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State-space or transfer function plant augmentation for
use in weighted mixed-sensitivity *H*_{∞} and *H*_{2} loopshaping
design

P = AUGW(G,W1,W2,W3)

`P = AUGW(G,W1,W2,W3)`

computes
a state-space model of an augmented LTI plant *P*(*s*)
with weighting functions *W*_{1}(*s*), *W*_{2}(*s*),
and *W*_{3}(*s*)
penalizing the error signal, control signal and output signal respectively
(see block diagram) so that the closed-loop transfer function matrix
is the weighted mixed sensitivity

$$T{y}_{1}{u}_{1}\triangleq \left[\begin{array}{c}{W}_{1}S\\ {W}_{2}R\\ {W}_{3}T\end{array}\right]$$

where *S, R* and *T* are given
by

$$\begin{array}{c}S={(}^{I}\\ R=K{(}^{I}\\ T=GK{(}^{I}\end{array}$$

The LTI systems *S* and *T* are
called the *sensitivity* and *complementary
sensitivity,* respectively.

**Plant Augmentation**

For dimensional compatibility, each of the three weights *W*_{1}, *W*_{2} and *W*_{3} must
be either empty, a scalar (SISO) or have respective input dimensions *N*_{y}, *N*_{u},
and *N*_{y} where *G* is *N*_{y}-by-*N*_{u}.
If one of the weights is not needed, you may simply assign an empty
matrix [ ]; e.g., `P = AUGW(G,W1,[],W3)`

is *P*(*s*)
as in the Algorithms section
below, but without the second row (without the row containing `W2`

).

The transfer functions *G*, *W*_{1}, *W*_{2} and *W*_{3} must
be proper,
i.e., bounded as $$s\to \infty $$ or, in the discrete-time
case, as $$z\to \infty $$. Additionally, *W*_{1}, *W*_{2} and *W*_{3} should
be stable. The plant *G* should be stabilizable and
detectable; else, `P`

will not be stabilizable by
any `K`

.

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