## Second-order leader-following consensus of nonlinear multi-agent systems via pinning control.(English)Zbl 1213.37131

The authors consider a nonlinear multi-agent system composed of $$N$$ coupled autonomous agents with second-order dynamics:
$\dot x_i(t)=v_i(t), \quad \dot v_i(t)=f(t, x_i(t), v_i(t)), \quad i=1,\dots,N, \tag{1}$ where $$n$$-vectors $$x_i$$ and $$v_i$$ are the position and velocity states of agent $$i$$, $$f$$ is a nonlinear vector-valued continuous function, $$u_i$$ is the control input for agent $$i$$. The virtual leader for multi-agent system (1) is described by
$\dot x_r(t)=v_r(t), \quad \dot v_r(t)=f(t, x_r(t), v_r(t)). \tag{2}$
The multi-agent system (1) is said to achieve second-order leader-following consensus, if its solution satisfies $$\lim_{t\to\infty}\|x_i(t)-x_r(t)\|=0$$, $$\lim_{t\to\infty}\|v_i(t)-v_r(t)\|=0$$, $$i=1,\dots,n$$, for any initial conditions.
With the use of graph theory, matrix theory, and LaSalle’s invariance principle, a consensus algorithm based on pinning control is developed in the paper for second-order multi-agent systems with time-varying and constant reference velocities $$v_r$$. A pinned-agent selection scheme is provided to determine what kind of followers and how many followers should be informed by the virtual leader, sufficient conditions are derived to guarantee the global asymptotic stability of the second-order leader-following consensus.
Numerical simulation results are presented for systems with 10 and 100 agents.

### MSC:

 37N35 Dynamical systems in control 93C15 Control/observation systems governed by ordinary differential equations 34H05 Control problems involving ordinary differential equations

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### References:

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