Main Content

Mixed-sensitivity *H*_{∞} synthesis
method for robust control loop-shaping design

`[`

computes a controller that minimizes the `K`

,`CL`

,`gamma`

,`info`

] = mixsyn(`G`

,`W1,W2,W3`

)*H*_{∞}
norm of the weighted closed-loop transfer function

$$M\left(s\right)=\left[\begin{array}{c}{W}_{1}S\\ {W}_{2}KS\\ {W}_{3}T\end{array}\right],$$

where *S* = (*I* +
*GK*)^{–1} and *T* = (*I* –
*S*) is the complementary sensitivity of the following control system.

You choose the weighting functions `W1,W2,W3`

to shape the frequency
responses for tracking and disturbance rejection, controller effort, and noise reduction and
robustness, respectively. For details about how to choose weighting functions, see Mixed-Sensitivity Loop Shaping.

`mixsyn`

computes the controller `K`

that yields
the minimum ||*M*(*s*)||_{∞}, which is
returned as `gamma`

. For the returned controller
*K*,

$$\begin{array}{c}{\Vert S\Vert}_{\infty}\le \gamma \left|{W}_{1}^{-1}\right|\\ {\Vert KS\Vert}_{\infty}\le \gamma \left|{W}_{2}^{-1}\right|\\ {\Vert T\Vert}_{\infty}\le \gamma \left|{W}_{3}^{-1}\right|.\end{array}$$

`[`

calculates a controller for the target performance level `K`

,`CL`

,`gamma`

] = mixsyn(`G`

,`W1,W2,W3`

,`gamTry`

)`gamTry`

.
Specifying `gamTry`

can be useful when the optimal controller performance
is better than you need for your application. In that case, a less-than-optimal controller
can have smaller gains and be better conditioned numerically. When
`W1,W2,W3`

capture the desired limits on the gains of
*S*, *KS*, and *T*, use
`gamtry`

= 1 to just enforce those limits.

If `gamTry`

is not achievable, `mixsyn`

returns
`[]`

for `K`

and `CL`

, and
`Inf`

for `gamma`

.

`[`

searches the range `K`

,`CL`

,`gamma`

] = mixsyn(`G`

,`W1,W2,W3`

,`gamRange`

)`gamRange`

for the best achievable performance.
Specify the range with a vector of the form `[gmin,gmax]`

. Limiting the
search range can speed up computation by reducing the number of iterations performed by
`mixsyn`

to test different performance levels.

`[`

specifies additional computation options. To create `K`

,`CL`

,`gamma`

] = mixsyn(___,`opts`

)`opts`

, use `hinfsynOptions`

.
Specify `opts`

after all other input arguments.

`mixsyn`

uses your weighting functions to generate an augmented plant
`P = augw(G,W1,W2,W3)`

. It then invokes `hinfsyn`

to
find a controller that minimizes the *H*_{∞} norm of the
closed-loop transfer function *M*(*s*) =
*LFT*(*P*,*K*). For details, see Mixed-Sensitivity Loop Shaping.