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

Ty1u11

Ty1u1=def[W1SW3T].

The term Ty1u1 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 W31. Otherwise, your performance and robustness requirements will not be achievable.

MIXSYN H Mixed-Sensitivity Loop Shaping Ty1 u1

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