Homework 3
Advanced Topics in Control: Distributed Systems and Control
Homework set #3
Note: The homework is due Friday, April 17 before 10:15 am.
Exercise 1: The Green matrix of a Laplacian matrix (i.e., alternative definition
and property of the pseudoinverse Laplacian matrix)
Assume L is the Laplacian matrix of a weighted connected undirected graph with n nodes. Show
that
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1. the matrix L + n1 1n 1Tn is positive definite,
2. the so-called Green matrix
−1 1
1
− 1n 1Tn
X = L + 1n 1Tn
n
n
is the unique solution to the system of equations:
(
LX = In − n1 1n 1Tn ,
1Tn X = 0Tn ,
3. X = L† , where L† is defined in Exercise 6.6.
Exercise 2: Properties of saddle points
Prove Lemma 7.3:
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Let L = LT ∈ Rn×n be a symmetric Laplacian associated to an undirected, connected, and weighted
graph, and consider the Lagrangian L, where each fi is strictly convex and twice continuously
differentiable for all i ∈ {1, . . . , n}. Then
1. if (y ∗ , z ∗ ) ∈ Rn × Rn is a saddle point of L, then so is (y ∗ , z ∗ + α1n ) for any α ∈ R;
2. if (y ∗ , z ∗ ) ∈ Rn × Rn is a saddle point of L, then y ∗ = x∗ 1n where x∗ ∈ R is a solution of the
original optimization problem (7.3); and
3. if x∗ ∈ R is a solution of the original optimization problem (7.3), then there are z ∗ ∈ Rn and
∂ ˜ ∗
y ∗ = x∗ 1n satisfying Lz ∗ + ∂y
f (y ) = 0n so that (y ∗ , z ∗ ) is a saddle point of L.
Exercise 3: The edge Laplacian matrix
For an unweighted undirected graph, analogously to the Laplacian matrix of L = BB T ∈ Rn×n , we
can define the edge Laplacian matrix Ledge = B T B ∈ Rm×m . Show that
1. ker(Ledge ) = ker(B);
2. for an acyclic graph Ledge is nonsingular; and
3. neglecting distinct eigenvalue multiplicities, L and Ledge have the same eigenvalues and, therefore, the same rank.
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Advanced Topics in Control: Distributed Systems and Control
Homework set #3
Exercise 4: Continuous distributed estimation from relative measurements
Consider the continuous distributed estimation algorithm given by the affine Laplacian flow (8.4).
Show that for an undirected and connected graph G and appropriately initial conditions x
ˆ(0) = 0n ,
the affine Laplacian flow (8.4) converges to the unique solution x
ˆ∗ of the estimation problem given
in Lemma 8.5.
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Exercise 5: Averaging with distributed integral control
Consider a Laplacian flow implemented as a relative sensing network over a connected and undirected graph with incidence matrix B ∈ Rn×|E| and weights aij > 0 for i, j ∈ E, and subject to a
constant disturbance term η ∈ R|E| , as shown in Figure 1.
..
u
_
.
x
x˙ i = ui
..
BT
B
⌘
+
.
+
..
z
.
aij
..
.
y
Figure 1: A relative sensing network with a constant disturbance input η ∈ R|E| .
1. Derive the dynamic closed-loop equations describing the model in Figure 1.
2. Show that asymptotically all states x(t) converge to some constant vector x∗ ∈ Rn depending
on the value of the disturbance η, i.e., x∗ is not necessarily a consensus state.
Consider the system from Figure 1 with a distributed integral controller forcing convergence to
consensus, as shown in Figure 2. Recall that 1s is the the Laplace symbol for the integrator.
3. Derive the dynamic closed-loop equations describing the model in Figure 2.
4. Show that the distributed integral controller in Figure 1 asymptotically stabilizes the set of
steady states (x∗ , p∗ ), where x∗ ∈ span(1n ) corresponds to consensus.
Hint: To show stability, use Lemma 7.4.
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Advanced Topics in Control: Distributed Systems and Control
..
u
_
.
Homework set #3
x
x˙ i = ui
..
.
BT
B
⌘
+
+
+
p
z
..
..
.
aij
..
.
..
.
y
.1
s
Figure 2: Relative sensing network with a disturbance η ∈ R|E| and distributed integral action.
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