A Survey of Observers for Nonlinear Dynamical Systems - Data Assimilation for Atmospheric, Oceanic and Hydrologic Applications - page 16

For linear systems, ( 1.22 ) reduces to a Riccati differential equation. Consider

x D Ax C B

w
y D Cx C D

v

x.0/ D 0

Its H 1 filter has the following form

x.t/ D Ax C K.t/.y.t/ C x.t//
K.t/ D Q.t/C T

where Q.t/ is a solution of the Riccati differential equation

Q.t/ D AQ.t/ C Q.t/A T C BB T Q.t/.C T C 2 I/Q.t/
Q.0/ D 0

This is an observer similar to Kalman filter. However, this filter can be modified to
estimate a linear combination of the state variables, z D Lx . There are many topics

and papers on H 1 filters, for instance ( Green and Limebeer 1995 ).

1.4.3 Minimum Energy Estimation

L 2 functions. We Consider a system model ( 1.21 ) in which the noises are unknown

seek the initial state error x 0 and the noises w .t/ and v .t/ of “minimum energy”

Z t

1
2

jj x 0 jj 2 C 1
2

./ jj 2 Cjj v ./ jj 2 d jj w

0

where v is consistent with the observation. The quality of estimation is defined

by an optimal control problem

.t/

1
2

Z t

jj x 0 jj 2 C 1
2

./ jj 2 Cjj y./ h.x.// jj 2 d Q.x;t/ D inf

x 0 ; w . /

jj w

0

in which x./ is subject to ( 1.21 )and x.t/ D x . The optimal estimation is given by

the one that minimizes Q.x;t/ ,

x.t/ D argmin x Q.x;t/

This approach is similar to and predates a H 1 estimation, except that it does not

require the searching for gain . The dynamic programming approach yields a

partial differential equation for Q.x;t/