I. Performance Specification

Consider the multivariable feedback control system shown in Fig. 3.1.53 The multivariable stability margins, and the performance of such systems, are stated in terms of the singular values of the closed-loop transfer function matrices from r to e and y. These relationships are14

(3.4)

(3.5)

where

(3.6)

The two matrices S( s) and T( s) are known respectively as the sensitivity and complementary sensitivity matrices. The singular value Bode plots for the transfer function matrices S( j ?) and T( j ?) play an important role in robust multivariable control system design.16

Figure 3.1: Block diagram of the multivariable feedback control system.

The singular values of S( s) determine the disturbance attenuation, since S( s) is in fact the closed-loop transfer function from disturbance d to plant output y. A disturbance attenuation performance specification may therefore be written as

(3.7)

where ? ( j ?) ? is the desired disturbance attenuation factor. Allowing W ₁( j ?) to depend on the frequency ? enables one to specify a different attenuation factor for each frequency.7

The singular value Bode plots of T(s) are used to measure the stability margins of multivariable feedback designs in the face of additive ( ? _A) and multiplicative ( ? _M) plant perturbations.16 These are shown in Fig. 3.2.

Figure 3.2: Additive/multiplicative uncertainty.

The multiplicative stability margin is...