PaperlayerView exact revision on arXiv

Source-linked AI summary

Distributed Control of Networked Dynamical Systems: Static Feedback, Integral Action and Consensus

Martin Andreasson, Dimos V. Dimarogonas, Henrik Sandberg, Karl H. Johansson

arXiv:1310.8620v2math.DSeess.SY

TL;DR

The paper studies how distributed controllers can achieve consensus in first- and second-order networked systems while handling nonlinearities and constant disturbances. It develops nonlinear feedback and distributed PI protocols, proves stability and equilibrium properties, and demonstrates applications in satellites, power systems, buildings, and mobile robots.

  • Problem

    The paper addresses distributed consensus and rejection of constant unknown disturbances in networked systems with single- and double-integrator dynamics.

  • Method

    It proposes nonlinear gain–interaction consensus protocols, analyzes them with integral Lyapunov functions, and designs distributed PI controllers for single- and damped double-integrator agents.

  • Results

    The protocols achieve asymptotic consensus, explicitly characterize convergence points or equilibria, and attenuate constant disturbances under stated stability conditions.

  • Takeaways & Limitations

    The results support distributed control applications in autonomous satellites, power systems, building temperature regulation, and mobile robots.

Abstract

from arXiv · show

This paper analyzes distributed control protocols for first- and second-order networked dynamical systems. We propose a class of nonlinear consensus controllers where the input of each agent can be written as a product of a nonlinear gain, and a sum of nonlinear interaction functions. By using integral Lyapunov functions, we prove the stability of the proposed control protocols, and explicitly characterize the equilibrium set. We also propose a distributed proportional-integral (PI) controller for networked dynamical systems. The PI controllers successfully attenuate constant disturbances in the network. We prove that agents with single-integrator dynamics are stable for any integral gain, and give an explicit tight upper bound on the integral gain for when the system is stable for agents with double-integrator dynamics. Throughout the paper we highlight some possible applications of the proposed controllers by realistic simulations of autonomous satellites, power systems and building temperature control.

I. INTRODUCTION

The paper addresses distributed control for nonlinear networked systems, disturbance rejection, and applications where communication or reference measurements are limited. It develops nonlinear consensus and distributed PI protocols, with theoretical analysis and application demonstrations.

  • A. General motivation: Distributed PI control remains difficult to characterize generally, motivating protocols that attenuate unknown disturbances in networked systems.Power-system frequency control illustrates the challenge when centralized communication is unavailable or future system scale makes existing architectures unsuitable.
  • A. General motivation: The paper introduces nonlinear consensus protocols that separate each agent’s input into a state-dependent gain and neighbor-interaction functions.The framework covers single- and double-integrator dynamics, including state-dependent damping, and explicitly characterizes equilibria.
  • A. General motivation: Integral action is applied to consensus problems because constant disturbances can cause formations using only relative measurements to drift.The paper positions its PI controllers as a response to disturbance propagation and limited prior disturbance-rejection results.
  • C. Main contributions: The paper proposes distributed PI controllers for single-integrator and damped double-integrator agents, deriving necessary and sufficient stability conditions for uniform controller gains.The stated applications include satellite control, mobile robots, green buildings, and power-system frequency control.
  • A. Thermal energy storage in smart buildings: Nonlinear building models represent temperature-dependent heat capacity and heat conductivity, making asymptotic room temperature a distributed-control question.Thermal energy storage creates nonlinear total room heat capacity despite approximately constant air heat capacity.
  • B. Autonomous space satellites: Satellite models use second-order dynamics with relative position and velocity measurements, motivating nonlinear interaction protocols for coordinated autonomous systems.The control signal is engine power, and the inertial-frame acceleration depends on velocity magnitude.

III. PROBLEM FORMULATION

The paper formulates distributed consensus over connected graphs for single- and double-integrator agents, then analyzes nonlinear feedback using invariant quantities and integral Lyapunov functions. For single-integrator agents, the proposed protocol converges asymptotically to a uniquely characterized agreement point.

  • A. Notation: The model uses a connected graph G with vertex set V, neighbor sets N_i, adjacency matrix B, and Laplacian L.For undirected graphs, the Laplacian satisfies L = BB^T.
  • B. System model: Agents are modeled with either single-integrator or double-integrator dynamics, with d_i representing a constant disturbance.The formulation supports consensus of positions or velocities depending on the agent dynamics.
  • C. Objective: The objectives are to characterize nonlinear-feedback stability and design protocols robust to constant unknown disturbances while driving agents to a common state.For single integrators the target is common x_i, while for double integrators it is common v_i.
  • IV. DISTRIBUTED CONTROL WITH STATIC NONLINEAR FEEDBACK: The nonlinear consensus class factors each input into a nonlinear gain function and a sum of nonlinear neighbor-interaction functions.The paper studies single-integrator, double-integrator, and state-dependent-damping cases under static nonlinear feedback.
  • A. Consensus for single-integrator dynamics: The gain functions are continuous and uniformly bounded above and below by positive constants.Specifically, 0 < γ ≤ γ_i(x) ≤ γ̄ for every agent and state.
  • A. Consensus for single-integrator dynamics: The interaction assumptions impose Lipschitz continuity, odd symmetry, and positive response to positive relative states.These conditions ensure motion toward neighbors and symmetry in the flow, while making consensus points equilibria.
  • A. Consensus for single-integrator dynamics: For single-integrator agents, an invariant integral quantity uniquely determines the agreement point, and an integral Lyapunov function proves asymptotic convergence.The Lyapunov derivative is negative unless all agents have equal states.
  • A. Consensus for single-integrator dynamics: Theorem 1 establishes asymptotic convergence to the uniquely determined agreement point, while the proof characterizes the equilibrium set without nonsmooth analysis.The integral Lyapunov construction improves the proof relative to a maximal-minus-minimal-state Lyapunov function.

B. Consensus for double-integrator dynamics

Under mild conditions, the nonlinear controller for double-integrator agents achieves asymptotic consensus in both positions and velocities, with the common velocity and equilibrium set explicitly characterized.

  • The controller extends linear second-order consensus analysis to nonlinear gains and interaction functions while preserving explicit equilibrium characterization.The proof uses an integral Lyapunov function and establishes consensus under mild conditions.
  • Agents achieve consensus in position and velocity from any initial condition, with all velocities converging to a uniquely determined common value v∗.Theorem 2 states that |x_i−x_j| and |v_i−v_j| approach zero, while v_i approaches v∗.
  • The Lyapunov derivative is nonpositive and vanishes only when relative velocities are zero, enabling LaSalle-based convergence analysis.The invariant-set argument excludes non-consensus trajectories and identifies the limiting common velocity.
  • The common equilibrium velocity is uniquely determined by an invariant integral quantity, and the resulting position and velocity limits are explicitly identified.Existence and uniqueness follow from the stated assumptions.
  • The gain interpretation treats 1/γ_i(v_i) as a velocity-dependent mass, linking the invariant quantity to total relativistic momentum.This provides a physical interpretation of the conserved quantity used to determine v∗.

C. Consensus for double-integrator dynamics with state-dependent damping

State-dependent damping and nonlinear interaction functions yield consensus for double-integrator agents from arbitrary initial positions, with a uniquely characterized consensus point.

  • The framework generalizes average consensus by incorporating nonlinear state-dependent damping and nonlinear interaction functions.It extends earlier results to a broader class of consensus functions.
  • Agents converge to a common point from any initial position when the interaction integrals diverge at infinity.The divergence condition is required for every edge interaction function.
  • Integral Lyapunov analysis and LaSalle’s invariance principle identify the largest invariant set as common position with zero velocity.Compactness and positive invariance support the convergence argument.
  • The consensus point is uniquely determined by the equilibrium equation because each damping function κ_i is positive.The uniqueness argument follows after convergence to zero velocity.
  • Starting from rest, agents converge to a common point for any initial positions, with the point specified by the corresponding integral condition.This is the stated corollary of the general double-integrator result.

V. DISTRIBUTED CONTROL WITH INTEGRAL ACTION

Distributed integral action is introduced to reject constant disturbances while retaining consensus properties, with stability analyzed separately for single- and double-integrator dynamics.

  • Static distributed controllers generally cannot reject constant disturbances, motivating a distributed integral-action protocol for single- and double-integrator agents.The proposed approach compensates for disturbances using only distributed information.
  • For single-integrator agents, the PI controller achieves convergence to a common value for any constant disturbance and any initial condition.Without absolute position measurements, relative consensus remains while absolute states may diverge.
  • The single-integrator stability proof uses integral-state augmentation, coordinate transformations, and Laplacian-based eigenvalue analysis.The transformed system separates unobservable and uncontrollable modes from the stable disagreement dynamics.
  • In the disturbance-free single-integrator case, agents converge to the average of their initial positions.This holds under the stated protocol and initial conditions.
  • For systems with absolute-position feedback, the stability test reduces to positivity conditions on the characteristic-polynomial coefficients.The Routh-Hurwitz criterion is used to establish that the system matrix is Hurwitz.

B. Consensus by distributed integral action for double-integrator dynamics with damping

A velocity-damped distributed PI controller achieves disturbance-robust consensus for double-integrator agents when its gains satisfy a strict stability condition.

  • The protocol requires each agent to integrate relative differences; communication of the integral state itself is unnecessary.Agents need only measure neighboring states and integrate those relative differences.
  • Agents converge to a common value under arbitrary constant disturbances provided that a < bγ.Without absolute position measurements, relative positions converge but absolute states are generally unbounded.
  • In the disturbance-free case, agents converge to the average of their initial positions for arbitrary initial velocities.The result applies to the velocity-damped PI protocol.
  • When δ = 0, stability holds if and only if a < bγ, establishing the gain condition as both necessary and sufficient.The characteristic polynomial reduces to cubic factors indexed by positive Laplacian eigenvalues.
  • For disturbances without absolute-position feedback, disagreement converges to zero even though absolute states generally diverge.Boundedness occurs only when the aggregate disturbance satisfies 1^T d = 0.

A. Thermal energy storage in smart buildings

The building example models room temperatures with nonlinear thermal properties and evaluates distributed temperature regulation under a floor topology.

  • A. Thermal energy storage in smart buildings: The example assumes uniform heat conductivity and nonlinear heat capacities in Rooms 2 and 5 to represent thermal energy storage.Other rooms and the corridor use a fixed heat-capacity parameter, while initial temperatures vary across the floor.
  • A. Thermal energy storage in smart buildings: The floor topology in Figure 1 defines the room and corridor interconnections used for the temperature-control example.The example sets a desired maximum temperature of 23°C and assigns room-specific thermal parameters.
  • A. Thermal energy storage in smart buildings: Thermal energy storage is represented through phase-transition materials whose nonlinear heat capacities help regulate building temperature near a desired maximum.The paper motivates this model by contrasting the approximately constant heat capacity of air with the nonlinear total heat capacity of rooms containing storage.

B. Autonomous space satellites

The paper applies distributed and PI control protocols to satellite consensus and power-system frequency regulation, with stability and disturbance-attenuation guarantees under stated gain conditions.

  • B. Autonomous space satellites: Satellites seek one-dimensional consensus using a distributed control law based only on relative position and velocity measurements.The satellite model accounts for engine power and inertial-frame acceleration.
  • B. Autonomous space satellites: The satellite communication topology is represented by an undirected graph, with example floor and communication topologies shown for the space-satellite application.Figure 1 distinguishes the physical floor topology from the satellites’ communication topology.
  • B. Autonomous space satellites: a = 1 yields asymptotic consensus under a constant disturbance, whereas a = 0 fails to reach consensus because the disturbance persists.The example uses five mobile robots with a string communication graph and prescribed initial conditions.
  • B. Autonomous space satellites: a = 15 = bγ makes the disturbed system marginally stable, while stability is guaranteed if and only if a < bγ.The result identifies the integral-gain boundary for the second-order example.
  • Power-system frequency control: Both centralized and decentralized controllers guarantee limt→∞ωi(t) = ωref for every bus and any initial condition when a, b > 0.The centralized guarantee is stated in Proposition 1, and the decentralized guarantee in Proposition 2.
  • Power-system frequency control: Power-system frequency control is analyzed for both centralized and decentralized architectures, including local control based on phase and frequency measurements.The decentralized architecture avoids sending control signals or reference values to buses and may help address security and network-splitting concerns.

2) Decentralized PI control:

The decentralized PI controller uses local phase and frequency measurements to regulate power-system frequencies without communicating control signals or reference values. Its stability analysis establishes convergence to the reference frequency, while practical feasibility is limited by measurement-error sensitivity.

  • 2) Decentralized PI control:: The decentralized controller regulates each bus using only local phase and frequency measurements, eliminating the need to send control signals or reference values.This architecture may be favorable for security concerns and when line trips split the network.
  • 2) Decentralized PI control:: The decentralized integral action satisfies zi = ωreft − δi, linking the controller state to the reference-frequency trajectory and bus phase.This relation follows by integrating the decentralized controller equation.
  • 2) Decentralized PI control:: With only frequency measurements, even slight measurement error can be integrated and cause instability, although PMU technology makes phase measurements increasingly available.The text identifies this as a practical feasibility constraint for the decentralized controller.
  • 2) Decentralized PI control:: For any initial conditions, decentralized control guarantees limt→∞ωi(t) = ωref for every bus when a, b > 0.The proof establishes stability through the transformed system and shows that the translated origin is the only equilibrium.
  • 2) Decentralized PI control:: The IEEE 30-bus test system was used to test centralized and decentralized frequency controllers with common parameters a = 0.8 and b = 0.04.The setup assumes line admittances from the cited test system and 132 kV voltages at all buses.

3) Simulations:

The simulations examine power-system frequency control and broader applications of the proposed consensus protocols. They also note that identical controller parameters may be restrictive for heterogeneous generators.

  • Power-system frequency control: A 200 kW load increase at buses 2, 3, and 7 tests centralized and decentralized frequency controllers targeting ωref = 50 Hz.Both controllers use a = 0.8 and b = 0.04 in this simulation.
  • Simulation assumption: Identical controller parameters for centralized and decentralized control may be restrictive when generators are more heterogeneous.The assumption may be less restrictive for homogeneous generators.
  • Applications: The proposed controllers are demonstrated in simulations of satellites, mobile robots, building temperature control, and electrical power-system frequency control.

APPENDIX

The appendix establishes boundedness of the network state by combining bounded relative states with a contradiction argument for the average position.

  • Boundedness argument: Bounded relative states imply that all pairwise position differences remain bounded on the connected graph.The proof uses a bound proportional to (n − 1)M′ in the infinity norm.
  • Contradiction argument: If the average position tends to positive infinity, every agent position eventually becomes positive and the gain-weighted position sum becomes unbounded.The analogous negative-infinity and nonconvergent cases are treated similarly.
  • Conclusion: The contradiction shows that the average position is bounded, so the full position state is bounded and the relevant set is compact.The conclusion invokes the Heine-Borel theorem.
Loading 1310.8620v2…