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Characterization and optimization of the smoothed spectral abscissa for time‐delay systems
Author(s) -
Gomez Marco A.,
Michiels Wim
Publication year - 2019
Publication title -
international journal of robust and nonlinear control
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.361
H-Index - 106
eISSN - 1099-1239
pISSN - 1049-8923
DOI - 10.1002/rnc.4631
Subject(s) - pointwise , mathematics , computation , norm (philosophy) , ordinate , matrix (chemical analysis) , lyapunov function , measure (data warehouse) , mathematical optimization , algorithm , control theory (sociology) , computer science , mathematical analysis , physics , control (management) , geometry , materials science , nonlinear system , quantum mechanics , database , artificial intelligence , political science , law , composite material
Summary We introduce a new robust stability measure for systems with multiple pointwise delays, which is called smoothed spectral abscissa, consistently with the existing measure for delay‐free systems. Its main characteristics are that it is smooth with respect to the system parameters and it provides a trade‐off between the decay rate of the system solution and theH 2norm of a transfer matrix related with the system. The smoothed spectral abscissa is implicitly defined in terms of theH 2norm of an auxiliary system, and its computation is based on the so‐called delay Lyapunov matrix. We show that these features make the smoothed spectral abscissa suitable for the design of robust controllers by using standard gradient‐based optimization techniques and exploiting a novel characterization of the derivatives of the delay Lyapunov matrix with respect to the system parameters.