Distributed Prescribed-Time Convex Optimization: Cascade Design and Time-Varying Gain Approach

TL;DR

This paper addresses the distributed prescribed-time convex optimization (DPTCO) problem for a class of nonlinear multi-agent systems (MASs) under undirected connected graph and utilizes the proposed framework to solve robust DPTCO problem for a class of chain-integrator MASs with external disturbances.

Abstract

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.

Figures (5)

• Figure 1: Cascaded system $\Sigma=[\Sigma_{1}^{\hbox{\tiny {T}}},\Sigma_{2}^{\hbox{\tiny {T}}}]^{\hbox{\tiny {T}}}$ with $d=\left[(d^{1})^{\hbox{\tiny {T}}},\cdots,(d^{N})^{\hbox{\tiny {T}}}\right]^{\hbox{\tiny {T}}}$.
• Figure 2: Trajectories of positions $x_{1}^{i}$ of the six robots for $0\leq t<T$, where $\bullet$ and $\blacktriangle$ denote the initial and final position, $\bigcirc$ denotes the equipotential lines of $P$.
• Figure 3: The trajectories of $e_{r}(t)$ and $\dot{e}_{r}(t)$
• Figure 4: The trajectories of tracking error between each agent's output and optimum
• Figure 5: The trajectories of $\hat{\theta}^{1}(t)$, $\Vert x_{2}^{1}(t)\Vert$, $\Vert x_{3}^{1}(t)\Vert$.