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# Documentation

## Mixed-Sensitivity Loop Shaping

A popular alternative approach to loopsyn loop shaping is H mixed-sensitivity loop shaping, which is implemented by the Robust Control Toolbox™ software command:

```K=mixsyn(G,W1,[],W3)
```

With mixsyn controller synthesis, your performance and stability robustness specifications equations (2-2) and (2-4) are combined into a single infinity norm specification of the form

${‖{T}_{{y}_{1}{u}_{1}}‖}_{\infty }\le 1$

${T}_{{y}_{1}{u}_{1}}\stackrel{def}{=}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\left[\begin{array}{cc}{W}_{1}& S\\ {W}_{3}& T\end{array}\right].$

The term ${‖{T}_{{y}_{1}{u}_{1}}‖}_{\infty }$ is called a mixed-sensitivity cost function because it penalizes both sensitivity S(s) and complementary sensitivity T(s). Loop shaping is achieved when you choose W1 to have the target loop shape for frequencies ω < ωc, and you choose 1/W3 to be the target for ω > ωc. In choosing design specifications W1 and W3 for a mixsyn controller design, you need to ensure that your 0 dB crossover frequency for the Bode plot of W1 is below the 0 dB crossover frequency of 1/W3, as shown in Singular Value Specifications on L, S, and T, so that there is a gap for the desired loop shape Gd to pass between the performance bound W1 and your robustness bound ${W}_{3}^{-1}$. Otherwise, your performance and robustness requirements will not be achievable.

MIXSYN H Mixed-Sensitivity Loop Shaping Ty1 u1