PaperlayerView exact revision on arXiv

Source-linked AI summary

Containment Control of Multi-Agent Systems with Dynamic Leaders Based on a $PI^n$-Type Approach

Yunpeng Wang, Long Cheng, Wei Ren, Zeng-Guang Hou, Min Tan

arXiv:1411.4346v3eess.SYcs.SY

TL;DR

Containment control with polynomial-trajectory leaders is challenging when follower dynamics differ from leader dynamics and when discrete-time operation is required. The paper proposes PI^n-type containment control, extends it to high-order integral followers and discrete time, and proves containment under stated conditions.

Abstract

from arXiv · show

This paper studies the containment control problem of multi-agent systems with multiple dynamic leaders in both the discrete-time domain and the continuous-time domain. The leaders' motions are described by $(n-1)$-order polynomial trajectories. This setting makes practical sense because given some critical points, the leaders' trajectories are usually planned by the polynomial interpolations. In order to drive all followers into the convex hull spanned by the leaders, a $PI^n$-type ($P$ and $I$ are short for {\it Proportion} and {\it Integration}, respectively; $I^n$ implies that the algorithm includes high-order integral terms) containment algorithm is proposed. It is theoretically proved that the $PI^n$-type containment algorithm is able to solve the containment problem of multi-agent systems where the followers are described by any order integral dynamics. Compared with the previous results on the multi-agent systems with dynamic leaders, the distinguished features of this paper are that: (1) the containment problem is studied not only in the continuous-time domain but also in the discrete-time domain while most existing results only work in the continuous-time domain; (2) to deal with the leaders with the $(n-1)$-order polynomial trajectories, existing results require the follower's dynamics to be $n$-order integral while the followers considered in this paper can be described by any-order integral; and (3) the "sign" function is not employed in the proposed algorithm, which avoids the chattering phenomenon. Furthermore, in order to illustrate the practical value of the proposed approach, an application, the containment control of multiple mobile robots is studied. Finally, two simulation examples are given to demonstrate the effectiveness of the proposed algorithm.

I. INTRODUCTION

The introduction motivates containment control through mobile-robot coordination and identifies gaps in handling polynomial-trajectory leaders, general follower dynamics, and discrete-time systems. It proposes a PI^n-type approach and highlights its scope and distinctions from earlier PI consensus work.

I. INTRODUCTION

  • A safe-tunnel robot task motivates the problem: master robots follow interpolated reference trajectories while slave robots move within the area they span.Reference points must be reselected if interpolation produces a trajectory outside the tunnel because of Runge’s phenomenon.Source: §I · p. 2
  • Existing bounded-input leader models cannot generate the polynomial trajectories considered, while matching leader dynamics to follower dynamics restricts applicability and prior results leave discrete-time containment unclear.The introduction also notes that non-smooth sign functions in earlier controllers can cause chattering.Source: §I · p. 2
  • The paper proposes PI^n-type containment control for dynamic polynomial-trajectory leaders in both continuous and discrete time, extending the approach to high-order integral followers.The single-integrator case requires at least one leader to have a directed path to each follower; the high-order case uses a modified PI^n−mDm-type algorithm.Source: §I · p. 2
  • The proposed controllers omit the discontinuous sign function and consider disturbance and measurement-noise effects.The paper also states that its algorithms address followers with any-order integral dynamics.Source: §I · pp. 2–3
  • Unlike PI consensus algorithms for leaderless systems,, this paper uses integral terms to eliminate polynomial-trajectory containment error and addresses polynomial-type disturbances in both time domains.The cited PI consensus algorithms address constant disturbances and present continuous-time results only.Source: §I · p. 3

II. PRELIMINARIES & PROBLEM FORMULATION

This section formulates containment for a directed multi-agent network with polynomial-trajectory leaders and integral-dynamics followers. It defines the target as convergence of follower positions into the leaders’ convex hull and gives a corresponding error criterion.

II. PRELIMINARIES & PROBLEM FORMULATION

  • The network is a weighted digraph with leaders and followers, and each follower must be reachable from at least one leader while measuring only relative positions to its neighbors.The paper assumes no self-loops and defines edge direction, adjacency weights, and the graph Laplacian.Source: §II · p. 3
  • Leaders follow nth-order polynomial trajectories, while followers are modeled with first-order dynamics and polynomial-type disturbances.The polynomial trajectory is motivated by robot paths planned through polynomial interpolation.Source: §II · pp. 3–4
  • Containment is achieved when every follower’s position converges into the convex hull of the leaders’ positions.The hull consists of convex combinations of leader positions with nonnegative weights summing to one.Source: §II · p. 4
  • The paper defines containment error and states that containment is solved if and only if that error converges to zero.The section also introduces the distance from a vector to a set as the infimum of its Euclidean distances to points in that set.Source: §II · p. 4

III. CONTAINMENT CONTROL OF MASS IN CONTINUOUS-TIME DOMAIN

The continuous-time controller uses relative-state feedback with proportional and integral terms, then proves containment under a bound on polynomial disturbance order and a gain condition. The integral terms target containment error induced by polynomial leader trajectories.

III. CONTAINMENT CONTROL OF MASS IN CONTINUOUS-TIME DOMAIN

  • The PI^n-type controller combines a proportional term with integrals up to order n of weighted relative-position errors.The paper identifies eliminating containment error caused by polynomial trajectories as the purpose of these integral terms.Source: §III · p. 4
  • The analysis augments agent states with position derivatives through order n and expresses the leader and follower dynamics in that state representation.The follower’s augmented dynamics include control input, and the closed-loop system is written in compact form.Source: §III · p. 4
  • Containment is guaranteed when disturbance order r is at most n − 1 and K = εB^T P with ε ≥ 0.5 max{1, σ_min^-1}, under the theorem’s stated eigenvalue conditions.The theorem assumes polynomial-trajectory leaders and single-integrator followers; its proof establishes closed-loop stability and convergence.Source: §III · pp. 4–5

IV. EXTENSIONS TO FOLLOWERS WITH HIGH-ORDER INTEGRAL DYNAMICS

The section extends containment control to followers with high-order integral dynamics, using a generalized controller and state estimation for derivative information. The analysis proves containment under the stated leader-trajectory and follower-dynamics assumptions, and the simulation reports followers converging into the leaders’ convex hull and moving with them.

IV. Extensions to Followers with High-Order Integral Dynamics

  • Theorem 2 establishes that algorithm (13), with the specified gain K, solves containment for followers with high-order integral dynamics.The gain is defined using the positive definite matrix P from the stated matrix inequality.Source: §IV · p. 5
  • The algorithm is a generalized PID-type controller, combining proportional, integral, and differential terms.The differential terms may be difficult to obtain, motivating follower-state estimation; leaders directly transmit their position and derivatives to connected neighbors.Source: §IV · p. 5
  • The follower-state estimator’s error decays exponentially, supporting the convergence analysis for the high-order follower case.The section gives an exponential bound for the estimated-state error and uses it in the containment proof.Source: §IV · pp. 5–6
  • Theorem 3 establishes containment with modified algorithm (15) for high-order integral followers moving under the assumed polynomial leaders.Its gain is specified using the matrices from Theorems 1 and 2.Source: §IV · p. 6
  • In the eight-agent simulation, followers converge into the leaders’ convex hull and move along with them.Agents 1–4 are leaders and agents 5–8 are followers.Source: §IV · p. 6

V. EXTENSIONS TO MASS IN DISCRETE-TIME DOMAIN

The discrete-time setup models leader motion with polynomial trajectories and defines the difference operator used in that domain. The supplied figure caption identifies moving profiles for all agents in the Section IV simulation example.

V. Extensions to MAS in Discrete-Time Domain

  • The discrete-time subsection assumes each leader follows a polynomial trajectory.The displayed trajectory includes terms through the nth power of the time index.Source: §V · p. 6
  • The difference operator is defined as Dx[k] = x[k + 1] − x[k], with its inverse given by cumulative summation.The text states that applying D to D^-1x[k] recovers x[k].Source: §V · p. 6
  • Fig. 3 presents the moving profiles of all agents in the Section IV simulation example.The caption identifies the figure’s scope but does not specify axes or a discrete-time result.Source: Figure 3 · p. 7

A. Followers with Single-Integrator Dynamics

For discrete-time systems with single-integrator followers, the paper proposes a PIn-type containment algorithm for leaders following polynomial trajectories. Theorem 4 gives a condition under which containment is solved, and the paper also presents a uniform-gain variant.

A. Followers with Single-Integrator Dynamics

  • The proposed discrete-time PIn-type algorithm is introduced for the single-integrator follower case.The coefficient 1/(1 + di) normalizes the Laplacian; the paper also considers replacing it with a uniform constant.Source: A. Followers with Single-Integrator… · pp. 7–8
  • When the disturbance order satisfies r ≤ n−1, Theorem 4 guarantees discrete-time containment using the PIn-type algorithm with K = (B^T PB)^−1B^T PÂ.P must be a positive definite solution to the stated matrix inequality.Source: A. Followers with Single-Integrator… · p. 7
  • The uniform-gain variant solves discrete-time containment with µ = µmin and K = (B^T PB)^−1B^T PÂ.The stated bound is εmax = maxi{|1−µminλi|} < 1.Source: A. Followers with Single-Integrator… · p. 8
  • The discrete-time algorithm does not use the sign function, which the authors say can cause chattering and is difficult to use in discrete time.The paper describes its discrete-time discussion as a preliminary attempt because existing literature rarely addresses containment with dynamic leaders in that domain.Source: A. Followers with Single-Integrator… · p. 8

B. Containment Control of MASs with Measurement Noises

For single-integrator followers with measurement noise, the paper modifies the containment algorithm and defines containment in a stochastic sense. It proves this objective is achieved when the polynomial disturbance order is at most n−1.

B. Containment Control of MASs with Measurement Noises

  • The noise model treats relative-state measurements as corrupted by bounded-intensity, mutually independent standard white-noise vectors.The algorithm replaces the relative-state term with its noise-corrupted version.Source: B. Containment Control of MASs with… · p. 8
  • Stochastic containment requires the distance between the expected follower position and the leaders’ convex hull to tend to zero, with bounded expected squared distance.The definition applies to each follower.Source: B. Containment Control of MASs with… · p. 8
  • When r ≤ n−1, the modified algorithm solves containment with measurement noise in the stochastic sense.The result assumes polynomial leader trajectories and single-integrator follower dynamics.Source: B. Containment Control of MASs with… · p. 8

C. Followers with High-Order Integral Dynamics

The paper extends containment control to followers with high-order integral dynamics and introduces a PIlm−mDm−1-type algorithm. It gives a state-estimation approach for unavailable relative-state differences and proves containment results for the high-order follower case.

C. Followers with High-Order Integral Dynamics

  • The proposed algorithm is named PIlm−mDm−1-type, with lm = max{n + 1, m}.The followers’ dynamics are given as a high-order difference equation.Source: C. Followers with High-Order Integral… · p. 9
  • For polynomial-trajectory leaders and high-order integral followers, Theorem 6 guarantees containment using algorithm (33) with the specified gain K.The gain is K = (F^T PF)^−1F^T PÊ, where P satisfies the stated matrix inequality.Source: C. Followers with High-Order Integral… · p. 9
  • When relative-state differences through order m−1 are unavailable, the paper introduces an estimator and replaces those differences in the containment algorithm with estimated values.The estimator’s stated error term decays exponentially, bounded by M5β5^k.Source: C. Followers with High-Order Integral… · p. 9
  • Theorem 7 also guarantees containment for high-order integral followers using algorithm (35) with its specified gain.The gain is K = (F^T PF)^−1F^T PÊ.Source: C. Followers with High-Order Integral… · p. 9

D. Applications: Coordinated Control of a Group of Mobile Robots

The application models differential-drive robots, plans master trajectories through reference points in a safe tunnel, and uses slave control based on the interaction topology. A simulation reports that the slaves enter and move with the masters’ convex hull despite disturbance and measurement noise.

Application setup

  • The scenario has three self-navigating master robots and three slave robots that measure relative positions with neighboring robots while crossing a narrow safe tunnel.The setting is a partially unknown area.Source: D. Applications: Coordinated Control… · p. 10

Robot model

  • Each robot is differential drive, with nonlinear kinematics that require feedback linearization before the containment algorithm can be applied.The robot is represented by the coordinate of its center point.Source: D. Applications: Coordinated Control… · p. 10

Simulation setup

Trajectory planning

Slave control

Simulation

  • Despite disturbance and measurement noise, all slave robots converge into the convex hull spanned by the master robots and move along with them.The moving profiles of all robots are shown in Fig. 6.Source: D. Applications: Coordinated Control… · p. 11

VI. CONCLUSIONS

The paper proposes a PIn-type containment algorithm for continuous- and discrete-time systems with dynamic leaders. It states a directed-path condition for containment and reports that the approach covers single- and high-order integral follower dynamics, without a nonsmooth sign function.

Contributions

  • The proposed algorithm addresses containment in both continuous-time and discrete-time domains, with leaders following polynomial trajectories.Containment is guaranteed when each follower has at least one leader with a directed path to it.Source: §VI · p. 11
  • The approach covers dynamic leaders even when followers have single-integrator dynamics, and also considers high-order integral follower dynamics.The conclusion contrasts this scope with prior requirements for n-order integral follower dynamics.Source: §VI · p. 11
  • The proposed algorithm contains no nonsmooth sign function and takes disturbance and measurement-noise effects into account.Source: §VI · p. 11

Application

  • The paper presents coordinated control of multiple mobile robots as a potential application of the algorithm.The conclusion includes moving profiles of all robots.Source: §VI · p. 11, Figure 6 · p. 11

APPENDIX A

Appendix A establishes finite-difference identities and properties of finite differences of standard white noise. These results include bounded covariance and vanishing cross-covariance at sufficiently separated time indices.

Finite-difference identity

  • The finite-difference identity in (43) is established for every positive integer n by mathematical induction.Source: Appendix A · p. 11

Noise covariance

  • For the vector of white-noise differences through order n, a finite constant G bounds the norm of its cross-covariance for lags m from 0 to n.The vector is defined as ν[k] = (η[k], Dη[k], …, D^nη[k])^T.Source: Appendix A · p. 12
  • The cross-covariance of the white-noise difference vectors is zero when the lag m is at least n + 1.Source: Appendix A · p. 12

Noise expectation

  • Every finite difference D^lη[k] of standard white noise has zero expectation for l = 0, 1, 2, ….The appendix proves the property by induction.Source: Appendix A · p. 12

Related papers

Loading 1411.4346v3…