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The distributed average consensus problem of l-th order multiRobot systems with multiple communication delays and switching topologies is discussed in this paper. Using the idea of state decomposition, the condition for guaranteeing average consensus is converted into verifying the stability of zero equilibrium of disagreement system. Considering multiple time-varying communication delays, common Lyapunov-Krasovskii functional is employed to analyze the stability of zero equilibrium. In order to relax the conservativeness, Free-weighting Matrices method is employed in the main results. After matrix order-reduced treatment, the tolerant upper bounds on communication delays can be obtained through solving feasible linear matrix inequalities (LMIs). Numerical examples and simulations are given to demonstrate the effectiveness of the proposed method. Different with the existing literature, the proposed LMI based delay-dependent stability criterion is characterized by lower conservativeness, simple formation of the solution, and wide range of time-varying delays.