leqr
====

H-infinity LQ gain (full state)



Calling Sequence
~~~~~~~~~~~~~~~~


::

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




Arguments
~~~~~~~~~

: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`_ LQ compensator (full state)


.. _lqr: lqr.html