This paper studies the decentralized adaptive tracking control problem for a class of discretetime multi-agent systems with unknown parameters and high-frequency gains using multi-model method. Each agent is strong coupling with its neighbors by the historical outputs. All agents are interacted either directly or indirectly. In the face of uncertainties, the projection algorithm as a normal adaptive method is adopted. In order to improve quality of identification, the multi-model method is taken to identify unknown parameters and high-frequency gains using switching sets of the multiple parameters' and high-frequency gains' estimates, and the index switching functions. Using the certainty equivalence principle, the control law for the hidden leader agent is designed by the desired reference signal; the control law for each follower agent is devised by neighbors' historical outputs. Moreover, the proposed decentralized adaptive control laws can guarantee the following performances of the system: (1) the leader agent tracks the reference trajectory and each follower agent follows the average value of its neighborhood historical outputs; (2) the synchronization of all the follower agents to the leader agent is achieved; (3) all the agents track the reference trajectory, and the closed-loop system eventually achieves strong synchronization. Finally, simulations validate the effectiveness on improving control performance of multi-model adaptive algorithm by comparing with the projection algorithm.

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PurposeThe purpose of this paper is to investigate the pricing strategy and the impact of agents' risk preference in a dual-channel supply chain in which both agents are risk-averse.Design/methodology/approachThe authors make use of the mean-variance (MV) method to measure the risk aversion of the agents and apply Stackelberg game to obtain the optimal strategies of the proposed models. Furthermore, the authors compare the optimal strategies with that in the benchmark model in which no agent is risk-averse.FindingsThe authors find that the pricing decisions can be divided into four categories according to the risk attitudes of the agents: the decisions that are independent of two agents' risk attitudes, the decisions that depend on only one agent’s risk attitude (i.e. depend on only manufacturer's risk attitude and depend on only retailer's risk attitude) and the decisions that depend on both agents' risk attitudes. In addition, the authors find that the retail price will be lower and the wholesale price in most cases will be lower than that in the benchmark when at least one agent's risk control is effective; the demand will be always increasing as long as one agent's risk control is effective. Furthermore, compared to the benchmark, a win-win strategy (i.e. Pareto improvement) for the supply chain members can be obtained in a certain range where the agents' risk controls are appropriate.Originality/valueThis research provides a theoretical reference for the managers to make the pricing decisions and the risk control in dual-channel supply chains with heterogeneous preference consumers.

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