CHECKING A FREQUENCY DOMAIN TEMPLATE

Performance can be defined as the minimization of the weighted H00 norm of a closed loop transfer matrix. Let e. g. the sensitivity function S, i. e. the transfer between the reference input and the tracking error. A template W is defined to reflect the design specifications. Nominal performance is achieved if:

In an alternative way, performance can be defined as the minimization of an error signal z(t) which is the response to an exogenous input signal w(t). Here again, if M\ denotes the transfer between w and z, nominal performance is ensured if:

ЦЛГ11Ц00 < 1 (1-42)

A structured model perturbation F 2 is introduced in the closed loop (see Figure 1.13). The transfer between w and z now corresponds to the LFT ЩМ, А2):

Fi(M, Д2) = Mu + МігДг(/ — M22A2)~l M21 (L43)

where M is partitioned as in equation (1.39).

Frequency и is fixed. Performance is guaranteed at this frequency despite the model uncertainty Д2 inside the unit ball BA2 if and only if:

(/) Robust stability of the closed loop is ensured:

/WM22 CM) < 1

(ii) The transfer between w and z remains lower than 1 despite the model uncertainty Д2:

max a(Fi(M{juj), Д2)) < 1 а2єв&2

A fictitious full complex block Ді is added (see Figure 1.12). With reference to subsection 4.3, it can be remarked that:

a{Ft(M(juj),A2)) = (Fi(M(jw), Д2)) (1-46)

Let the augmented model perturbation Д of equation (1.38). Using the Main Loop Theorem, it can be claimed that the two above conditions are satisfied if and only if:

Мд(M(juj)) < 1 (1.47)

As a consequence, the robust performance problem reduces to an aug­mented robust stability problem, in which a fictitious performance block Дх is added (Doyle, 1985). Note that this block is possibly struc­tured: if signals w and z are decomposed as w = [uq,… ,wm]T and z = [zi,… ,zm]T, and if we are just interested in the transfer functions between scalar signals w% and z*, the performance block Дх will be chosen as a complex diagonal model perturbation.

Remark: equation (1.47) corresponds to a test ("Is robust performance guaranteed inside the unit ball ЯД2 ?"). As proved later, the compu­tation of the robustness margin (i. e. the maximal value of k, such that robust performance is guaranteed inside кВД2) is a skewed ц problem.