1. faculty of mathematics and
natural sciences
Date 28.03.2012
Controllability of Diffusively-coupled
Multi-agent Systems with Switching
Topologies
Shuo Zhang, M. Kanat Camlibel and Ming Cao
1

2. faculty of mathematics and
natural sciences
Date 28.03.2012
Outline
• Diffusively Coupled Multi-Agent Systems with Switching
Topologies
• Review: Controllability of Linear Switched Systems
• Graph Partitions
• Main Results
• Concluding Remarks
2

3. faculty of mathematics and
natural sciences
Date 28.03.2012
Diffusively Coupled Multi-Agent
System with Switching Topologies
• p simple, undirected graphs: G = (V, E), 1  p
• Vertex set: V = {1, 2, . . . , n}
• Edge set of G: E ⊆ V × V
• Leaders of G
• Set of leaders: V
L
= {ℓ
1
, . . . , ℓ
m
} ⊆ V
• Set of followers: V
F
= V V
L
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4. faculty of mathematics and
natural sciences
Date 28.03.2012
• Diffusively coupled multi-agent system with G and V
L
˙j = −
(j,k)∈E
(j − k), if j ∈ V
F
˙j = −
(j,k)∈E
(j − k) + j, if j ∈ V
L
j: agent j’s state; j: external input to agent j
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5. faculty of mathematics and
natural sciences
Date 28.03.2012
• Compact form
˙ = −L + M
 = 1 . . . n
T
,  = 1 . . . n
T
, L: the Laplacian
matrix of G, M ∈ Rn×n defined by
[M]jk =
1 if j = ℓ
k
0 otherwise.
• Switched multi-agent system
˙(t) = −Lσ(t)(t) + Mσ(t)(t), (1)
σ(t) : R+ → {1, 2, . . . , p}: a right-continuous piecewise
constant switching signal
5

6. faculty of mathematics and
natural sciences
Date 28.03.2012
Controllability of Linear Switched
Systems: Review
• A linear switched system
˙(t) = Aσ(t)(t) + Bσ(t)(t), (0) = 0 (2)
 ∈ Rn: the state,  ∈ Rm: the input, σ(t): switching signal
• W0 =
p
=1
im B, Wk+1 =
p
=1
〈A | Wk〉 for all k 0
• ∃ s.t. Wq = W for all q 
• the controllable subspace of (2): W = W
• W is the smallest of the subspaces that are invariant
under A’s for all  ∈ {1, . . . , p} and contain
p
=1
im B
• (2) is controllable if and only if W = Rn
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7. faculty of mathematics and
natural sciences
Date 28.03.2012
Graph Partitions
• A cell: any nonempty subset of the vertex set V
• A partition of V: a collection of mutually disjoint cells
π = {C1, . . . , Cr} if
r
j=1
Cj = V
• Characteristic matrix P(π) ∈ Rn×r of π
[P(π)]j =
1 if  ∈ Cj
0 otherwise
1  n, 1 j r.
7

8. faculty of mathematics and
natural sciences
Date 28.03.2012
• An example of π of V and P(π)
P(π) =


1 0 0
0 1 0
0 0 1
0 1 0
0 0 1


1
2 43 5
• : the set of all the partitions of V
8

9. faculty of mathematics and
natural sciences
Date 28.03.2012
Partial Order on 
• π1 finer than π2, π1 π2: each cell of π1 is a subset of
some cell of π2
• π1 π2 ⇐⇒ im P(π2) ⊆ im P(π1)
• Example: π1 = {C1, . . . , C5}, C = {}, 1  5
π2 = {{1}, {2, 4}, {3, 5}}
1
2 3 4 5
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10. faculty of mathematics and
natural sciences
Date 28.03.2012
: a Complete Lattice
•  with “ ”: a complete lattice
• Any subset  of  has its greatest lower bound ∧
and its least upper bound ∨ in 
• Lemma 1 Given  = {π1, . . . , π} ⊆ , it holds that

=1
im P(π) = im P(

=1
π)
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11. faculty of mathematics and
natural sciences
Date 28.03.2012
Almost Equitable Partitions
• An almost equitable partition π = {C1, . . . , Cr} of a given
graph G: for every 1  = j n, ∃ a number bj s.t. any
vertex  ∈ C has bj neighbors in Cj
• Lemma 2 A partition π of G is almost equitable if and
only if im P(π) is L-invariant.
D. M. Cardoso et al. Laplacian eigenvectors and eigenvalues and
almost equitable partitions, European Journal of Combinatorics,
28:665-673, 2007
• AEP(G): the set of all the almost equitable partitions of
G
11

12. faculty of mathematics and
natural sciences
Date 28.03.2012
• For the collection of graphs {G1, . . . , Gp} and π of G,
define
AEP(π1, . . . , πp) = {π | π ∈ AEP(G) and π π for all }
• Lemma 3 There exists a unique partition, say
π∗ = π∗(π1, . . . , πp) s.t.
π∗
∈ AEP(π1, . . . , πp)
π π∗
for all π ∈ AEP(π1, . . . , πp)
π π for all π ∈ AEP(π1, . . . , πp) =⇒ π∗
π .
12

13. faculty of mathematics and
natural sciences
Date 28.03.2012
Distance Partitions
• For a connected G = (V, E), the distance partition relative
to a vertex : a partition where each cell is of the form
{ ∈ V | dist(, ) = } for some , 0  dim G
1
2 4 6
3 5
13

14. faculty of mathematics and
natural sciences
Date 28.03.2012
Main Results
• K: controllable subspace of the switched multi-agent
system ˙(t) = −Lσ(t)(t) + Mσ(t)(t)
• G (1  p): set of leaders V
L
= {ℓ
1
, . . . , ℓ
m
}, set of
followers: V
F
= V V
L
, a given partition
πL

= {{ℓ
1
}, . . . , {ℓ
m
}, V
F
}
• The collection of G’s:
AEP(πL
1
, . . . , πL
p
) = {π | π ∈ AEP(G) and π πL

for all }
• Theorem 1 K ⊆ im P(π∗(πL
1
, . . . , πL
p
))
• π∗(πL
1
, . . . , πL
p
): the least upper bound of
AEP(πL
1
, . . . , πL
p
)
14

15. faculty of mathematics and
natural sciences
Date 28.03.2012
• Theorem 2 Suppose that all G’s (1  p) are
connected. Then
dim(K) mx
 ∈ {1, 2, . . . , p}
k ∈ {1, 2, . . . , m}
crd(πD(ℓ
k
; G))
• πD(ℓ
k
; G): the distance partition relative to ℓ
k
in G
• Suppose a given G is connected. Then
dim K mxℓk∈VL crd(πD(ℓk))
– VL: set of leaders
– K: the controllable subspace of the diffusively
coupled multi-agent system associated with G
S. Zhang et al. Controllability of diffusively-coupled multi-agent
systems with general and distance regular coupling topologies.
Proc. of the 50th IEEE CDC, 759-764, 2011
15

16. faculty of mathematics and
natural sciences
Date 28.03.2012
1 1
2 3 4 5
6 2
3
4
5
6
Figure 1: Upper bound achieved
1
12 3 4 5
6
2
3
4
5
6
Figure 2: Lower bound achieved
16

17. faculty of mathematics and
natural sciences
Date 28.03.2012
Concluding Remarks
• Tight bounds of the controllable subspace of a diffusively
coupled multi-agent system with switching topologies
• Open questions
• When each agent has more complex dynamics, say
general linear dynamics
• Choice of leaders to guarantee controllability
• Choice of topologies to guarantee controllability...
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