Cooperative control of multi-agent systems has garnered significant attention as a prominent research direction in the field of artificial intelligence, drawing considerable interest from both academia and industry in recent years. Consensus theory serves as the foundation of multi-agent cooperative control, while consensus-based formation control is a critical research branch within this theory framework. Currently, multi-agent formation control has achieved substantial advancements in various applications, such as collaborative positioning, persistent surveillance, payload transportation, and target tracking, finding extensive implementation in both civilian and military sectors. This paper addresses the optimization problem of achieving time-varying formation control for high-order continuous linear multi-agent systems under sampling communication with global minimum-energy consumption. Traditional continuous communication methods among agents frequently encounter challenges, such as data packet loss and communication congestion, due to bandwidth limitations. In practical multi-agent systems, communication among agent is predominantly handled by digital computer systems with sampling, communication, and computation at fixed intervals according to operational clocks. Although sampling communication can effectively mitigates these issues, it may lead to increased control energy consumption, particularly with smaller sampling periods. Numerical simulations for both scenarios, with and without minimum-energy constraints, demonstrate that under sampled-data communication conditions, the proposed control strategy effectively achieves time-varying formation in multi-agent systems while significantly reducing global control energy consumption, thereby validating the superiority of the proposed design criterion.

The methodology begins with constructing global equations for the multi-agent system, utilizing local neighborhood information at discrete sampled instants to design piece-wise constant control inputs, thereby establishing a control protocol with global energy consumption being considered. In this designed protocol, controllers update at each sampling period with a positive lower bound between triggering events, which can effectively prevent Zeno behavior and align with practical digital computer system requirements. This approach reduces communication bandwidth usage, decreases computational frequency, and minimizes unnecessary energy consumption. Furthermore, as the control inputs depend solely on local neighbor information without requiring global information, the design is a fully distributed cooperative strategy. By the construction of a time-varying delay model, a mathematical relationship was established among global control energy consumption, communication topology matrix, and the control gain. Employing the state-space decomposition method, the multi-agent system under sampling communication was transformed into two linearly independent subsystems: consensus subsystems characterizing the macroscopic motion and non-consensus subsystems capturing relative movements among agents. This transformation converted the formation control problem into a stability problem of the inconsistent subsystem, significantly simplifying system analysis and controller design complexity. A Lyapunov-Krasovskii functional candidate was constructed. Based on the formation feasible condition, utilizing generalized eigenvalue approach and LMI (linear matrix inequality) methods, an upper bound for global energy consumption under sampling communication was constructed, and a minimum-energy constraint was designed. Through Schur complement theory, sufficient conditions for achieving time-varying formation with minimum-energy constraints were established. By variable substitution techniques, multiple nonlinear unknown terms in the sufficient conditions were eliminated, yielding a practical design criterion for determining the control gain. Leveraging the convex properties of LMI systems, the design criterion was optimized using the minimum non-zero and maximum eigenvalues of the communication topology matrix, substantially reducing computational complexity. Numerical simulations comparing scenarios with and without minimum-energy constraints demonstrated that the proposed control strategy effectively reduces global energy consumption while maintaining formation accuracy under sampling communication, validating the superiority of the design criterion.

To verify effectiveness and advantages of the proposed theoretical approach, with the guarantee of time-varying formation achievement, comprehensive simulations were conducted comparing three key aspects: global control energy consumption, tracking error, and motion trajectories. The results demonstrate that: 1) The actual global control energy consumption with minimum-energy constraints is significantly lower than that without minimum-energy constraints; 2) The tracking error sum with minimum-energy constraints exhibits smoother convergence characteristics compared to fluctuating error characteristics without minimum-energy constraints; 3) Both control methods can ensure the achievement of the time-varying formation, with significant trajectory differences of each individual agent before the achievement of the formation, while post-formation trajectories become essentially identical with consistent relative positions at the same time point. This phenomenon arises because different control gains result in different motion trajectories for all the agents,whereas the motion trajectory of each agent is the same after the achievement of the formation because of the same reference function, the same desired formation function, and the control input being zero.

This study presents a novel theoretical framework for time-varying formation control with minimum-energy constraints under sampling communication. A mathematical model characterizing the upper bound of the global energy consumption under discrete control inputs is established, providing a theoretical foundation for energy optimization of multi-agent systems operating under sampling communication. By constructing a minimum energy constraint and employing generalized eigenvalue theory, sufficient conditions for analysis and design of time-varying formation control are derived in the form of LMIs，which can ensure both the achievement of the desired formation and minimal global control energy consumption. The proposed sufficient conditions are independent of the number of agents, ensuring scalability and broad applicability. This work offers significant theoretical support and methodological guidance for reducing control energy consumption in multi-agent systems，such as warehouse management, environmental exploration, and swarm combat.