. INTRODUCTION OF THE FREQUENCY

Figwe 10.2. Introduction of frequency as a parametric uncertainty.

In order to apply the previous Lemma, an LFT model is to be derived for the dynamic system M(s), in which the frequency appears as a real parameter. The issue is more precisely to determine a complex matrix H such that, for a given strictly positive frequency uiq:

M(j(uj0 + 6u>)) = Fi(H(loq), Swim) V6u> > – loq (10.19)

Let (A, B,C, D) a state-space representation of M (s), and m the dimen­sion of matrix A. It is well known that (see Figure 10.2 with s = j(uiQ + 6ш)):

/6u>-ui0, M(j(uJo + Sui)) = F,(Mo, 7™ ) (Ю.20)

V шо + дш/

. INTRODUCTION OF THE FREQUENCY

with:

. INTRODUCTION OF THE FREQUENCY

Further note that:

with:

Подпись:T= — ( Ir? Ijp )

CUq V lm *m)

As a consequence:

. INTRODUCTION OF THE FREQUENCY

. INTRODUCTION OF THE FREQUENCY

where H(h ) is the star product of Mo and T (see subsection 3.1 of chapter 3):

. INTRODUCTION OF THE FREQUENCY

Interestingly, the LFT Fi(H(wo),SwIm) remains well-posed for Sw =

 

Fi(H(u0), – щіт) = D — CA~lB = M{0) (10.26)

Equation (10.19) is thus valid for all 8w > —wo.

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