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By Alexandre Seuret, Laurentiu Hetel, Jamal Daafouz, Karl H. Johansson

This edited monograph contains state of the art contributions on non-stop time dynamical networks with delays. The booklet is split into 4 components. the 1st half provides instruments and strategies for the research of time-delay platforms with a specific awareness on keep an eye on difficulties of enormous scale or infinite-dimensional platforms with delays. the second one a part of the e-book is devoted to using time-delay versions for the research and layout of Networked keep an eye on platforms. The 3rd a part of the booklet makes a speciality of the research and layout of platforms with asynchronous sampling periods which happen in Networked regulate platforms. The final a part of the booklet exposes numerous contributions facing the layout of cooperative keep watch over and remark legislation for networked keep an eye on structures. the objective viewers basically includes researchers and specialists within the box of keep watch over concept, however the e-book can also be helpful for graduate scholars.

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Automatica 48(1), 109–114 (2012) 17. M. J. Loiseau, in On the Stability of Positive Difference Equations, ed. by R. Sipahi, T. I. Niculescu, P. Pepe. Time Delay Systems - Methods, Applications and New Trends, Series LNCIS, vol. 423. (Springer, 2012), pp. 125–147 18. F. Di Meglio, R. Vazquez, M. Krstic, Stabilization of a system of n + 1 coupled first-order hyperbolic linear PDEs with a single boundary input. IEEE Trans. Autom. Control 58(12), 3097–3111 (2013) 19. S. Elaydi. , 3rd edn (Springer, 2005) 20.

This procedure will be discussed more in detail in Sect. 3. Let us now consider the optimality conditions for the problem L2 in the SISO case. We then have the following result initially proposed in [13]: Theorem 3 (Condition for L2 optimality) Let H ∈ L2 (iR) be a given system and define its decomposition as H = H+ + H− where H+ ∈ H (C+ ) and H− ∈ H (C− ). 29) s=−λˆ i s=−λˆ j hold for all i = 1, . . , k and all j = k + 1, . . , r . The conditions presented in Theorem 3 have the particularity that one should know the decomposition of a transfer function H (s) into its stable and purely unstable parts H + (s) and H − (s).

The above theorem states that a local minimum is a bitangential Hermite interpolant of the original model evaluated at the mirror image of the low-order model poles with respect to its tangential directions bˆk and cˆk , given by its residues (see [7, 22]). Note also that the poles and residues are not known a priori and must be computed using the iterative algorithm proposed in [3]. This procedure will be discussed more in detail in Sect. 3. Let us now consider the optimality conditions for the problem L2 in the SISO case.

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