{'fraction_non_alphanumeric': 0.023002421307506054, 'fraction_numerical': 0.0, 'mean_word_length': 5.261363636363637, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 0, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 0, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Design of distributed delay-controllers for multi-agent systems

The time delay is introduced positively into controllers and the method of designing the distributed delay controllers for multi-agent systems is investigated. Distributed single and double proportional-retarded (SPR and DPR) controllers which include time delay are both considered. By using only two branches of the Lambert W function, the SPR controllers are designed and the stabilizing set of the control parameters is presented analytically for first-order multi-agent systems. In particular, the distributed SPR controllers satisfying the pole assignment index are presented for single-integrator multi-agent systems. The SPR controllers have only two adjustable parameters and may not be applicable to complex systems. Hence, for high-order multi-agent systems, the Η DPR controllers are further studied. The multi-agent system is firstly decomposed into several independent subsystems. Then, the problem of finding the set of the Η DPR controllers is converted to the one of stabilizing a group of quasi-polynomials for each subsystems. On the basis of triangle property, two proportional parameters (k) in the DPR controller are obtained. Moreover, the delay parameters ( τ ) in the DPR controllers are determined for the proportional parameters and then searching for the intersection of delay parameters in the resultant stabilizing region for each characteristic value of Laplacian matrix. The simulation results demonstrate the validity of the proposed control method. And the results in the paper provide an effective way for the distributed control of multi-agent system.