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Ajuda Scilab >> CACSD > lqe

# lqe

### 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)```