Controllability of complex networks using perturbation theory of extreme singular values

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by  Stephane Chretien

CONTROLLABILITY OF COMPLEX NETWORKS USING
PERTURBATION THEORY OF EXTREME SINGULAR VALUES
STÉPHANECHRÉTIEN AND SÉBASTIEN DARSES
6
1 Abstract. Pinning control on complex dynamical networks has emerged as a very impor-
0 tanttopic in recent trendsof control theoryduetotheextensivestudyof collective coupled
2 behaviorsandtheirroleinphysics,engineeringandbiology. Inpractice,real-worldnetworks
n consists of a large number of vertices and one may only be able to perform a control on a
a fraction of them only. Controllability of such systems has been addressed in [6], where it
J was reformulated as a global asymptotic stability problem. The goal of this short note is to

2 refinetheanalysis proposed in [6]usingrecent resultsin singular valueperturbation theory.
2

] 1. Introduction
C
O In recent years, extensive efforts have been devoted to the control of complex dynamical
networks. One major issue is that real world networks usually consist of a very large number
.
h
of nodes and links which makes it impossible to apply control actions to all nodes.
t
a Pinning control is a new way to address this problem by placing local feedback injections
m
on a small fraction of the nodes.
[
Controllability of such systems has been addressed in [6], where it was reformulated as

1 a global asymptotic stability problem. The goal of this short note is to refine the analysis
v proposed in [6] using recent results in singular value perturbation theory.
2
4
1.1. The model. One considers a set of N n-dimensional oscillators governed by a system of
0
6 nonlinear differential equations. Moreover, we assume that each oscillator is coupled with a
0 restricted set of other oscillators. This coupling relationship can be efficiently described using
.
1 a graph where the vertices are indexed by the oscillators and there is an edge between two
0
oscillators if they are coupled. The overall dynamical system is given by the following set of
6
differential equations
1
:
v N
i (1.1) x′(t) = f(x (t)) σB l x (t)+u (t), t t ,
X i i − ij j i ≥ 0
j=1
r X
a
i = 1,...,N, where x (t) Rn is the state of the ith oscillator, σ > 0, B Rn×n, f : R R
i
∈ ∈ →
describes the dynamics of each oscillator, L = (l ) is the graph Laplacian of the
ij i,j=1,...,N
underlying graph, and u (t), i = 1,...,N are the controls. For the system to be well defined,
i
we have to specify some initial conditions x (t ) = x for i = 1,...,N.
i 0 i0
1.2. Thecontrolproblem. Assumethatwehaveareferencetrajectorys(t),t t satisfying
0
≥
the differential equation
s′(t) = f(s(t)).
1
2 STÉPHANECHRÉTIENANDSÉBASTIENDARSES
Our goal is to control the system using a limited number of nodes. The selected nodes are
called the "pinned nodes". For this purpose, we use a linear feedback law of the form
u (t) = p Ke (t),
i i i
where e (t)= s(t) x (t), K is a feedback gain matrix, and where
i i
−
1 if node i is pinned
p =
i
(0 otherwise.
Let P denote the diagonal matrix with diagonal (p ,...,p ).
1 N
1.3. Controllability. In [6], the authors propose a definition for (global pinning-) controlla-
bility (based on Lyapunov stability criteria):
Definition 1.1. We say that the system (1.1) is controllable if the error dynamical system
e := (e (t)) is Lyapunov stable around the origin, i.e. there exists a positive definite
i 1≤i≤N
function V such that dV(e(t)) < 0 when e(0) = 0.
dt 6
The following result, [6, Corollary 5], provides a sufficient condition for a system to be
controllable:
Proposition 1.2 ([6]). Assume that f is such that there exists a bounded matrix F e, whose
ξ,ξ
coefficients depend on ξ and ξ˜, which satisfies
(1.2) F e ξ ξ = f(ξ) f(ξ), ξ,ξ Rn.
ξ,ξ − − ∈
(cid:16) (cid:17)
Let Q Rn×n be a positive definiteematrix such that e e
∈
QK +KtQt = κ QB +BtQt
QB+BtQt 0
(cid:0) (cid:1)
(cid:23)
and (cid:0) (cid:1)
1
(1.3) 2 λN (σL+κP) λn QB +BtQt > suepkFξ,ξek kQk.
ξ,ξ
(cid:0) (cid:1)
Then the system is controllable.
Many systems of interest satisfy the constraint specified by (1.2); see [?]. This proposition
is very useful for node selection via the matrix P. Indeed, assume that Q is selected, then
one may try to maximise λ (σL+κP) as a function of P, under the constraint that no
N
more than r nodes can be pinned. This is a combinatorial problem that can be relaxed using
semi-definite programming or various heuristics [3].
1.4. Goal of the paper. Our goal in the present note is to propose an easy controllability
conditionrefining[6,Corollary7],basedonthealgebraicconnectivityofthegraph,thenumber
of pinned nodes, the coupling strengh and the feedback gain. Our approach is based on
perturbation theory of the extreme singular values of a matrix after appending a column. The
basic results of this theory are given in the appendix.
CONTROLLABILITY OF COMPLEX NETWORKS USING PERTURBATION THEORY 3
2. Main result
2.1. Notations. The Kronecker symbol is denoted by δ , i.e. δ = 1 if i = j and is equal to
i,j i,j
zerootherwise. Wedenote by x theeuclidian norm ofavector xand by A theassociated
2
k k k k
operator norm (spectral norm) of a matrix A.
For any symmetric matrix B Rd×d we will denote its eigenvalues by λ (B)
1
∈ ≥ ··· ≥
λ (B). The largest eigenvalue will sometimes also be denoted by λ (B) and the smallest by
d max
λ (B). The smallest nonzero eigenvalue of apositive semi-definite matrix B will be denoted
min
by λ (B).
min>0
2.2. A simple criterion for controllability. Our main result is the following theorem.
Theorem 2.1. Let Q Rn×n be a positive definite symmetric matrix that satisfies
∈
QK +KtQt = κ QB+BtQt
QB +BtQt 0,
(cid:0) (cid:1)
(cid:23)
and assume that (cid:0) (cid:1)
σλ (L) λ QB +BtQt
min>0 min
(2.4) F < .
k ξ,ξ˜k 2 Q
k (cid:0)k (cid:1)
If κ satisfies
r
deg
κ i=1 i +σλ (L),
≥ σλmin>0(LP)− λm2ink(FQξB,ξ˜+kBkQtQkt) min>0
then the system is controllable.
Proof. We follow the same steps as for the proof of Corollary 7 in [6]. We assume without loss
of generality that the first r nodes are the pinned nodes. We may write P as
r
P = e et,
i i
i=1
X
where e is the ith member of the canonical basis of RN, i.e. e (j) = δ . We will try to
i i i,j
compare λ (σL+κP) with λ (σL) and use Proposition 1.2 to obtain a sufficient condition
N N
for controllability based on L, i.e. the topology of the network. For this purpose, let us recall
that L can be written as
L = t,
I ·I
where is the incidence matrix of any directed graph obtained from the system’s graph by
I
assigning an arbitrary sign to the edges [1]. Of course L will not depend on the chosen
assignment. Using this factorization of L, we obtain that
r
σL+κ e et = √κ e ,...,√κ e ,√σ √κ e ,...,√κ e ,√σ t.
i i r 1 r 1
I I
i=1
X (cid:2) (cid:3) (cid:2) (cid:3)
Moreover, λ (σL+κP) can be expressed easily as the smallest nonzero eigenvalue of
min>0
the rth term of a sequence of matrices with shape (A.8) for which we can use Theorem A.2
iteratively. Indeed, we have
t
λ (σL+κe ) = λ √κ e ,√σ √κ e ,√σ .
min>0 1 min>0 1 1
I I
(cid:16) (cid:17)
(cid:2) (cid:3) (cid:2) (cid:3)
4 STÉPHANECHRÉTIENANDSÉBASTIENDARSES
Let us denote by x the vector √κ e and by X the matrix [√σ ]. Then, we have that
1
I
t xtx xtX
√κ e ,√σ √κ e ,√σ = .
1 I 1 I Xtx XtX
(cid:20) (cid:21)
(cid:2) (cid:3) (cid:2) (cid:3)
Therefore, Theorem A.2 gives
deg
λ σL+κe et σλ (L) 1 ,
min>0 1 1 ≥ min>0 − (κ σλ (L))
min>0
−
(cid:0) (cid:1)
where deg is the degree of node number 1.
1
Let us now consider λ (σL+κ e +δ e ). We have that
min>0 1 2 2
t
λ (σL+κ e +δ e ) = λ √κ e ,√κ e ,√σ √κ e ,√κ e ,√σ .
min>0 1 2 2 min>0 2 1 2 1
I I
(cid:16) (cid:17)
(cid:2) (cid:3) (cid:2) (cid:3)
Let us denote by x the vector √κ e and by X the matrix [√κ e ,√σ ]. Then, we have that
2 1
I
t xtx xtX
√κ e ,√κ e ,√σ √κ e ,√κ e ,√σ =
2 1 I 2 1 I Xtx XtX
(cid:20) (cid:21)
(cid:2) (cid:3) (cid:2) (cid:3)
and using Theorem A.2 again, we obtain
deg
λ σL+κe et +κe et λ (σL+κe et) 2 .
min>0 1 1 2 2 ≥ min>0 1 1 − (κ λ (σL+κe et))
− min>0 1 1
(cid:0) (cid:1)
Since λ (σL+κe et) λ (σL), we thus obtain
min>0 1 1 ≤ min>0
deg
λ σL+κe et +κe et λ (σL+κe et) 2 .
min>0 1 1 2 2 ≥ min>0 1 1 − (κ σλ (L))
min>0
−
(cid:0) (cid:1)
We can repeat the same argument r times and obtain
r
deg
(2.5) λ (σL+κP) σλ (L) i=1 i .
min>0 min>0
≥ − κ σλ (L)
−P min>0
Finally, by Proposition1.2, weknowthatthefollowing constraint issufficient forpreserving
controllability
r 2 F Q
(2.6) λ σL+κ e et k ξ,ξ˜k k k .
min>0 i i ≥ λ (QB +BtQt)
! min
i=1
X
By (2.5), it is sufficient to garantee the controllability of our system to impose
σλ (L) ri=1degi 2 kFξ,ξ˜k kQk .
min>0 − κ σλ (L) ≥ λ (QB+BtQt)
−P min>0 min
Then, combining (2.6) with (2.4) implies that
r
deg
κ i=1 i +σλ (L)
≥ σλmin>0(LP)− λm2ink(FQξB,ξ˜+kBkQtQkt) min>0
is a sufficient condition for controllability. (cid:3)
CONTROLLABILITY OF COMPLEX NETWORKS USING PERTURBATION THEORY 5
Appendix A. Perturbation theory of extreme singular values after
appending a column
A.1. Framework. Let d be an integer. Let X Rd×n be a d n-matrix and let x Rd be
∈ × ∈
column vector. We denote by a subscript t the transpose of vectors and matrices. There exist
at least two ways to study the singular values of the matrix (x,X) obtained by appending the
column vector x to the matrix X:
(A1) Consider the matrix
xt xtx xtX
(A.7) A = x X = ;
Xt Xtx XtX
(cid:20) (cid:21) (cid:20) (cid:21)
(cid:2) (cid:3)
(A2) Consider the matrix
xt
A = x X = XXt+xxt.
Xt
(cid:20) (cid:21)
(cid:2) (cid:3)
On one hand, one may setudy in (A1) the eigenvalues of the (n+1) (n+1) hermitian
×
matrix A, i.e. the matrix XtX augmented with an arrow matrix.
On the other hand, one will deal in (A2) with the eigenvalues of the d d hermitian matrix
×
A, which may be seen as a rank-one perturbation of XXt. The matrices A and A have the
same non-zeros eigenvalues, and in particular λ (A) = λ (A). Moreover, the singular
max max
vealues of the matrix (x,X) are the square-root of the eigenvalues of the matrix A.e
Equivalently, the problem of a rank-one perturbation can be reephrased as the one of con-
trolling the perturbation of the singular values of a matrix after appending a column.
A.2. A theorem of Li and Li. In this paper, we useaslightly more general framework than
(A1), that is the case of a matrix
c at
(A.8) A = ,
a M
(cid:20) (cid:21)
where a Rd, c R and M Rd×d is a symmetric matrix.
∈ ∈ ∈
The following theorem provides sharp upper bounds for λ (A), and lower bounds on
max
λ (A), depending on various information on the sub-matrix M of A. As discussed above,
min
this problem has close relationships with our problem of appending a column to a given
rectangular matrix, because λ (A)= λ (A).
1 1
Theorem A.1 (Li-Li’s inequality and a lower bound). Let d be a positive integer and let
e
M Cd×d be an Hermitian matrix, whose eigenvalues are λ λ with corresponding
1 d
eige∈nvectors (V , ,V ). Set c R, a Cd. Let A be given by≥(A··.·8≥). Therefore:
1 d
··· ∈ ∈
2 a,V 2 2 a 2
1
(A.9) h i λ (A) max(c,λ ) k k ,
1 1
η + η2+4 a,V 2 ≤ − ≤ η + η2+4 a 2
1 1 h 1i 1 1 k k
with p p
η = c λ .
1 1
| − |
6 STÉPHANECHRÉTIENANDSÉBASTIENDARSES
A.3. Perturbation of the smallest nonzero eigenvalue. Thesametechnics usedtoprove
Theorem A.1 also give lower bounds for the smallest nonzero eigenvalue, which are also direct
consequences of Li-Li’s inequality. For more details, we refer the reader to [2].
Theorem A.2. Let d be a positive integer and let M Cd×d be a positive semi-definite
∈
Hermitian matrix, whose eigenvalues are λ λ with corresponding eigenvectors
1 d
(V , ,V ). Set c R, a Cd. Let A be give≥n b·y··(A≥.8). Assume that M has rank r d.
1 d
··· ∈ ∈ ≤
Therefore:
2 a 2
(A.10) λ (A) min(c,λ ) k k ,
r+1 r
≥ − η + η2+4 a 2
r r
k k
with
p
η = c λ .
r r
| − |
In particular, the following perturbation bounds of Weyl and Mathias hold:
Corollary A.3.
(A.11) λ (A) min(c,λ ) a
r+1 r 2
≥ −k k
a 2
(A.12) λ (A) min(c,λ ) k k2 .
r+1 r
≥ − c λ
r
| − |
Acknowledgement – We thank Maurizio Porfiri for pointing out a missing assumption in
Theorem 2.1.
References
1. Brouwer, AndriesE. andHaemers, Willem H.Spectraof graphs. Universitext.Springer,NewYork,2012.
2. Chrétien, S. and Darses, S., Perturbation bounds on the extremal singular values of a matrix after ap-
pending a column. arXiv preprint arXiv:1406.5441,2014.
3. Ghosh, A., and Boyd, S., Growing well connected graphs, Proceedings of the 45th IEEE Conference on
Decision and Control, (2006), p. 6605–6611.
4. Horn, Roger A.; Johnson, Charles R. Matrix analysis. Cambridge University Press, Cambridge, 1985.
5. Li, Chi-Kwong; Li, Ren-Cang. A note on eigenvalues of perturbed Hermitian matrices. Linear Algebra
Appl. 395 (2005), 183–190.
6. Porfiri, Maurizio and di Bernardo, Mario, Criteria for global pinning-controllability of complex networks.
Automatica J. IFAC 44 (2008), no. 12, 3100–3106.
National Physical Laboratory, Hampton Road, Teddington, UK
E-mail address: [e-mail]
LATP, UMR 6632, Université Aix-Marseille, Technopôle Château-Gombert, 39 rue Joliot
Curie, 13453 Marseille Cedex 13, France, and, Laboratoire de Mathématiques, UMR 6623,
Université de Franche-Comté, 16 route de Gray,, 25030 Besancon, France
E-mail address: [e-mail]