lqe

lqe — linear quadratic estimator (Kalman Filter)

Calling sequence

[K,X]=lqe(P21)

`P21`	: `syslin` list
`K, X`	: real matrices

lqe returns the Kalman gain for the filtering problem in continuous or discrete time.

P21 is a syslin list representing the system P21=[A,B1,C2,D21]

The input to P21 is a white noise with variance:



     [B1 ]               [Q  S]
BigV=[   ] [ B1' D21'] = [    ]
     [D21]               [S' R]

X is the solution of the stabilizing Riccati equation and A+K*C2 is stable.

In continuous time:



(A-S*inv(R)*C2)*X+X*(A-S*inv(R)*C2)'-X*C2'*inv(R)*C2*X+Q-S*inv(R)*S'=0



K=-(X*C2'+S)*inv(R)

In discrete time:



X=A*X*A'-(A*X*C2'+B1*D21')*pinv(C2*X*C2'+D21*D21')*(C2*X*A'+D21*B1')+B1*B1'

K=-(A*X*C2'+B1*D21')*pinv(C2*X*C2'+D21*D21')

xhat(t+1)= E(x(t+1)| y(0),...,y(t)) (one-step predicted x) satisfies the recursion:



xhat(t+1)=(A+K*C2)*xhat(t) - K*y(t).

F. D.