Controllability of complex networks using perturbation theory of extreme singular values

0.13 MB·

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: [email protected]
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: [email protected]