Average-consensus tracking of multi-agent systems with additional interconnecting agents
Introduction
Recently, distributed coordination control strategies have been widely applied in various engineering fields including sensor networks, coordination of unmanned vehicles, smart grid, etc. Consensus problem has been one of the most fundamental and important issues of distributed coordination control of multi-agent systems, and requires that the multiple agents achieve an agreement on their states of interest by exchanging information locally. General consensus problem has been thoroughly analyzed for multi-agent systems under fixed topology [1], [2], switching topologies [3], [4], time delays [5], [6], [7], external disturbances [8], fractional-order dynamics [9], [10], hybrid dynamics [11], etc.
As a special consensus algorithm, average-consensus problem means that the agents agree on the average value of agents’ initial values, and the average-consensus seeking is always reached for multi-agent systems with undirected and connected topology [2], [3], [13] or directed and balanced topology [5], [12]. Generally, average-consensus seeking can be easily destroyed by topology’s unbalance [14], quantization [15] and noises [16], [17]. Regarding non-balanced digraphs, Cai et al. designed some innovative average-consensus algorithms for discrete-time first-order multi-agent systems [18], [19], [20], and average-consensus convergence was proved under fixed topology and switching topologies, respectively.
Average-consensus tracking problem analyzed in this paper requires the agents to asymptotically reach the average value of reference inputs via local interactions, e.g., the average value of sensors’ measuring values in sensor network. Average-consensus tracking problem has attracted more and more researchers in the fields of distributed mapping [21], and distributed estimation and data fusion of sensor networks [22], [23]. Different agents’ reference inputs with usual proportional consensus algorithm lead to the estimation errors, so the proportional-integral (PI) consensus algorithms are designed to deal with the average-consensus tracking problem of first-order multi-agent systems with different constant reference inputs [23], [24]. Based on the internal model principle, Bai et al. constructed a dynamical average-consensus tracking algorithms for first-order multi-agent systems with different time-varying reference inputs [25]. Besides, Li and Guo [26] proposed another simplified PI average-consensus tracking algorithm and analyzed the error bounds of average consensus convergence under switching topologies for constant and time-varying reference inputs respectively. Chen et al. [28], [29] designed the dynamical average-consensus tracking algorithms via signum functions for time-varying reference inputs with bounded derivatives and accelerations respectively, and consensus convergence analysis were presented under fixed and switching topologies. George et al. [27] proposed a robust dynamical average-consensus tracking algorithm for arbitrary reference inputs with bounded derivatives. The proposed algorithm in [27] did not require knowing the full dynamics or the derivatives of the references inputs, but did possess robustness to the changes of network topology.
In the aforementioned works on average-consensus tracking problem, it has been assumed that each agent accesses a reference input. In real engineering applications, however, some agents in the network cannot get the reference values for their disabled measuring instruments, and they just possess abilities of computing and communicating. Hence, we are motivated to investigate the average-consensus tracking problem of constant reference inputs for the first-order multi-agent systems, and the multi-agent systems consist of valid agents and additional interconnecting agents, in which the valid agents have assigned constant reference inputs, while the additional interconnecting agents does not have reference inputs. The average-consensus tracking control goal requires all the agents reaching the average value of valid agents’ reference inputs. On the basis of the proportional-integral consensus algorithm, we propose the average-consensus algorithms for the valid agents and the additional interconnecting agents respectively. According to frequency-domain analysis method, average-consensus convergence is analyzed under symmetric and connected topology without and with identical communication delay, respectively.
Section snippets
Agents’ dynamics and topology
Consider the first-order dynamic agents given bywhere xi(t) ∈ R and ui(t) ∈ R are the state and the control input of agent i, respectively. Among the n agents (1), some agents called “valid agents” measure the real value correctly to get the reference inputs, but a few agents called “additional interconnecting agents” do not have the reference input since they cannot measure the real values. It is worth emphasizing that all the agents have the capabilities of computing and
Delay-free case
With the algorithm (3), the closed-loop form of agents (1) is
Defining and and the multi-variable form of system (4) iswhere La and Ld are the Laplacian matrices of the digraphs
Simulation
In this section, we investigate the multi-agent system of nine first-order agents (1) (see Fig. 1), in which the valid agents are labeled as 1,2,3,4,5,6 and the additional interconnecting agents are labeled as 7,8,9. Apparently, the interconnection topology in Fig. 1 has a globally reachable node. For simplicity, the adjacent weights are chosen as 1, e.g., so the topology is
Conclusion
In this paper, the average-consensus tracking problem is studied for the first-order multi-agent systems with constant reference inputs. In the multi-agent system, the valid agents measure the real values correctly and get constant reference inputs, but the additional interconnecting agents does not have the reference inputs. The average-consensus tracking requires all the agents to converge to the average value of valid agents’ reference inputs, and the proportional-integral consensus
Acknowledgment
This work was supported by the China Scholarship Council Fund (Grant no. 201606845005), National Natural Science Foundation of China (Grant no. 61473138), Natural Science Foundation of Jiangsu Province (Grant no. BK20151130), and Six Talent Peaks Project in Jiangsu Province (Grant no. 2015-DZXX-011).
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