# COMPUTATIONAL METHOD

The following Lemma is introduced as a preliminary. It can be proved in the same way as Lemma 2..1 of chapter 8.

LEMMA 3..1 Let M a complex matrix, Ді a mixed model perturbation which is to be maintained inside its unit ball, and Д2 a mixed model perturbation whose size is free. Then:

We now come back to the initial problem. Remember that Д 1 is a mixed model perturbation which gathers the parametric uncertainties and neglected dynamics, while Д2 = diag(8{,… ,6^) is a structured complex model perturbation corresponding to time delays. The associated interconnection structure M – Д, with Д = diag(AьДг), is presented in Figure 11.8. Note that matrices M and Д are the values of the transfer matrices M(s) and Д(й) at s = ju. Frequency to is fixed in the following.

The idea is simply to use the first v upper bound of chapter 8 in order to compute a lower bound of the robust delay margin. A is to be maintained inside its unit ball, while the size of Д2 is free. The difficulty is to prove that a lower bound of the robust delay margin is indeed obtained with this approach.

Let mi the dimension of Д*. In the same way as in chapter 8 (section L), scaling matrices D_{ associated to perturbations A{ are introduced (*.e DiAi = AiDi)- D and £>2 are then defined as:

so that D = D + D2 is a scaling matrix associated to Д. a is an upper bound of і’д(М) if the following LMI is satisfied:

When multiplying the above inequality on the left and on the right by:

The above LMI becomes: XMfDi + Di)MX – XDX + j(XGMX — XM’GX) < a2P2 (11.38) |

Let:

Di = XDiX M = X~lMX G = XGX

Equation (11.38) can be rewritten as:

M*{DX + P2)M – Di+ j(GM – M*G) < a2P2

The key point is to note that the above LMI implies that a is also an upper bound of і’д(М), with:

A = diag(A ЬД2) (11.42)

and A2 is a full complex block with the same dimensions as A2.

As a consequence, using Lemma 3..1 and the fact that 2(H) = o(H) for any complex matrix H, it can be claimed that:

Noting that FU{M, A) = D^2Fu(M, Ai)D21^2, the above property is rewritten as:

(t(D2^2Fu(M, A)D2 l^2) < a (11.44)

Remember that FU{M, A{) is the transfer matrix seen by the complex structured perturbation A2, which models the time delays (see Figure 11.8). Following subsections 3.1 and 3.2, the above equation means that the sufficient condition of robust stability, with respect to uncertain time delays, is satisfied for all Ai Є BA 1. a can thus be used in equations (11.25) and (11.27) to compute lower bounds of the robust MIMO phase and delay margins.

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