Robust Consensus in a Class of Fractional-Order Multi-Agent Systems with Interval Uncertainties Using the Existence Condition of Hermitian Matrices

Abstract

This study outlines the necessary and sufficient criteria for swarm stability asymptotically, meaning consensus in a class of fractional-order multi-agent systems (FOMAS) with interval uncertainties for both fractional orders 0 < α < 1 and 1 < α < 2. The constraints are determined by the graph topology, agent dynamics, and neighbor interactions. It is demonstrated that the fractional-order interval multi-agent system achieves consensus if and only if there are some Hermitian matrices that satisfy a particular kind of complex Lyapunov inequality for all of the system vertex matrices. This is done by using the existence condition of the Hermitian matrices in a Lyapunov inequality. To do this, at first it is shown under which conditions a multi-agent system with unstable agents can still achieve consensus. Then, using a lemma and a theory, the Lyapunov inequality regarding the negativity of the maximum eigenvalue of an augmented matrix of a FOMAS is used to find some Hermitian matrices by checking only a limited number of system vertex matrices. As a result, the necessary and sufficient conditions to reach consensus in a FOMAS in the presence of internal uncertainties are obtained according to the Lyapunov inequalities. Using the main theory of the current paper, instead of countless matrices, only a limited number of vertex matrices need to be used in Lyapunov inequalities to find some Hermitian matrices. As a confirmation of the notion, some instances from numerical simulation are also provided at the end of the paper.

Hybrid finite‐time fault‐tolerant consensus control of non‐linear fractional order multi‐agent systems based on fault detection and estimation

“…We present this control law in the form of the following theorem and prove its stability: Theorem 4.1. By applying the fractional order sliding mode control law U = (T −1 Q T ⊗ I m ) Û in the fractional order MAS exposed to the fault in (7), consensus is obtained in a finite time, which Û is obtained from the following equation:…”

“…In [5], the leader–follower problem of multiple mobile robots is solved using the Lyapunov redesign method and [6] has designed a robust distributed observer for non‐linear MAS where the graph among the agents will not remain connected in all of the time. The consensus issue pertains to achieving the convergence of all agents’ states to a certain value [7]. To achieve consensus, distributed algorithms are typically preferred.…”

Chaotic vibration control of a composite cantilever beam

“…In order to control vibrations of various nonlinear/linear dynamic systems derived from engineering 2 , 14 , 17 – 20 , different control strategies have been developed 21 . Among those control strategies, one strategy, namely the sliding mode control (SMC), was proposed in 1992 by Utkin 22 , and has been wildly applied in engineering vibration control along with other SMC based strategies.…”

The Stability Analysis of Linear Systems with Cauchy—Polynomial-Vandermonde Matrices

“…An extensive amount of work has been done in the direction of discussing both necessary and sufficient criteria for swarm stability asymptotically. The system under consideration achieves the asymptotically if and only if there exists Hermitian matrices satisfying complex Lyapunov inequality for all of the system vertex matrices [55].…”