# . TWO NEGLECTED DYNAMICS

• Two neglected dynamics Ai(s) and A2(s) are now introduced at the plant inputs and outputs. For the sake of simplicity, associated weighting functions Wi(s) are not accounted for. These neglected dynamics may especially enable to handle simultaneously uncertainties in the actuators and sensors dynamics.

The aim is here again to transform the uncertain closed loop of Fig­ure 1.4 into the standard interconnection structure M(s) — A(s). As in the previous subsection, M(s) is the transfer matrix between the inputs (wi, w2) and outputs (z, Z2), whereas the structured model perturbation A(s) is:

In the previous subsection, A(s) was an unstructured model perturb­ation, since it corresponded to any transfer matrix satisfying the Hoo inequality (1.3). In the above equation, both Ai(s) and A2(s) are as­sumed to satisfy the Hoo inequality (1.3). There’s however an additional structural information in equation (1.9).

Remark: an unstructured model perturbation A(s) becomes at s = joj a complex matrix A(ju>) without specific structure, which simply satis­fies <( Д (juj) ) < 1 . Д « ) is said to be a full complex block. [3]

At frequency w, this unit ball becomes:

BA(jw) = {ACM = diag(Ai(ju),…, Am(jcu)) | a(A(ju)) < 1K1.12)

Because of the block diagonal structure of A, the inequalities )| A.(s) ||co < 1 andCT(A(ju;)) < 1 reduce to ||Aj(s)||oo < 1 and a(Aj(jw)) < 1 for each block of neglected dynamics.

• As in the previous subsection, the aim is to compute the maximal amount of neglected dynamics, for which the closed loop remains stable. Assuming that the nominal closed loop is asymptotically stable, the ro­bustness margin kmax is the maximal value of k, for which the closed loop remains stable in the presence of neglected dynamics Aj(s) satisfy­ing ||Aj(s)||oo < A; for j =

It is still possible to apply the small gain theorem to the above prob­lem. The result is nevertheless conservative, i. e. only a lower bound of the robustness margin kmax is obtained. Indeed, when applying the small gain theorem to the interconnection structure M(s) – A (s), the block diagonal structure of the model perturbation A(s) is not taken into ac­count, i. e. A (s) is considered as a full block of neglected dynamics.