Exercise session 5

Advanced Topics in Control: Distributed Systems and Control
Exercise Class #5
1 Continuous Time Linear Systems
Exercise 2.12: Cyclic pursuit
Consider four mobile robotic vehicles, indexed by i ∈ {1, 2, 3, 4}. We model each robot as fullyactuated kinematic point mass, that is, we write p˙i = ui , where pi ∈ C is the position of robot i
in the plane and ui ∈ C is its velocity command. The robots are equipped with onboard cameras
as sensors. The task of the robots is rendezvous at a common point (while using only onboard
sensors). A simple strategy to achieve rendezvous is cyclic pursuit: each robot i picks another
robot, say i + 1, and pursues it. This gives rise to the control ui = pi+1 − pi and the closed-loop
system
  
 
p˙1
−1 1
0
0
p1
p˙2   0 −1 1


0  p2 
 =
.
p˙3   0
0 −1 1  p3 
p˙4
1
0
0 −1 p4
A simulation of the cyclic-pursuit dynamics is shown in Figure 1. Your tasks are as follows.
0.3
0.2
0.1
y
0
-0.1
-0.2
-0.3
-0.4
-0.3
-0.2
-0.1
x
0
0.1
0.2
0.3
Figure 1: Four robots with initial positions that perform a cyclic pursuit to rendezvous at
•.
1. Prove that the center of mass
average(p(t)) =
4
X
pi (t)
i=1
4
is constant for all t ≥ 0. Notice that this is equivalent to saying d/dt average(p(t)) = 0.
2. Prove that the robots asymptotically rendezvous at the initial center of mass, that is,
lim pi (t) = average(p(0)) for i ∈ {1, . . . , 4} .
t→∞
3. Prove that if the robots are initially arranged in a square formation, they remain in a square
formation under cyclic pursuit.
1
Advanced Topics in Control: Distributed Systems and Control
Exercise Class #5
Hint: Recall that for a matrix A with semisimple
eigenvalues, the solution to the equation x˙ = Ax
P
is given by the modal expansion x(t) = i eλi t vi wiT x(0), where λi is an eigenvalue, and vi and wi
are the associated right and left eigenvectors pairwise normalized to wiT vi = 1.
Solution:
distributed as hard copy in class
2 Review of Laplacian Matrices
1. The Laplacian matrix of the weighted digraph, G is
L = Dout − A.
2. In components, L = (`ij )i,j∈{1,...,n}
(
−aij ,
`ij = Pn
h=1,h6=i aih ,
if i 6= j,
if i = j,
or, for an unweighted undirected graph,


if {i, j} is an edge, not self-loop,
−1,
`ij = d(i),
if i = j,


0,
otherwise.
3. The graph, G is undirected (i.e., symmetric adjacency matrix) if and only if L is symmetric.
In this case, Dout = Din = D and A = AT .
4. For L = LT (i.e., aij = aji ), we have:
T
x Lx =
n
X
xi (Lx)i =
i=1
X
aij (xi − xj )2
{i,j}∈E
Exercise 6.3: The Laplacian of an undirected graph is positive semidefinite
Give an alternative proof, without relying on the Gerˇsgorin disks Theorem 2.5, that the Laplacian
matrix L of an undirected weighted graph is symmetric positive semidefinite. (Note that the proof
of Lemma 6.3 relies on Gerˇsgorin disks Theorem 2.5).
Solution:
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Exercise 6.4: The Laplacian of a weight-balanced digraph
Prove that, if the digraph G is weight-balanded, then L + LT is positive semidefinite.
Hint: Recall the proof of Lemma 6.3.
2
Advanced Topics in Control: Distributed Systems and Control
Solution:
Exercise Class #5
distributed as hard copy in class
Exercise 6.5: Disagreement function
The quadratic form associated with B = B T ∈ Rn×n is the function x 7→ xT Bx. On a weighted
digraph graph, G with n nodes, the disagreement function 7→ ΦG Rn R is
ΦG (x) =
n
1 X
aij (xj − xi )2 .
2
i,j=1
Show the following are true.
1. ΦG is the quadratic form associated with the symmetric positive-semidefinite matrix
1
P = (Dout + Din − A − AT );
2
2. P =
Solution:
1
2
L + L(rev) , where the Laplacian of the reverse digraph is L(rev) = Din − AT .
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Exercise 6.8: Monotonicity of Laplacian eigenvalues
Consider a symmetric Laplacian matrix L ∈ Rn×n associated to a weighted and undirected graph
G = V, E, A. Assume G is connected and let λ2 (G) > 0 be its algebraic connectivity, i.e., the
second-smallest eigenvalue of L. Show that
1. λ2 (G) is a monotonically non-decreasing function of each the weights aij , {i, j} ∈ E; and
2. λ2 (G) is monotonically non-decreasing function in the edge set in the following sense: λ2 (G) ≤
λ2 (G0 ) for any graph G0 = (V, E 0 , A0 ) with E ⊂ E 0 and aij = a0ij for all {i, j} ∈ E.
Hint: Use the disagreement function.
Solution:
distributed as hard copy in class
3