Show PDF File - robotics - Universidad de Zaragoza

Σˆx cen
V
Qii = Pii − 1
n
n
j=1
(Pij + Pji) + 1
n 2
n
n
j=1 j ′ =1
Pjj ′,
i ∈ V
blkTr Σˆx cen
n n
= Qii = Pii − V
i=1 i=1
1
n n
Pij
n
i=1 j=1
− (1n ⊗ Ip) T Σˆx a V (1n ⊗ Ip)/n.
= blkTr Σˆx a V
Σˆx a V (1n ⊗ Ip) T Σˆx a V (1n ⊗ Ip) 0
blkTr Σˆx cen
− blkTr Σˆx V
a
0
V
A Tr(blkTr(A)) = Tr(A)
blkTr Σˆx cen
− blkTr Σˆx V
a
0 blkTr Σˆx V
cen
V
blkTr Σˆx a
V
Tr(Σˆx cen)
= Tr(blkTr(Σˆx cen
V V )) ≤ Tr(blkTr(Σˆx a V )) = Tr(Σˆx a V ).
Tr(Σˆx a V
) − Tr(Σˆx cen
V
) = 1
n
n n i=1 j=1 Tr(Pij).
ˆx a a
0
k Σˆx a V = [Pij] Pij = 0
i, j ∈ {1, . . . , k} blkTr(Σˆx cen
V ) = blkTr(Σˆx a V ) − n i=k+1
n
i=k+1
n
j=k+1 Pij/n 0
n
j=k+1 Pij/n,
i ∈ V ˆx cen
i
ˆx cen
V = ((ˆxcen 1 ) T , . . . , (ˆx cen
n ) T ) T
ˆx cen
1 + · · · + ˆx cen
n = 0 (1n ⊗
Ip)(I − Hcen) = 0 i j ˆx cen
i = ˆx cen
j − ˆx a j + ˆx a i
ˆx a V = ((ˆxa 1) T , . . . , (ˆx a n) T ) T
ˆx a V
ˆx cen
i (0) = 0 i ∈ V
t
ˆx cen
i (t + 1) =
Wi,j(ˆx cen
j (t) − ˆx a j + ˆx a i )
j∈Ni∪{i}
i ∈ V W = [Wi,j] Wi,j > 0
(i, j) ∈ E Wi,j = 0 j /∈ Ni Wi,i ∈ [α, 1] Wi,j ∈ {0} ∪ [α, 1]
i, j ∈ V α ∈ (0, 1]
−ˆx a j + ˆx a i ze
e = (j, i)
ˆze ˆz = (A ⊗ Ip) T ˆx a V , ˆz = ((ˆz1) T , . . . , (ˆzm) T ) T
t
ˆx a i , ˆx a j ˆx a i (t), ˆx a j (t)