Consensus Problem of Delayed Linear Multi-agent Systems by Cheng-Lin Liu & Fei Liu

Author:Cheng-Lin Liu & Fei Liu
Language: eng
Format: epub
Publisher: Springer Singapore, Singapore

Proof

The system (3.9) is rewritten as , where . Since the union of n agents’ interconnection topologies across has a spanning tree, is from Lemma 3.3. Because and the possible topologies of the n agents (3.4) and (3.6) are finite, the set of all possible is finite. According to Lemma 3.1, we get , where is a constant vector. Then, is a stochastic matrix with Preconditions 3.1, 3.2, and 3.3, and we obtain . Hence, , i.e., the agents (3.4) and (3.6) achieve an asymptotic stationary consensus.

Remark 3.1

Theorem 3.1 just presents sufficient consensus criterions, which depend on Preconditions 3.1, 3.2 and 3.3 strictly. However, Preconditions 3.1, 3.2, and 3.3 present a constructive design criterion for coupling weights and control parameter of the consensus algorithms (3.3) and (3.5) with given sampling interval.

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