lqe

linear quadratic estimator (Kalman Filter)

Calling Sequence

[K,X]=lqe(P21)

Arguments

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

Description

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] P21=syslin(‘c’,A,B1,C2,D21) or P21=syslin(‘d’,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).

Examples

//Assume the equations
//.
//x = Ax + Ge
//y = Cx + v
//with
//E ee' = Q_e,    Evv' = R,    Eev' = N
//
//This is equivalent to
//.
//x = Ax  + B1 w
//y = C2x + D21 w
//with E { [Ge ]  [Ge v]' } = E { [B1w ] [B1w D21w]' } = bigR =
//         [ v ]                  [D21w]
//
//[B1*B1'  B1*D21';
// D21*B1'  D21*D21']
//=
//[G*Q_e*G' G*N;
// N*G' R]

//To find (B1,D21) given (G,Q_e,R,N) form bigR =[G*Q_e*G' G*N;N'*G' R].
//Then [W,Wt]=fullrf(bigR);  B1=W(1:size(G,1),:);
//D21=W(($+1-size(C2,1)):$,:)
//
//P21=syslin('c',A,B1,C2,D21);
//[K,X]=lqe(P21);

//Example:
nx=5;ne=2;ny=3;
A=-`diag`_(1:nx);G=`ones`_(nx,ne);
C=`ones`_(ny,nx); Q_e(ne,ne)=1; R=`diag`_(1:ny); N=`zeros`_(ne,ny);
bigR =[G*Q_e*G' G*N;N'*G' R];
[W,Wt]=`fullrf`_(bigR);B1=W(1:`size`_(G,1),:);
D21=W(($+1-`size`_(C,1)):$,:);
C2=C;
P21=`syslin`_('c',A,B1,C2,D21);
[K,X]=lqe(P21);
//Riccati check:
S=G*N;Q=B1*B1';
(A-S*`inv`_(R)*C2)*X+X*(A-S*`inv`_(R)*C2)'-X*C2'*`inv`_(R)*C2*X+Q-S*`inv`_(R)*S'

//Stability check:
`spec`_(A+K*C)

See Also

• lqr LQ compensator (full state)
• observer observer design