Soft Sensing - Advanced Control and Supervision of Mineral Processing Plants - page 186

But since is minimum, by the Projection Theorem ε is orthogonal to the space

V m spanned by the elements of ϕ T

ε

( e.g. , see Figures 4.11 and 4.12) so

θ 1 θ k θ m

ε = ϕ 1 ( θ 1 − )+···+ ϕ k ( θ k − )+...+ ϕ m ( θ − ), (4.113)

θ 1 θ

ε , ϕ =(( θ 1 − )) ϕ k , ϕ 1 +···+( θ n −

) +

ϕ k , ϕ m = 0 , (4.114)

for k m ,sinceε is orthogonal to ϕ k . A system of m equations is then

formed which in matrix form is
⎡

= 1 , 2 , ...,

⎤

ϕ 1 , ϕ 1 ··· ϕ m , ϕ 1

⎣ . . ⎦

θ

ϕ 1 , ϕ k ··· ϕ m , ϕ k ( θ − )= 0 . (4.115)

. .

, ··· ,

ϕ 1 ϕ m ϕ m ϕ m

P xx

Hence, if P xx is nonsingular, a conditions already required in (4.107) for finding
θ, then the only solution is

θ θ and from (4.112) the error

( θ − )= 0. Thus, θ =

=

ε w in this general LIP case. Figure 4.13 is an illustration of this case in an inner

product space.

y

ε

w

ϕ

ϕ θ

T

y

=

Figure 4.13 When w is orthogonal to ϕ, parameter estimation using the projection of y on ϕgives
thetruevectorθ in the LIP case

4.3.2 Estimation of Expected Values Through Time Averages

The solution to many problems in engineering is given in terms of expected values.
The difficulty is that the probability functions needed to determine those expected