Singular value loop shaping

The singular values of a complex n x m matrix A, denoted by cr,(A) are the nonnegative square-roots of the eigenvalues of A A ordered such that 

The maximum singular value rr of A and the minimum singular value o of A are defined by 

As with a SISO system, a sensitivity function may be defined 

The singular value of the sensitivity funetion irfSfja )) and of the eomplementary sensitivity funetion (r(T(ja )) ean be displayed as Bode plots and play an important role in robust multivariable eontrol system design. 

The singular values of S determine the disturbanee attenuation, and thus a performanee speeifieation may be written 

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Multivartable robust control using H infinity %Singular value loop shaping using the weighted mixed %sensitivity approach nug=200 ... 

For larger attractive interactions, however, the stationary-state locus becomes S-shaped with a hysteresis loop (Fig. 12.2(b)). The critical value of a at which the hysteresis loop first appears can be determined using the singularity theory equations of 7.3.1. The appropriate form of the stationary-state condition F is... 

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