Delays and Networked Control Systems by Alexandre Seuret Laurentiu Hetel Jamal Daafouz & Karl H. Johansson

Author:Alexandre Seuret, Laurentiu Hetel, Jamal Daafouz & Karl H. Johansson
Language: eng
Format: epub
Publisher: Springer International Publishing, Cham

(7.36)

Thus, L depends nonlinearly on the LMI variables and and it can a priori not be directly minimized. To overcome this issue, we impose the following upper bound

(7.37)

for some . As a result, minimizing and may help to minimize L. We translate inequality (7.37) into an LMI and we state the following claim.

Claim

Assume that LMI (7.29), (7.30) are verified. Then, there exist such that

(7.38)

which implies that inequality (7.37) holds.

We note that (7.38) does not introduce additional constraints on system (7.26) compared to (7.29), (7.30). This comes from the fact that there always exist (eventually large) such that (7.38) holds, in view of Schur complement of (7.38).

In conclusion, we formulate the problem as a multiobjective optimization problem as we want to minimize under the constraint (7.29), (7.30) and (7.38). Several approaches have been proposed in the literature to handle such problems. We choose the weighted sum strategy among others and we formulate the LMI optimization problem as follows

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