860(119905minus120591)

=

119873minus1

sum119894=0

120572119894

(119905 minus 120591) 119860119894 (8)


theorem




[0 0 120572 0 0⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟119895th element





119860 =

[[[[[[[[

[

minus11988611

0 sdot sdot sdot 0

11988621

minus11988622

d 0

1198861198731

1198861198732


]]]]]]]]

]

119861 = [1198871


(9)

Let 119908119894

= [1199081198941 119908119894



119908119879

119894119861 = 0 lArrrArr 119908

119894

11198871

= 0 lArrrArr 119908119894

1= 0 (10)






119894119860 = 120582

119894119908119894as follows

1205821

= minus11988611

997904rArr 1199081

1= 1

1205822

= minus11988622

997904rArr 1199082

1=

11988621

11988611

minus 11988622

1205823

= minus11988633

997904rArr 1199083

1=

11988631

11988611

minus 11988633

+11988632

11988621

(11988622

minus 11988633

) (11988611

minus 11988633

)

120582119873

= minus119886119873119873

997904rArr 119908119873

1=

1198861198731

11988611

minus 119886119873119873

+

119873minus1

sum1198971=2

1198861198731198971

11988611989711

(11988611989711198971

minus 119886119873119873

) (11988611

minus 119886119873119873

)

+

119873minus1

sum1198971=3

1198971minus1

sum1198972=2

1198861198731198971

11988611989711198972

11988611989721

(11988611989711198971

minus 119886119873119873

) (11988611989721198972

minus 119886119873119873

) (11988611

minus 119886119873119873

)

+

119873minus1

sum1198971=119873minus1

sdot sdot sdot


sum119897119873minus1=2

1198861198731198971

11988611989711198972


(11988611989711198971

minus 119886119873119873


minus 119886119873119873

) (11988611

minus 119886119873119873

)

(11)




119908119894

1= 1205721

11198861198941

+

119894minus1

sum1198971=1

1205721198971

21198861198941198971

11988611989711⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟


+

119894minus1

sum1198971=1

1198971minus1

sum1198972=1

12057211989711198972

31198861198941198971

11988611989711198972

11988611989721⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟


+ sdot sdot sdot

+

119894minus1

sum1198971=119894minus1

sdot sdot sdot


sum119897119894minus1=2

1205721

119894minus11198861198941198971

11988611989711198972



(12)



1= 0 Hence 119908119894119861 = 0 for 119894 = 1 119873





= 119860119909 + 119861119906 (13)

where 119861 = 119903 diag1198871 1198872 119887

119898 isin 119877






119861 =

[[[[[[[[[[[

[

1198871

0 sdot sdot sdot 0

0 1198872

d 0

0 sdot sdot sdot 0 119887119898

0(119873minus119898)times119898

]]]]]]]]]]]

]

(14)


[0 0 120572 0 0⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟119895th element




119860 =[[

[

1198601

0d

0 119860119898

]]

]

119861 =[[

[

1198611

0d

0 119861119898

]]

]

(15)


=

[0 0 119887119896 0 0⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟


obtain

Rank [120582119868 minus 119860 119861]

= Rank [[

[

1205821198681

minus 1198601

1198611

0d

0 120582119868119898

minus 119860119898

119861119898

]]

]

(16)


119896for


119896minus 119860119896



Rank [120582119868 minus 119860 119861] = 119873 (17)


2 1199093(see Figure 1(b))





A =

0 0 0

a21 0 0

a31 0 0

lfloorlceil

rfloorrceil

B =

b1

0


u1

b1x1

x2

a31a21

x3

b1 0 0 0

0 b1a21 0 0


rfloorrceil

C =

n = 3 m = 1 srank(119966) = 2

(a)

A =

0 0 0

a21 0 0

a31 0 0

lfloorlceil

rfloorrceil

B =

b1 0

0 b2

0 0


u1

u2

b1

b2

x1

x2

a31a21

x3

b1 0 0 0 0 0

0 b2 b1a21 0 0 0


rfloorrceil

C =

n = 3 m = 2 srank(119966) = 3


B =

b1

0


u1

b1x1

x2

a31a21

x3

A =

a11 0 0

a21 a22 0

a31 0 a33

lfloorlceil

rfloorrceil

N = 3 m = 1 srank(119966) = 3

b1 b1a11 b1a211


rfloorC =

rceil












=

infin

sum119894=0

119905119894A119894

119904 (18)



119904for 119896 = 1 2 infin leads




119905 = (max119894

119873

sum119895=1

119895 =119894

119886119894119895

)

minus1

minus 120576 (19)




A119904

=

[[[[[

[

0 1 1 1

0 0 0 1

1 1 0 1

0 1 0 0

]]]]]

]

(20)


1

2

3

4

a14 = 03 a12 = 02a13 = 08

a24 = 05

a42 = 07

a34 = 04 a32 = 01a31 = 06




[[[[[

[


0 lowast 0 lowast


0 lowast 0 lowast

]]]]]

]

(21)











1

2

3

4

5

6

7

8

910

11

12

13

14

15

1617

18

19 20

21

22

23

24

25

26

27 28

29

30



119904








6 Simulations




Input A119904


119873minus A119904119905)minus1




119873 119871 119873119863

119899119889

30 41 6 02

[1 29 30]272523211917151311975 6 8 10 12 14 16 18 20 22 24 26 283 42

[30

1]2

34

56

78

910

1112

1314

1516

1718

1920

2122

2324

2526

2728

29




119889=

119873119889




1

2

3

4

5

6

7

8

910

11

12

13

14

15

1617

18

19 20

21

22

23

24

25

26

27 28

29

30



119873 119871 119873119863

119899119889

1000 1361 96 0096


119873 119871 119873119863

119899119889

2000 2929 128 006405000 6931 408 0081610000 13951 748 00748


7 Conclusion




[1 1000]800600400200[100

01]

200

400

600

800




Appendix


11 11988622

119886119873119873


P =

119873minus1

sum119894=1

120573119894L119894 (A1)



119894





(L2)119894119895

=

119873

sum1198961=1

1198961=119895

1198861198941198961

1198861198961119895

= (A2)119894119895

(A2)



(L3)119894119895

=

119873

sum1198961=1

1198961=119894119895

119873

sum1198962=1

1198962=119894119895

1198861198941198961

11988611989611198962

1198861198962119895

= (A3)119894119895

(A3)



119894119895= 0 for

119896 = 1 2 119873 minus 1





(119860 + 119861)minus1

= 119860minus1

infin

sum119894=0

(minus1)119894

(119861119860minus1

)119894

(A4)





References











































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












119908119894

1= 1205721

11198861198941

+

119894minus1

sum1198971=1

1205721198971

21198861198941198971

11988611989711⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟


+

119894minus1

sum1198971=1

1198971minus1

sum1198972=1

12057211989711198972

31198861198941198971

11988611989711198972

11988611989721⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟


+ sdot sdot sdot

+

119894minus1

sum1198971=119894minus1

sdot sdot sdot


sum119897119894minus1=2

1205721

119894minus11198861198941198971

11988611989711198972



(12)



1= 0 Hence 119908119894119861 = 0 for 119894 = 1 119873





= 119860119909 + 119861119906 (13)

where 119861 = 119903 diag1198871 1198872 119887

119898 isin 119877






119861 =

[[[[[[[[[[[

[

1198871

0 sdot sdot sdot 0

0 1198872

d 0

0 sdot sdot sdot 0 119887119898

0(119873minus119898)times119898

]]]]]]]]]]]

]

(14)


[0 0 120572 0 0⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟119895th element




119860 =[[

[

1198601

0d

0 119860119898

]]

]

119861 =[[

[

1198611

0d

0 119861119898

]]

]

(15)


=

[0 0 119887119896 0 0⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟


obtain

Rank [120582119868 minus 119860 119861]

= Rank [[

[

1205821198681

minus 1198601

1198611

0d

0 120582119868119898

minus 119860119898

119861119898

]]

]

(16)


119896for


119896minus 119860119896



Rank [120582119868 minus 119860 119861] = 119873 (17)


2 1199093(see Figure 1(b))





A =

0 0 0

a21 0 0

a31 0 0

lfloorlceil

rfloorrceil

B =

b1

0


u1

b1x1

x2

a31a21

x3

b1 0 0 0

0 b1a21 0 0


rfloorrceil

C =

n = 3 m = 1 srank(119966) = 2

(a)

A =

0 0 0

a21 0 0

a31 0 0

lfloorlceil

rfloorrceil

B =

b1 0

0 b2

0 0


u1

u2

b1

b2

x1

x2

a31a21

x3

b1 0 0 0 0 0

0 b2 b1a21 0 0 0


rfloorrceil

C =

n = 3 m = 2 srank(119966) = 3


B =

b1

0


u1

b1x1

x2

a31a21

x3

A =

a11 0 0

a21 a22 0

a31 0 a33

lfloorlceil

rfloorrceil

N = 3 m = 1 srank(119966) = 3

b1 b1a11 b1a211


rfloorC =

rceil












=

infin

sum119894=0

119905119894A119894

119904 (18)



119904for 119896 = 1 2 infin leads




119905 = (max119894

119873

sum119895=1

119895 =119894

119886119894119895

)

minus1

minus 120576 (19)




A119904

=

[[[[[

[

0 1 1 1

0 0 0 1

1 1 0 1

0 1 0 0

]]]]]

]

(20)


1

2

3

4

a14 = 03 a12 = 02a13 = 08

a24 = 05

a42 = 07

a34 = 04 a32 = 01a31 = 06




[[[[[

[


0 lowast 0 lowast


0 lowast 0 lowast

]]]]]

]

(21)











1

2

3

4

5

6

7

8

910

11

12

13

14

15

1617

18

19 20

21

22

23

24

25

26

27 28

29

30



119904








6 Simulations




Input A119904


119873minus A119904119905)minus1




119873 119871 119873119863

119899119889

30 41 6 02

[1 29 30]272523211917151311975 6 8 10 12 14 16 18 20 22 24 26 283 42

[30

1]2

34

56

78

910

1112

1314

1516

1718

1920

2122

2324

2526

2728

29




119889=

119873119889




1

2

3

4

5

6

7

8

910

11

12

13

14

15

1617

18

19 20

21

22

23

24

25

26

27 28

29

30



119873 119871 119873119863

119899119889

1000 1361 96 0096


119873 119871 119873119863

119899119889

2000 2929 128 006405000 6931 408 0081610000 13951 748 00748


7 Conclusion




[1 1000]800600400200[100

01]

200

400

600

800




Appendix


11 11988622

119886119873119873


P =

119873minus1

sum119894=1

120573119894L119894 (A1)



119894





(L2)119894119895

=

119873

sum1198961=1

1198961=119895

1198861198941198961

1198861198961119895

= (A2)119894119895

(A2)



(L3)119894119895

=

119873

sum1198961=1

1198961=119894119895

119873

sum1198962=1

1198962=119894119895

1198861198941198961

11988611989611198962

1198861198962119895

= (A3)119894119895

(A3)



119894119895= 0 for

119896 = 1 2 119873 minus 1





(119860 + 119861)minus1

= 119860minus1

infin

sum119894=0

(minus1)119894

(119861119860minus1

)119894

(A4)





References











































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










A =

0 0 0

a21 0 0

a31 0 0

lfloorlceil

rfloorrceil

B =

b1

0


u1

b1x1

x2

a31a21

x3

b1 0 0 0

0 b1a21 0 0


rfloorrceil

C =

n = 3 m = 1 srank(119966) = 2

(a)

A =

0 0 0

a21 0 0

a31 0 0

lfloorlceil

rfloorrceil

B =

b1 0

0 b2

0 0


u1

u2

b1

b2

x1

x2

a31a21

x3

b1 0 0 0 0 0

0 b2 b1a21 0 0 0


rfloorrceil

C =

n = 3 m = 2 srank(119966) = 3


B =

b1

0


u1

b1x1

x2

a31a21

x3

A =

a11 0 0

a21 a22 0

a31 0 a33

lfloorlceil

rfloorrceil

N = 3 m = 1 srank(119966) = 3

b1 b1a11 b1a211


rfloorC =

rceil












=

infin

sum119894=0

119905119894A119894

119904 (18)



119904for 119896 = 1 2 infin leads




119905 = (max119894

119873

sum119895=1

119895 =119894

119886119894119895

)

minus1

minus 120576 (19)




A119904

=

[[[[[

[

0 1 1 1

0 0 0 1

1 1 0 1

0 1 0 0

]]]]]

]

(20)


1

2

3

4

a14 = 03 a12 = 02a13 = 08

a24 = 05

a42 = 07

a34 = 04 a32 = 01a31 = 06




[[[[[

[


0 lowast 0 lowast


0 lowast 0 lowast

]]]]]

]

(21)











1

2

3

4

5

6

7

8

910

11

12

13

14

15

1617

18

19 20

21

22

23

24

25

26

27 28

29

30



119904








6 Simulations




Input A119904


119873minus A119904119905)minus1




119873 119871 119873119863

119899119889

30 41 6 02

[1 29 30]272523211917151311975 6 8 10 12 14 16 18 20 22 24 26 283 42

[30

1]2

34

56

78

910

1112

1314

1516

1718

1920

2122

2324

2526

2728

29




119889=

119873119889




1

2

3

4

5

6

7

8

910

11

12

13

14

15

1617

18

19 20

21

22

23

24

25

26

27 28

29

30



119873 119871 119873119863

119899119889

1000 1361 96 0096


119873 119871 119873119863

119899119889

2000 2929 128 006405000 6931 408 0081610000 13951 748 00748


7 Conclusion




[1 1000]800600400200[100

01]

200

400

600

800




Appendix


11 11988622

119886119873119873


P =

119873minus1

sum119894=1

120573119894L119894 (A1)



119894





(L2)119894119895

=

119873

sum1198961=1

1198961=119895

1198861198941198961

1198861198961119895

= (A2)119894119895

(A2)



(L3)119894119895

=

119873

sum1198961=1

1198961=119894119895

119873

sum1198962=1

1198962=119894119895

1198861198941198961

11988611989611198962

1198861198962119895

= (A3)119894119895

(A3)



119894119895= 0 for

119896 = 1 2 119873 minus 1





(119860 + 119861)minus1

= 119860minus1

infin

sum119894=0

(minus1)119894

(119861119860minus1

)119894

(A4)





References











































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










1

2

3

4

a14 = 03 a12 = 02a13 = 08

a24 = 05

a42 = 07

a34 = 04 a32 = 01a31 = 06




[[[[[

[


0 lowast 0 lowast


0 lowast 0 lowast

]]]]]

]

(21)











1

2

3

4

5

6

7

8

910

11

12

13

14

15

1617

18

19 20

21

22

23

24

25

26

27 28

29

30



119904








6 Simulations




Input A119904


119873minus A119904119905)minus1




119873 119871 119873119863

119899119889

30 41 6 02

[1 29 30]272523211917151311975 6 8 10 12 14 16 18 20 22 24 26 283 42

[30

1]2

34

56

78

910

1112

1314

1516

1718

1920

2122

2324

2526

2728

29




119889=

119873119889




1

2

3

4

5

6

7

8

910

11

12

13

14

15

1617

18

19 20

21

22

23

24

25

26

27 28

29

30



119873 119871 119873119863

119899119889

1000 1361 96 0096


119873 119871 119873119863

119899119889

2000 2929 128 006405000 6931 408 0081610000 13951 748 00748


7 Conclusion




[1 1000]800600400200[100

01]

200

400

600

800




Appendix


11 11988622

119886119873119873


P =

119873minus1

sum119894=1

120573119894L119894 (A1)



119894





(L2)119894119895

=

119873

sum1198961=1

1198961=119895

1198861198941198961

1198861198961119895

= (A2)119894119895

(A2)



(L3)119894119895

=

119873

sum1198961=1

1198961=119894119895

119873

sum1198962=1

1198962=119894119895

1198861198941198961

11988611989611198962

1198861198962119895

= (A3)119894119895

(A3)



119894119895= 0 for

119896 = 1 2 119873 minus 1





(119860 + 119861)minus1

= 119860minus1

infin

sum119894=0

(minus1)119894

(119861119860minus1

)119894

(A4)





References











































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Input A119904


119873minus A119904119905)minus1




119873 119871 119873119863

119899119889

30 41 6 02

[1 29 30]272523211917151311975 6 8 10 12 14 16 18 20 22 24 26 283 42

[30

1]2

34

56

78

910

1112

1314

1516

1718

1920

2122

2324

2526

2728

29




119889=

119873119889




1

2

3

4

5

6

7

8

910

11

12

13

14

15

1617

18

19 20

21

22

23

24

25

26

27 28

29

30



119873 119871 119873119863

119899119889

1000 1361 96 0096


119873 119871 119873119863

119899119889

2000 2929 128 006405000 6931 408 0081610000 13951 748 00748


7 Conclusion




[1 1000]800600400200[100

01]

200

400

600

800




Appendix


11 11988622

119886119873119873


P =

119873minus1

sum119894=1

120573119894L119894 (A1)



119894





(L2)119894119895

=

119873

sum1198961=1

1198961=119895

1198861198941198961

1198861198961119895

= (A2)119894119895

(A2)



(L3)119894119895

=

119873

sum1198961=1

1198961=119894119895

119873

sum1198962=1

1198962=119894119895

1198861198941198961

11988611989611198962

1198861198962119895

= (A3)119894119895

(A3)



119894119895= 0 for

119896 = 1 2 119873 minus 1





(119860 + 119861)minus1

= 119860minus1

infin

sum119894=0

(minus1)119894

(119861119860minus1

)119894

(A4)





References











































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











[1 1000]800600400200[100

01]

200

400

600

800




Appendix


11 11988622

119886119873119873


P =

119873minus1

sum119894=1

120573119894L119894 (A1)



119894





(L2)119894119895

=

119873

sum1198961=1

1198961=119895

1198861198941198961

1198861198961119895

= (A2)119894119895

(A2)



(L3)119894119895

=

119873

sum1198961=1

1198961=119894119895

119873

sum1198962=1

1198962=119894119895

1198861198941198961

11988611989611198962

1198861198962119895

= (A3)119894119895

(A3)



119894119895= 0 for

119896 = 1 2 119873 minus 1





(119860 + 119861)minus1

= 119860minus1

infin

sum119894=0

(minus1)119894

(119861119860minus1

)119894

(A4)





References











































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










References











































Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









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