..
1 xn
??????????????»
?·a1
a2 ??, Yn = MnA,
46
Chapter 2 Algorithms for Robotic Deployment of WSN
Fig. 4. Sampling sequence for a Gaussian field distribution (original - (a), estimated
-(b)) is generated to minimize error variance using the infinity norm (c), and the
2-norm (d). First fifteen sampling locations are shown.
47
(a) (b)
(c)
(d)
Dan O. Popa and Frank L. Lewis
Fig. 5. Sampling sequence for a Gaussian field distribution where the Gaussian
centers are unknown, generated using fmincon and the 2-norm of the linear field
parameters. Convergence of all parameters, including the Gaussian centers to their
true values is observed.
48
(a) (b)
(c) (d)
(e) (f
(g) (h)
Chapter 2 Algorithms for Robotic Deployment of WSN
in which Yn is the measurement vector after n samples, Mn is the basis function
matrix from equation (4), and A is the vector of unknown parameters.
Note that the measurement values yi are the corresponding measurement values
zi in equation (4). The least-square estimate of ???A??™ after n measurements
will be: ?† A = M+
n Yn = (MT
n Mn)??’1MT
n Yn,
?† A = ??±????????????
?· 1 1 ?· ?· ?· 1
x1 x2 ?· ?· ?· xn ????®???????????°
1 x1
1 x2
...
1 xn
??????????????»
???????????????
??’1
?· 1 1 ?· ?· ?· 1
x1 x2 ?· ?· ?· xn ????®???????????°
Z1
Z2
...
Zn
??????????????»
,
) ?† A =
1
nPn
i=1 x2
i
??’ (Pn
i=1 xi)2 ?·Pn
i=1 yiPn
i=1 x2
i ??’ (Pn
i=1 xiyi)Pn
i=1 xi
??’Pn
i=1 yiPn
i=1 xi + n(Pn
i=1 xiyi) ??
=
1
f0(x) ?·f1(x, y)
f2(x, y) ?? = ?·F1(x, y)
F2(x, y) ??.
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