leqr

Scilab Function
Last update : 12/01/2004

leqr - H-infinity LQ gain (full state)

### Calling Sequence

  `[K,X,err]=leqr(P12,Vx)`

### Parameters

- `P12` : `syslin` list
- `Vx` : symmetric nonnegative matrix (should be small enough)
- `K,X` : two real matrices
- `err` : a real number (l1 norm of LHS of Riccati equation)

### Description

`leqr` computes the linear suboptimal H-infinity LQ full-state gain for the plant `P12=[A,B2,C1,D12]` in continuous or discrete time.

`P12` is a `syslin` list (e.g. `P12=syslin('c',A,B2,C1,D12)` ).

          [C1' ]               [Q  S]
          [    ]  * [C1 D12] = [    ]
          [D12']               [S' R]

`Vx` is related to the variance matrix of the noise `w` perturbing `x` ; (usually `Vx=gama^-2*B1*B1'` ).

The gain `K` is such that `A + B2*K` is stable.

`X` is the stabilizing solution of the Riccati equation.

For a continuous plant:

    (A-B2*inv(R)*S')'*X+X*(A-B2*inv(R)*S')-X*(B2*inv(R)*B2'-Vx)*X+Q-S*inv(R)*S'=0

    K=-inv(R)*(B2'*X+S)

For a discrete time plant:

    X-(Abar'*inv((inv(X)+B2*inv(R)*B2'-Vx))*Abar+Qbar=0

    K=-inv(R)*(B2'*inv(inv(X)+B2*inv(R)*B2'-Vx)*Abar+S')

with `Abar=A-B2*inv(R)*S'` and `Qbar=Q-S*inv(R)*S'`

The 3-blocks matrix pencils associated with these Riccati equations are:

                   discrete                        continuous
       |I  -Vx  0|   | A    0    B2|       |I   0   0|   | A    Vx    B2|
      z|0   A'  0| - |-Q    I    -S|      s|0   I   0| - |-Q   -A'   -S |
       |0   B2' 0|   | S'   0     R|       |0   0   0|   | S'   -B2'   R|

### See Also

`lqr` ,

### Author

F.D.;