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Distributed Average Tracking for Multiple Signals Generated by Linear Dynamical Systems: An Edge-based Framework

Yu Zhao, Yongfang Liu, Zhongkui Li, Zhisheng Duan

arXiv:1606.09329v1eess.SYcs.SY

TL;DR

The paper addresses distributed average tracking when time-varying reference signals are generated by linear dynamics and their nonzero inputs are unavailable to agents. It designs continuous edge-based algorithms with static or adaptive coupling, reporting bounded adaptive tracking errors that converge exponentially to a small adjustable set.

  • Problem

    The paper studies how agents can track the average of multiple time-varying signals generated by linear dynamics when their nonzero reference inputs are unavailable to any agent.

    Source: §1 · p. 1

  • Method

    It develops a pair of continuous edge-based algorithms using static and adaptive coupling strengths, with boundary-layer design and adaptive control.

    Source: §1 · p. 2, §6 · p. 10

  • Results

    Adaptive tracking errors are uniformly ultimately bounded and, under the stated condition, exponentially converge to a bounded set.

    Source: §4 · pp. 7–8

  • Takeaways & Limitations

    The proposed algorithms address average tracking for multiple signals generated by general linear systems with bounded inputs.

    Source: §1 · p. 2, §6 · p. 10

    The analysis assumes the communication graph is undirected and connected.

    Source: §2.1 · p. 2

Abstract

from arXiv · show

This paper studies the distributed average tracking problem for multiple time-varying signals generated by linear dynamics, whose reference inputs are nonzero and not available to any agent in the network. In the edge-based framework, a pair of continuous algorithms with, respectively, static and adaptive coupling strengths are designed. Based on the boundary layer concept, the proposed continuous algorithm with static coupling strengths can asymptotically track the average of multiple reference signals without the chattering phenomenon. Furthermore, for the case of algorithms with adaptive coupling strengths, average tracking errors are uniformly ultimately bounded and exponentially converge to a small adjustable bounded set. Finally, a simulation example is presented to show the validity of theoretical results.

1 Introduction

Distributed average tracking extends cooperative tracking to the average of multiple reference signals, but earlier work often assumed constant or convergent signals. This paper addresses time-varying signals generated by general linear dynamics using continuous edge-based algorithms with static or adaptive coupling.

1 Introduction

  • Earlier approaches commonly treated reference signals as constant or converging, while later work considered nonlinear, non-smooth, or discontinuous methods for time-varying signals [18],,,,,.The paper positions its work as extending the signal dynamics addressed by prior integrator-based approaches, and linear-signal work.Source: §1 · pp. 1–2
  • The paper targets average tracking for time-varying signals generated by general linear systems, with nonzero reference inputs unavailable to network agents.The algorithms use relative states of neighboring agents and an edge-based framework.Source: §1 · pp. 1–2
  • The proposed algorithms use edge-based relative-state information and extend average tracking to signals generated by general linear systems with bounded inputs.The paper contrasts this with node-based algorithms applying a signum function across the whole neighborhood and.Source: §1 · pp. 1–2
  • Adaptive coupling removes the need for global information, while boundary-layer design with clock synchronization is intended to reduce chattering in continuous algorithms.The global quantities identified are the network size, the smallest nonzero Laplacian eigenvalue, and the input bound.Source: §1 · pp. 1–2

2 Preliminaries

The preliminaries specify the undirected graph model and its connectivity assumption, then state spectral properties of the graph Laplacian and a related matrix.

2.1 Graph Theory

  • The incidence matrix D(G) encodes an arbitrarily oriented graph: each edge column has 1 at its initial node, -1 at its terminal node, and zero elsewhere.The adjacency matrix is symmetric, with off-diagonal entries indicating adjacent node pairs.Source: §2.1 · p. 2
  • The graph is assumed to be undirected and connected.Connectivity means a path exists between each pair of distinct nodes.Source: §2.1 · p. 2
  • Under the graph assumption, the Laplacian has a simple zero eigenvalue with eigenvector 1, and all other eigenvalues are positive.The smallest nonzero Laplacian eigenvalue is denoted λ2.Source: §2.1 · p. 2
  • The matrix M has a simple zero eigenvalue with right eigenvector 1, and eigenvalue 1 with multiplicity N − 1.The passage also states 1^T M = 0.Source: §2.1 · p. 2

3 Distributed average tracking for multiple reference signals with general linear dynamics

This section develops continuous edge-based algorithms for tracking the average of signals generated by linear dynamics and establishes guarantees under stated input and design assumptions. It also presents a modified algorithm that removes an initialization requirement and a bounded-error option for clock-synchronization disturbance.

3 Distributed average tracking for multiple reference signals with general linear dynamics

  • If external disturbance may disrupt clock synchronization, setting ϕ=0 yields uniformly ultimately bounded tracking error rather than an error-free guarantee.The paper also states that the error exponentially converges to a bounded set.Source: §3 · pp. 3, 5
  • Lemma 2 links consensus of the filter states to average tracking, with s_i(t0)=0 required when A is not asymptotically stable.When A is asymptotically stable, the paper says the same conclusion holds for any initial condition.Source: §3 · pp. 3–4
  • Algorithm 1 synchronizes local clocks, derives the feedback gain from an algebraic Riccati equation, and selects coupling strengths using graph and input bounds.The clock synchronization device is specified to synchronize local times in finite time.Source: §3 · p. 4
  • Under Assumptions 1 and 2, Algorithm 1 makes each agent track the average of signals with bounded reference inputs when its gains are designed as specified.The theorem's design uses coupling strengths and a feedback gain selected by Algorithm 1.Source: §3 · pp. 2–4
  • The modified algorithm achieves distributed average tracking without requiring x_i(t0)=r_i(t0).This modification addresses the initial-state requirement of the original algorithm.Source: §3 · p. 5
  • The edge-based design uses network symmetry, while boundary-layer approximations provide continuous algorithms intended to reduce the chattering of discontinuous signum designs.The paper contrasts this framework with node-based algorithms that it says cannot solve average tracking.Source: §3 · pp. 3, 6

4 Distributed average tracking with distributed adaptive coupling strengths

Distributed adaptive coupling laws address reliance on global network information while supporting average tracking for signals generated by linear dynamics. The analysis establishes ultimate boundedness and exponential convergence to bounded sets, with a smaller tracking-error set under a tighter condition.

4 Distributed average tracking with distributed adaptive coupling strengths

  • Distributed adaptive coupling laws address reliance on global λ2, f0, and N information that may be difficult to obtain in large networks.The paper states that the adaptive strategy removes this global-information limitation.Source: §4 · pp. 6, 9
  • Under Assumption 1, ξ and adaptive gains αij(t), βij(t) are uniformly ultimately bounded, and ξ plus gain errors converge exponentially to a bounded set.The exponential-convergence statement covers ξ, eαij, and eβij.Source: §4 · pp. 7–8
  • When max{µϑ, νχ}<γ, tracking errors ξ exponentially converge to the smaller bounded set Ω2.The theorem states the condition for the smaller-set guarantee.Source: §4 · pp. 7–8
  • The algorithms extend tracking to signals generated by linear dynamics and use continuous boundary-layer redesign to reduce chattering relative to non-smooth methods.The comparison is with prior non-smooth algorithms for integrator-type and linear signals.Source: §4 · p. 9

5 Simulations

A six-agent simulation illustrates the adaptive algorithm’s tracking behavior and compares continuous and discontinuous algorithms. The agents’ tracking errors converge to a small bounded neighborhood of the origin, while the continuous algorithm greatly reduces chattering.

5 Simulations

  • The section presents an example to verify Theorem 2; solving ARE (10) with Q = I gives the gain matrices K and Γ.Source: §5 · p. 9
  • The agents’ states reach a small bounded neighborhood of the average of all signals.The state trajectories are shown in Fig. 2.Source: §5 · p. 9, Figure 2 · p. 9
  • Tracking errors converge to a small bounded neighborhood of the origin as t→∞.Fig. 3 shows the tracking-error trajectories.Source: §5 · p. 9, Figure 3 · p. 9
  • The continuous algorithm (27) greatly reduces the chattering effect of the discontinuous algorithm in [24].The comparison is shown in Fig. 5.Source: §5 · p. 9

6 Conclusions

The paper addresses average tracking of time-varying signals generated by general linear dynamics, with nonzero, bounded reference inputs unavailable to agents. It develops static- and adaptive-coupling continuous algorithms and identifies stabilizability as a condition for sufficient existence results.

6 Conclusions

  • The studied signals are generated by general linear dynamical systems and have nonzero, bounded reference inputs unavailable to network agents.The conclusion identifies tracking with relative output information of neighboring agents as future work.Source: §6 · pp. 9–10
  • The paper develops continuous distributed algorithms with static and adaptive coupling strengths using the boundary layer concept.The conclusion reports sufficient conditions for algorithm existence when each agent is stabilizable.Source: §6 · p. 10
  • Sufficient conditions for the existence of distributed algorithms are given when each agent is stabilizable.The paper identifies tracking with only neighboring agents’ relative output information as a future topic.Source: §6 · p. 10
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