Small-Gain Theorem Based Distributed Prescribed-Time Convex Optimization For Networked Euler-Lagrange Systems

Abstract

In this paper, we address the distributed prescribed-time convex optimization (DPTCO) for a class of networked Euler-Lagrange systems under undirected connected graphs. By utilizing position-dependent measured gradient value of local objective function and local information interactions among neighboring agents, a set of auxiliary systems is constructed to cooperatively seek the optimal solution. The DPTCO problem is then converted to the prescribed-time stabilization problem of an interconnected error system. A prescribed-time small-gain criterion is proposed to characterize prescribed-time stabilization of the system, offering a novel approach that enhances the effectiveness beyond existing asymptotic or finite-time stabilization of an interconnected system. Under the criterion and auxiliary systems, innovative adaptive prescribed-time local tracking controllers are designed for subsystems. The prescribed-time convergence lies in the introduction of time-varying gains which increase to infinity as time tends to the prescribed time. Lyapunov function together with prescribed-time mapping are used to prove the prescribed-time stability of closed-loop system as well as the boundedness of internal signals. Finally, theoretical results are verified by one numerical example.

Figures (4)

• Figure 1: Schematic diagram of $N$ controlled EL systems, where $\chi^{i}={\sum}_{j\in\mathcal{N}_{i}}(y^{j}-y^{i})$ is the relative information received by $i$th agent from its neighbors. Only system output $y^{i}$ is available for neighboring agents via communication network.
• Figure 2: Communication graph $\mathcal{G}$
• Figure 3: Trajectory of the norm of global objective function, where $\nabla\bar{f}^{i}(\bar{y}^{i})$ denotes the gradient of local objective function in (\ref{['eq:opti-prob']}).
• Figure 4: Trajectories of positions $y^{i}$ of the six robots in three-dimensional space for $0\leq t\leq5s$, where $\blacklozenge$, $\blacktriangle$ and $\bullet$ denote the initial position, position at the prescribed time, and final position, respectively, the red and black regular hexagons represent the formation of the six robots at the prescribed time and final time.