Perturbation of multiagent linear systems

M. I. GARC´

IA-PLANAS

Universitat Polit`

ecnica de Catalunya

Departament de Matem`

atica Aplicada

Mineria 1, 08038 Barcelona

SPAIN

Abstract: The main objective of this note is to explore if, making a small perturbation of an uncontrollable multi-

agent linear system with a previously interrelationship topology established, a controllable multi-agent system

with the same topology can be obtained. Arnold geometric techniques will be used for the objective, and versal

deformations will be constructed in the set of equivalent systems.

Key–Words: Multiagent linear systems, deformation, controllability.

Received: May 13, 2022. Revised: January 23, 2023. Accepted: February 20, 2023. Published: March 7, 2023.

1 Introduction

Recently, the study of multiagent systems has at-

tracted the attention of many researchers because this

class of systems appears in various areas of knowl-

edge, such as the cooperative control of unmanned

aerial vehicles, the consensus problem of communi-

cation networks, the training control of mobile robots,

neural networks modeling the brain structure, Etc.,

[5], [9], [10].

An interesting technique for analyzing pertur-

bations and investigating complicated objects such

as singularities and bifurcations in multiparamet-

ric dynamical multi-systems is the construction of

versal deformations since they provide a particular

parametrization of families of multi-systems.

Versal deformation permits speaking of generic

families relative to a generic property of interest as,

for example, controllability. In this case, generic fam-

ilies with controllable members have the property that

such members remain even when the family is per-

turbed. The generic property permits us that a small

perturbation may eliminate all the non-controllable

cases. In order to construct versal deformation, one

deﬁnes an equivalence relation in the space of multi-

agent systems preserving controllability character.

The knowledge of a versal deformation provides a

particular parametrization of the space of multi-agent

systems in a neighborhood of a ﬁxed point, which can

be effectively applied to the perturbation analysis of

this point.

(For more information about versal deformations,

see [1], [2], [3]).

2 Preliminaries

Let us consider a group of kagents. The following lin-

ear dynamical systems give the dynamic of each agent

˙x1(t) = A1x1(t) + B1u1(t) + C1v1(t)

˙xk(t) = Akxk(t) + Bkuk(t) + C1vk(t)









(1)

Ai∈Mn(IC),Bi∈Mn×m(IC),Ci∈Mn×p(IC),

xi(t)∈ICn,ui(t) = fi(x1(t), . . . , xk(t)∈ICm,

vi(t)∈ICp,1≤i≤k.

Sometimes, the considered internal controls ui

are given by means a communication topology de-

ﬁned by an undirected graph with

i) Vertex set: V={1, . . . , k}

ii) Edge set: E={i, j)|i, j ∈⊂ V×V

iii) Neighbor of i:Ni={j∈| (i, j)∈E}. (In the

case where j=ithe edge is called self-loop).

deﬁning the communication topology among agents:

ui(t) = X

j∈Ni

(xi(t)−xj(t)),1≤i≤k

for some Ki∈Mm×n(IC),1≤i≤k

Writing X= (xi, . . . , xk)t,U= (ui, . . . , uk)t

and V= (vi, . . . , vk)t, and considering the Laplacian

matrix associated with the graph that is deﬁned in the

following manner

L= (lij ) = 





|Ni|if i=j

−1if j∈ Ni

0otherwise

(2)

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To simplify notations, we will write the systems

in matrix language:

A=diag (A1, . . . , Ak)

∈Mn(IC) ×k

. . . ×Mn(IC) = M1

B=diag (B1, . . . , Bk)

∈Mn×m(IC) ×k

. . . ×Mn×m(IC) = M2

C=diag (C1, . . . , Ck)

∈Mn×p(IC) ×k

. . . ×Mn×p(IC) = M3

and X= (xi, . . . , xk)t,U= (L ⊗ In)X=

(ui, . . . , uk)tand V= (vi, . . . , vk)t, and in the case

where the communication topology for internal con-

trol is considered, the control is written as U= (L ⊗

In)X,

Finally. we will call by Mthe set M=M1×

M2×M3. This set is clearly, a differentiable mani-

fold.

3 Equivalence relation for differen-

tiable families of multi-systems

In the set Mwe consider the following equivalence

relation

Deﬁnition 1 Two systems (A1,B1,C1)and

(A2,B2,C2)in Mare said equivalent if and

only if, there exist P=diag(P1, . . . , Pk)∈

Gl(n; IC) ×k

. . . ×Gl(n; IC),Q=diag(Q1, . . . , Qk)∈

Gl(m; IC) ×k

. . . ×Gl(m; IC),R=diag(R1, . . . , Rk)∈

Gl(p; IC) ×k

. . . ×Gl(p; IC),K=diag(K1, . . . , Kk)∈

Mm×n(IC) ×k

. . . ×Mm×n(IC) and F=

diag(F1, . . . , Fk)∈Mp×n(IC) ×k

. . . ×Mp×n(IC), such

that

(A2,B2,C2) =

(P−1A1P+P−1B1K+P−1C1F,P−1BQ,P−1CR)

This equivalence relation can be seen as the action

by a Lie group in the following manner:

Let G={(P,Q,R,K,F)∈Gl(n, IC) ×

Gl(m; IC) ×Gl(p; IC) ×Mm×n(IC) ×Mp×n(IC). It is a

Lie group to the usual product of matrices in the form:





P10 0

K1Q10

F10R1



·



P20 0

K2Q20

F20R



=





P1P20 0

K1P2+Q1K2Q1Q20

F1P2+R1F20R1R2





and





P10 0

K1Q10

F10R1





−1

=



P−1

10 0

−Q−1

1K1P−1

1Q−1

−R−1

1F1P−1

10R−1





Gacts over Min the following manner

Calling G= (P, Q, R, K, F )and M=

(A,B,C)∈M,

φ:G × M−→ M

(G, M)−→ φ(G, M ) = ¯

M(3)

Where ¯

M= (P−1AP +P−1BK +

P−1CF,P−1BQ,P−1CR)

φis differentiable and surjective.

Fixing M0= (A0,B0,C0)∈Mwe have the

differentiable map

φM0:G −→ M

G−→ φ(G, M 0)(4)

The image of this map Im φM0is the set of equiv-

alent systems to M0and it is called orbit of M0and

it is denoted by O(X0). On the other hand, the sub-

set of Gleaving invariant the system M0,{G∈ G |

φM0(G) = M0}is called the stabilizer of M0.

The differentiability of the action allows a local

study of the orbit and the stabilizer computing the dif-

ferential of φM0.

Lemma 2 The differential dφM0,I:TIG −→ Mat

the identity point I∈ G, on any element G∈TIG, is

given by

dφM0,I(G)) =

([A,P] + BK +CF,BQ −PB,CR −PC)

Remark 3 TIG={(P,Q,R,K,F)∈Mn(IC) ×

Mm(IC) ×Mp(IC) ×Mm×n(IC) ×Mp×n(IC). and

TM0M=M.

Proof:

It sufﬁces to compute the linear approximation of

the map on the identity.

φM0(I+εG) =

M0+ε(([A,P] + BK +CF,BQ −PB,CR −PC))

+ε2. . .

⊓⊔

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4 Versal deformations

There is quite a lot of literature in which it can ﬁnd

the deﬁnition of deformation and versal deformation;

in this case, we take the one found in [6]

Deﬁnition 4 A deformation of an element M0∈M

is a family of elements of Mindexed by λ∈Λφ:

Λ−→ Mwhere Λ⊂ICmis a neighborhood of 0, and

where φ(0) = M0and φdepends smoothly on the

parameters.

Deﬁnition 5 A deformation φ(λ) = φ(λ1, . . . , λm)

of M0is versal if and only if for any deformation

φ′(µ1, . . . , φk)∈Mof M0,φ′(µ)is induced by

φ(λ), that is to say, there exists a neighborhood V

of 0 in ICk, a map ψ:V−→ ICmwith ψ(0) = 0, and

a map g:V−→ G with g(0) = Isuch that ∀µ∈V,

φ′(µ) = g(µ)φ(ψ(µ))g−1(µ)with ψand gholomor-

phic (smooth).

It is obvious that if we have a versal deforma-

tion of an element, automatically we have a versal

deformation of any element that is equivalent to it,

since if M=φ(G, M0)is an equivalent element

of M0and φ(λ)is a versal deformation of M0then

φ(G−1, φ(λ)) is a versal deformation of M′.

A versal deformation having a minimal number

of parameters is called miniversal.

4.1 Transversality

The versatility condition admits a useful geometric

characterization in terms of transversality. We begin,

then, by recalling the notion of transversality.

Deﬁnition 6 Let S⊂Wbe a differentiable subman-

ifold of a manifold W. Consider a differentiable map

ψΛ−→ W, of another manifold Son W. Let λ∈Λ

such that ψ(λ)∈S

It is said that the map ψis transversal to Sin λ,

if the tangent space to W, in ψ(λ)decomposes in the

way:

Tψ(λ)W=Im dψλ+Tψ(λ)S.

It is called mini transversal if said sum is direct.

Transversality allows obtaining local trivializa-

tions along the orbits:

Proposition 7 Let ψ: Λ −→ Mbe a deformation

of M0minitransversal to the orbit O(M0)in 0, and

G1⊂Ga submanifold minitransversal to the stabi-

lizer of M0in I. then the application

β: Λ ×G1−→ M

deﬁned by β(λ, G) = φ(G, M0(λ)is a local diffeo-

morphism at (0,I).

Proof: It sufﬁces to apply the inverse function theo-

rem and to prove that dβ is exhaustive at (0,I).⊓⊔

The following result was proved by Arnold [1],

in the case where Gl(n;C)acts on Mn(C), remark-

ing that what was important was the action of the Lie

group on the variety, thus generalizing the result and

providing the relationship between a versal deforma-

tion of M0and the local structure of the orbit.

Theorem 8 ([1]) 1. A deformation φ(λ)of (M0)is

versal if and only if it is transversal to the orbit

O(M0)at (M0).

2. The minimal number of parameters of a versal

deformation is equal to the codimension of the

orbit of M0in M,ℓ= codim O(M0).

Corollary 9 Then φ(λ) = M0+ (TM0O(M0))⊥for

some scalar product is a miniversal deformation.

Let {v1, . . . , vℓ}be a basis of any arbitrary com-

plementary subspace (TM0O(M0)cto TM0O(M0).

Corollary 10 The deformation

φ: Λ ⊂Cℓ−→ M, φ(λ) = M0+

ℓ

i=1

λivi

is a miniversal deformation.

The versatility condition admits a useful geomet-

ric characterization in terms of transversality. We be-

gin, deﬁning scalar products in Mand TIG, we can

consider the adjoint application of dφM0.

The Euclidean scalar product considered in this

paper is deﬁned as follows:

For all Mi= (Ai,Bi,Ci)∈M

⟨M1, M2⟩1=

trace(A1A∗

2) + trace(B1B∗

2) + trace(C1C∗

2),

(5)

where A∗denotes the conjugate transpose of a matrix

Theorem 11 The normal complementary subspace to

tangent space to the orbit of the system M0is deﬁned

by the set of elements (X,Y,Z)∈Mverifying

[X∗,A]−BY∗−CZ∗= 0

X∗B= 0

X∗C= 0

Y∗B= 0

Z∗C= 0











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Proof:

It sufﬁces to observe that:

⟨(([A,P]+BK+CF,BQ−PB,CR−PC),(X,Y,Z)⟩= 0

for any (P,Q,R,K,F)∈TIG, if and only if

trace 



[X∗,A]−BY∗−CZ∗X∗B X∗C

0Y∗B0

0 0 Z∗C





·



P⋆ ⋆

K Q ⋆

F⋆R



= 0

⊓⊔

After this theorem, it is easy to compute these

spaces.

5 Controllability

The importance of the qualitative property of dynamic

systems in the control theory, known as controllabil-

ity, is well known.

The controllability concept involves taking the

system from any initial state to any ﬁnal state in ﬁnite

time, regardless of the path or input. Let us consider

the multi-agent system 1

It is important to emphasize that various deﬁni-

tions of controllability are derived, depending to a

large extent on the class of dynamic systems and the

form of allowable controls, [7].

In our particular setup, the controllability charac-

ter can be described as

rank A−λIn×kB C =nk

Proposition 12 The controllability character is in-

variant under the equivalence relation considered.

Proof:

rank A−λIn×kB C =

rank P−1A−λIn×kB C 



K Q

F R





⊓⊔

For a ﬁxed B-feedback Kand the ﬁxed topology

comunication, the system 1 can be written as

X(t) = (A+BK(L ⊗ In))X(t) + CV(t).(6)

(See [8] for Kronecker product properties).

The controllability of the system can be analyzed

by computing the rank of the controllability matrix:

(C(A+BK)(L ⊗ In))C)

. . . (A+BK)(L ⊗ In))nk−1C)

The rank of this matrix is invariant under feed-

back, that is to say

Proposition 13 The matrix controllability of the sys-

tem 1 is invariant under external feedback

Proof:

rank (C(A+BK)(L ⊗ In) + CF)C). . .

(A+BK)(L ⊗ In) + CF)nk−1C)

=rank (C(A+BK)(L ⊗ In))C). . .

(A+BK)(L ⊗ In))nk−1C)

·





IFB . . .

......







⊓⊔

We are going to carry out the study for a particular

case in which all the systems have the same dynamics,

that is, Ai=A,Bi=B,Ci=Cand Ki=Kfor

all 1≤i≤k; and the graph deﬁning the topology

relating to the systems is undirected and connected.

For being un undirected graph the matrix Lis sym-

metric, then there exist an orthogonal matrix Psuch

that PLPt=D, and the connection ensures that 0 is

a simple eigenvalue of L.

Proposition 14 Under these conditions, the system

can be described as

X(t) =

((Ik⊗A)+(Ik⊗BK)(L ⊗ In))X(t)+(In⊗C)V(t)

(7)

In our particular setup, we have that there ex-

ists an orthogonal matrix P∈Gl(k, R)such that

PLPt=D=diag (λ1, . . . , λk), (λ1≥. . . ≥

λk−1> λk= 0).

Corollary 15 The system can be described in terms of

the matrices A,B,Cthe feedback Kand the eigen-

values of L.

Proof:

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Following the properties of Kronecker product,

we have that.

(P⊗In)(Ik⊗A)(Pt⊗In)=(Ik⊗A)

(P⊗In)(Ik⊗BK)(Pt⊗In) =

(Ik⊗BK)

(P⊗In)(Ik⊗C)(Pt⊗Ik) = (Ik⊗C)

(P⊗In)(L ⊗ In)(Pt⊗In)=(D ⊗ In)

and calling b

X= (P⊗In)X, and b

V= (P⊗Ik)Vwe

have

X=((Ik⊗A) + (Ik⊗BK)(D ⊗ In))b

X+ (Ik⊗C)b

Using this description, the analysis of controlla-

bility is easier.

Proposition 16 The system (7) is controllable if and

only if the systems (A+λiBK, C)are controllable

for each 1≤i≤k.

5.1 Perturbation

The controllability character is generic in multi-agent

systems’ space M.

Proposition 17 The subset C⊂Mof the control-

lable multi-agent systems is an open and dense set in

the space Mof multi-agent systems.

Proof:

Taking into account that if a nk-order minor is

non-zero, it is in the neighborhood, we conclude that

all small perturbations of this minor are non-zero, and

in particular for all perturbed minors corresponding to

a perturbed system, so the set Cis open in M, and for

density it sufﬁces to take into account the fact that the

function rank : ICm×n−→ IR is lower semicontinuous

in the space of rectangular matrices of size r×sfor

all pair of non-zero positive numbers rand s.⊓⊔

Then, the set of controllable systems is the

union of orbits of controllable systems. Each non-

controllable system is located on the frontier of one

of these orbits.

Proposition 18 For each non-controllable system,

there exists a neighborhood of this system containing

controllable systems. These controllable systems can

be described using the miniversal deformation of the

given system.

For the case of the non-controllable systems 6,

and in order to preserve the ﬁxed feedback K, the

equivalence relation is reduced to external feedback

in the following sense

Deﬁnition 19 Two systems ˙

X(t) = (A+BK(L ⊗

In))X(t) + CV(t)and ˙

X(t) = (A1+BK(L ⊗

In))X(t) + C1V(t)are equivalent if and only if

A1=A+CF,C1=CR

for some F∈QkMp×n(IC) and R∈QkGl(p, IC)

This relation can be given in a matritial expres-

sion:

A10C1=A0C



F R





where 



F R



∈ G1⊂ G

G1has the structure group with the same opera-

tion as G.

Proposition 20 For a non-controllable system of type

6, If in a neighborhood of it, there exists a controllable

one it can be found in (A+X,0,C+Z)with (X,0,Z)

in a neighborhood of 0∈Mverifying that X∗C= 0

and Z∗C= 0.

Proof: It sufﬁces to consider the miniversal deforma-

tion restricted to this case.

⊓⊔

6 Conclusion

In this work, we have explored whether, with a small

perturbation of a non-controllable multi-agent linear

system with a previously established interrelationship

topology, we can obtain a controllable multi-agent

system with the same topology. For this, we have used

geometric techniques deﬁning transversal families to

the set of equivalent systems under a previously de-

ﬁned equivalence relation that preserves controllabil-

ity.

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auser Boston.

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E-ISSN: 2224-2678

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