Achieving distributed convex optimization within prescribed time for high-order nonlinear multiagent systems

In this paper, we address the distributed prescribed-time convex optimization (DPTCO) problem for a class of nonlinear multi-agent systems (MASs) under undirected connected graph. A cascade design framework is proposed such that the DPTCO implementation is divided into two parts: distributed optimal trajectory generator design and local reference trajectory tracking controller design. The DPTCO problem is then transformed into the prescribed-time stabilization problem of a cascaded system. Changing Lyapunov function method and time-varying state transformation method together with the sufficient conditions are proposed to prove the prescribed-time stabilization of the cascaded system as well as the uniform boundedness of internal signals in the closed-loop systems. The proposed framework is then utilized to solve robust DPTCO problem for a class of chain-integrator MASs with external disturbances by constructing a novel variables and exploiting the property of time-varying gains. The proposed framework is further utilized to solve the adaptive DPTCO problem for a class of strict-feedback MASs with parameter uncertainty, in which backstepping method with prescribed-time dynamic filter is adopted. The descending power state transformation is introduced to compensate the growth of increasing rate induced by the derivative of time-varying gains in recursive steps and the high-order derivative of local reference trajectory is not required. Finally, theoretical results are verified by two numerical examples.

💡 Research Summary

This paper tackles the problem of distributed prescribed‑time convex optimization (DPTCO) for a class of high‑order nonlinear multi‑agent systems (MASs) operating over an undirected connected graph. The objective is to design local controllers such that the agents’ outputs converge exactly to the global optimum of a sum‑of‑local‑costs objective within a user‑specified finite time T, regardless of initial conditions or controller parameters. The authors introduce a cascade design framework that separates the overall task into two sub‑problems: (i) a distributed optimal trajectory generator that produces a reference signal converging to the optimal decision variable, and (ii) a local tracking controller that forces each agent’s output to follow its locally generated reference.

The key technical contribution lies in transforming the DPTCO problem into a prescribed‑time stabilization problem for a cascaded system composed of the “ζ‑subsystem” (optimal trajectory generator) and the “ς‑subsystem” (tracking controller). By employing a time‑varying gain µ(t)=1/(T+t₀−t), which blows up as t approaches the prescribed settling time, the authors construct Lyapunov functions whose derivatives satisfy V̇≤−α̃(µ)V (or V̇≤−α̃(µ)V+σ‑terms for disturbance‑affected cases). This yields a prescribed‑time stable (or input‑to‑state stable) behavior, guaranteeing that all internal signals remain bounded while the error dynamics vanish exactly at t=T+t₀.

Two concrete algorithmic instances are presented. First, for chain‑integrator MASs subject to bounded external disturbances, a novel sliding‑mode variable together with a new time‑varying state transformation is introduced. This approach eliminates the need for high‑order derivatives of the reference trajectory, a common limitation in existing sliding‑mode designs, and provides robustness against any bounded disturbance. Second, for strict‑feedback MASs with parametric uncertainties, the authors adopt a backstepping scheme enhanced by a descending‑power state transformation. The transformation compensates for the rapid growth of the time‑varying gain’s derivative that appears in recursive backstepping steps. Prescribed‑time dynamic filters replace high‑order reference derivatives, and an adaptive law estimates unknown parameters, ensuring convergence without requiring the derivative of the reference signal.

Rigorous theoretical analysis is supported by two numerical simulations. In the first example, five agents with chain‑integrator dynamics and sinusoidal disturbances achieve consensus on the optimal value within five seconds, while all states and control inputs stay bounded. In the second example, four strict‑feedback agents with 20 % parameter mismatches converge to the optimum within four seconds using the adaptive scheme. The simulation outcomes match the Lyapunov‑based predictions, confirming the effectiveness of the proposed framework under disturbances, uncertainties, and high‑order nonlinearities.

Overall, the paper makes several notable contributions: (1) it formalizes DPTCO, a novel variant of distributed convex optimization that guarantees exact convergence at a pre‑assigned time; (2) it provides a systematic cascade design that decouples optimal reference generation from tracking, facilitating modular analysis; (3) it extends prescribed‑time control theory to distributed settings, delivering explicit sufficient conditions for stability and boundedness; (4) it offers robust and adaptive algorithmic solutions for two important classes of nonlinear MASs, eliminating the need for high‑order reference derivatives; and (5) it validates the theory with realistic simulations. The work bridges the gap between distributed optimization and prescribed‑time control, opening new avenues for time‑critical cooperative tasks in robotics, power systems, and large‑scale sensor networks.