Scilab 5.5.2
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Please note that the recommended version of Scilab is 2025.0.0. This page might be outdated.
See the recommended documentation of this function
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:
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)
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