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.TH lqe 1 "April 1993" "Scilab Group" "Scilab Function"
.so ../sci.an
.SH NAME
lqe - linear quadratic estimator (Kalman Filter)
.SH CALLING SEQUENCE
.nf
[K,X]=lqe(P21)
.fi
.SH PARAMETERS
.TP 10
P21
: \fVsyslin\fR list
.TP 10
K, X
: real matrices
.SH DESCRIPTION
\fVlqe\fR returns the Kalman gain for the filtering
problem in continuous or discrete time.
.LP
\fVP21\fR is a \fVsyslin\fR list representing the system \fVP21=[A,B1,C2,D21]\fR
.LP
The input to \fVP21\fR is a white noise with variance:
.nf
[B1 ] [Q S]
BigV=[ ] [ B1' D21'] = [ ]
[D21] [S' R]
.fi
\fVX\fR is the solution of the stabilizing Riccati
equation and \fVA+K*C2\fR is stable.
.LP
In continuous time:
.nf
(A-S*inv(R)*C2)*X+X*(A-S*inv(R)*C2)'-X*C2'*inv(R)*C2*X+Q-S*inv(R)*S'=0
.fi
.nf
K=-(X*C2'+S)*inv(R)
.fi
.LP
In discrete time:
.nf
X=A*X*A'-(A*X*C2'+B1*D21')*pinv(C2*X*C2'+D21*D21')*(C2*X*A'+D21*B1')+B1*B1'
.fi
\fVK=-(A*X*C2'+B1*D21')*pinv(C2*X*C2'+D21*D21')\fR
.LP
\fVxhat(t+1)= E(x(t+1)| y(0),...,y(t))\fR (one-step predicted \fVx\fR)
satisfies the recursion:
.nf
xhat(t+1)=(A+K*C2)*xhat(t) - K*y(t).
.fi
.SH SEE ALSO
lqr
.SH AUTHOR
F. D.
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