Tutorial on dynamic average consensus: the problem, its applications, and the algorithms

Centralized solutions have drawbacks

• Centralized solutions are fragile, non-scalable, identifier-dependent, delayed, and privacy-limiting, while flooding can incur O(N^2) communication and storage costs per agent.Flooding removes the single failure point but still requires network-wide information sharing, motivating local distributed interactions and decisions.Source: Centralized solutions have drawbacks · p. 2

Objectives and roadmap of the article

• The tutorial unifies dynamic average consensus problem definitions, applications, distributed algorithms, performance trade-offs, and convergence requirements.Its roadmap covers formulation, network-system applications, static and dynamic algorithm development, continuous- and discrete-time methods, and perfect tracking using input knowledge.Source: Objectives and roadmap of the article · pp. 3–5

Multiple instantiations of static average consensus algorithms are not able to deal with dynamic problems

• Dynamic consensus tracks the changing average closely, whereas reinitializing static consensus at each sample does not converge before the next sampling time.Figure 2 compares both approaches; the dynamic method retains past actions and outperforms repeated static reinitialization even when communication frequency increases.Source: Multiple instantiations of static… · p. 3, Figure 2 · p. 4

Required mathematical background and available resources for implementation

• Dynamic algorithms treat reference signals as external inputs, requiring input-to-state stability analysis in addition to internal stability; graph theory and C or Matlab support implementation.The paper primarily considers strongly connected, weight-balanced directed graphs and provides relevant graph-theoretic and ISS background.Source: Required mathematical background and… · p. 5, Objectives and roadmap of the article · p. 5

• The problem is defined for continuous-time and discrete-time reference signals, with corresponding driving commands that achieve convergence of each agent’s state to the average.Discrete-time execution updates the local agreement state and internal variables before broadcasting the next message.Source: Dynamic Average Consensus: Problem… · p. 6
• Agents use local variables and out-neighbor information to compute an agreement state that converges asymptotically to the time-varying average, with proper initialization when needed.The driving command may be memoryless or generated by local internal dynamics; the out-neighbor convention is equivalent to using in-neighbors with reversed edge interpretation.Source: Dynamic Average Consensus: Problem… · p. 6
• A third class combines continuous-time agent dynamics with discrete-time, potentially asynchronous communication imposed by wireless-device restrictions.Transmission times need not be synchronized across agents.Source: Dynamic Average Consensus: Problem… · p. 7
• Desirable algorithms are scalable, robust to practical disturbances, and correct through exact convergence or formal error guarantees.Relevant disturbances include delays, packet drops, changing membership, and noisy measurements.Source: Dynamic Average Consensus: Problem… · p. 7
• Convergence requires sufficiently frequent network information flow, while arbitrarily fast average signals cannot be tracked with zero error without prior knowledge of their generating dynamics.Guarantees therefore depend on both network connectivity and the rate of change of the reference signals.Source: Dynamic Average Consensus: Problem… · pp. 7–8

Applications of Dynamic Average Consensus in Network Systems

• Dynamic average consensus applies broadly to networked dynamical-system problems, motivating applications in formation control, state estimation, optimization, and resource allocation.The paper presents a selected overview of applications illustrating the range of applicability.Source: Applications of Dynamic Average… · p. 8

Distributed formation control

• Dynamic consensus computes a time-varying formation center, enabling agents to track mobile targets while maintaining prescribed relative positions.A two-layer design uses consensus in the cyber layer to compute the geometric center and physical tracking of the average plus b_i; consensus can also estimate target-position variance to adjust formation scale and avoid collisions.Source: Distributed formation control · p. 8, Figure 3 · p. 9

Distributed state estimation

• Distributed state estimation uses dynamic consensus to replace fusion-center coupling in local Kalman-filter implementations, reducing reliance on a single centralized processor.The centralized Kalman filter is optimal but creates a single point of failure and high communication costs; distributed alternatives let sensors process local measurements and fuse neighbor estimates.Source: Distributed state estimation · pp. 9–10, Figure 4 · p. 10

Distributed unconstrained convex optimization

• Dynamic consensus estimates the cumulative global gradient required by centralized accelerated methods, enabling distributed convex optimization over connected graphs.The estimate is updated while the optimization iteration changes the coupling term, and related approaches extend to convex and nonconvex optimization.Source: Distributed unconstrained convex… · p. 11

Distributed resource allocation

• Dynamic consensus estimates demand mismatch in distributed resource allocation, allowing agents to coordinate cost-minimizing responses for power dispatch, routing, and economic systems.Agents can use x_i(t) − d/N as the reference signal when each knows the demand and network size; strictly convex local costs yield convergence of the centralized dynamics to x⋆.Source: Distributed resource allocation · pp. 11–12

A Look at Static Average Consensus Leading up to the Design of a Dynamic Average Consensus Algorithm

• Static consensus uses Laplacian-based linear feedback initialized by local references and converges exponentially to their average under constant, strongly connected, weight-balanced graphs.The continuous-time rate is no worse than ˆλ2, while the discrete-time rate is bounded by ρ ∈ (0, 1) under the theorem’s Laplacian condition.Source: A Look at Static Average Consensus… · pp. 13–14

A first design for dynamic average consensus

• Algorithm (11) injects reference signals continuously rather than as initial conditions, and establishes zero asymptotic tracking error for asymptotically constant inputs over connected graphs.The equivalent continuous-time forms either require reference derivatives or avoid differentiating them; the design assumes the special initialization xi(t0) = ui(t0).Source: A first design for dynamic average… · pp. 14–15, Figure 7 · p. 15
• The tracking bound depends on the disagreement component of reference derivatives, enabling exact tracking of unbounded common dynamics and bounded error when uncommon components have bounded rates.For strongly connected, weight-balanced digraphs, the convergence rate is governed by ˆλ2; larger connectivity improves the steady-state tracking bound.Source: A first design for dynamic average… · pp. 16–17

Implementation challenges and solutions

• The alternative implementation (16) removes derivative computation and makes initialization and bounded measurement or communication perturbations manageable, but agent arrivals and permanent departures can still induce bias.Under admissible switching graphs with dwell time and recurring joint strong connectivity, the algorithms remain bounded and converge to an error bound at rate no worse than λ.Source: Implementation challenges and solutions · pp. 17–19

Example: distributed formation control revisited

• In the mobile-target formation example, tracking error vanishes under input perturbations and switching topologies but retains a constant bias under agent changes or initialization errors.Figure 10 also shows that all agents converge at the same rate while tracking the geometric center.Source: Example: distributed formation control… · pp. 20–21
• The formation-control example evaluates algorithms (11) and (16) for 4 agents using reference inputs (18) in the tracking scenario described in Figure 8.The supplied passage identifies the evaluation scope but does not report quantitative outcomes.Source: Figure 10 · p. 22

Continuous-Time Dynamic Average Consensus Algorithms

• Continuous-time dynamic average consensus algorithms differ in driving commands and initialization requirements, with some derivative-dependent forms admitting alternative implementations without explicit reference-signal derivatives.Table 1 summarizes these driving-command arguments and initialization requirements.Source: Continuous-Time Dynamic Average… · p. 23, Table 1 · p. 23

Robustness to initialization and permanent agent dropout

• Algorithm (19) requires no special initialization and tracks bounded-rate dynamic averages with ultimately bounded error over strongly connected, weight-balanced digraphs.Its tracking performance depends on reference-signal derivative magnitude, graph connectivity, and the design parameter α; robustness enables recovery after agent departures and arrivals.Source: Robustness to initialization and… · pp. 24–25, Figure 12 · p. 26

Controlling the rate of convergence

• FOI-DC uses simultaneous fast and slow dynamics so β prespecifies the dominant convergence rate, while small ϵ reduces tracking error for signals with bounded first and second derivatives.The fast dynamics estimate the average of the input and its derivative, while the slow dynamics track the reference average without waiting for fast convergence.Source: Controlling the rate of convergence · pp. 26–27

An alternative algorithm for directed graphs

• Algorithm (25) is robust to reference-signal measurement perturbations and preserves each agent’s input privacy because an adversary observing all network messages cannot uniquely reconstruct that reference signal.Its convergence rate to the error bound is min{α, β Re(λ2)}; β controls tracking-error size, whereas α controls convergence rate.Source: An alternative algorithm for directed… · p. 28

Discrete-Time Dynamic Average Consensus Algorithms

• Discrete-time communication avoids the continuous-communication requirement of continuous-time algorithms, but parameters must be carefully selected to guarantee and optimize convergence.The section proposes root-locus techniques for parameter selection and introduces extra dynamics for further acceleration.Source: Discrete-Time Dynamic Average… · p. 29

Non-robust dynamic average consensus algorithms

• For non-robust algorithms, root-locus design minimizes the disagreement-pole radius by placing limiting poles at z = ρ for λ2 and z = −ρ for λN.The analysis assumes a constant, connected, undirected graph with known or bounded smallest and largest nonzero Laplacian eigenvalues.Source: Discrete-Time Dynamic Average… · p. 29, Non-robust dynamic average consensus… · p. 31
• Adding extra dynamics accelerates convergence while requiring agents to track two internal state variables instead of one.The accelerated design adds a second feedback-loop pole and can substantially improve convergence despite the small memory increase.Source: Non-robust dynamic average consensus… · p. 31

Robust dynamic average consensus algorithms

• Proportional-integral algorithms use quadratic root-locus dependence to obtain robustness to initial conditions, including in accelerated designs with closed-form parameter solutions.Their two Laplacian blocks complicate design relative to the linear root loci of the non-robust algorithms.Source: Robust dynamic average consensus… · pp. 32–33
• Robust proportional-integral algorithms converge more slowly than their non-robust counterparts but provide robustness to initial conditions; with parameters selected from λ2 and λN, the algorithms converge exponentially to uavg at rate ρ.The convergence guarantees assume a connected, undirected graph, constant scalar reference signals, and the required initialization for the non-robust algorithms.Source: Robust dynamic average consensus… · pp. 33–34

Signals with a known model (discrete time)

• Known input models enable zero steady-state error by embedding the signal model in feedback; degree m−1 polynomial signals require cascading m internal-model stages.The no-delay design is numerically stable but degrades on time-varying graphs, whereas the delayed design retains favorable tracking properties; both are not robust to arbitrary initial conditions unless a proportional-integral variant is used.Source: Signals with a known model (discrete… · pp. 35–37

Bandlimited signals (discrete time)

• Bandlimited references can achieve arbitrarily small error with a feedforward prefilter when the graph is connected and balanced, the scaled Laplacian contracts, and sufficiently many stages are used.The prefilter approximates the required delay over the passband; arbitrarily small error additionally requires exact arithmetic, because rounding errors introduce high-frequency components.Source: Bandlimited signals (discrete time) · pp. 37–38, Figure 17 · p. 38

Signals with bounded derivatives (continuous time)

• Continuous-time signum-feedback algorithms track differentiable references with bounded derivatives at zero error in finite time without requiring an input model.The gain can be selected using a known derivative bound; variants trade derivative information, initialization robustness, hop count, and convergence speed, with extra dynamics changing finite-time convergence to exponential convergence.Source: Signals with bounded derivatives… · pp. 39–41
• Finite-time sliding-mode tracking suffers from chattering, which increases control energy; smoothing reduces switching but introduces error proportional to the boundary-layer thickness.Higher-order sliding-mode control is identified as a potential remedy but has not been used for dynamic average consensus in the discussed work.Source: Signals with bounded derivatives… · pp. 41–42

• The tutorial covers applications, continuous- and discrete-time algorithms, convergence acceleration, and improved tracking using prior reference-signal information.It frames these methods as distributed solutions for dynamic average consensus across several scenarios.Source: Conclusions · p. 42
• Design must balance convergence rate, steady-state error, robustness to initial conditions, internal stability, memory, and communication demands.The article expects applications to expand in smart grids, autonomous vehicles, and distributed robotics.Source: Conclusions · p. 42
• Future work should develop opportunistic state-triggered sampling and control with convergence guarantees and measurable efficiency gains over periodic implementations.Such event-triggered approaches could support alternative communication and computation mechanisms, including cloud-mediated coordination.Source: Conclusions · pp. 42–43
• Initialization-free dynamic average consensus over directed graphs remains an important research direction, especially for time-varying signals and limited-capability agents using cloud computation.Cloud coordination can exchange estimates, plans, or fallback strategies indirectly while leveraging high-performance computation.Source: Conclusions · p. 43
• Interconnecting dynamic average consensus with other coordination layers enables estimates to influence, and be influenced by, the processes generating time-varying signals.Resource-allocation examples motivate analysis through stable interconnections, input-to-state stability, passivity, and self-tuning mechanisms.Source: Conclusions · p. 43
• Privacy-preserving dynamic average consensus is in its infancy, but signal variation may change how privacy guarantees evolve compared with static consensus.This direction is relevant to power networks, smart transportation, the Internet of Things, and manufacturing systems.Source: Conclusions · pp. 43–44

Sidebar: Further Reading

• Robustness studies address fixed delays, additive input disturbances, time-varying graphs, command saturation, agent-specific dynamics, and leader-follower roles.Source: Sidebar: Further Reading · p. 55
• For directed interactions, scalable deterministic synchronous averaging requires weight-balanced graphs, while alternatives use out-degrees, identifiers, randomization, or specialized asynchronous updates.Distributed balancing methods and push-sum protocols [S16] address unbalanced graphs, but both require agents to know their out-degree.Source: Sidebar: Further Reading · p. 55
• Time-varying methods require weight balance, but dynamic average consensus remains possible when failures destroy balance if the expected graph is balanced.Related work also designs topologies for random link failures and optimizes edge weights for fast consensus.Source: Sidebar: Further Reading · p. 55

• Solmaz S. Kia is a UC Irvine assistant professor whose research spans distributed optimization, coordination, estimation, nonlinear control, and probabilistic robotics.She received an NSF CAREER award in 2017.Source: Authors Information · p. 65
• Bryan Van Scoy is a University of Wisconsin–Madison postdoctoral researcher focused on distributed multi-agent algorithms and optimization theory.He earned his Ph.D. in electrical engineering and computer science from Northwestern University in 2017.Source: Authors Information · p. 65
• Jorge Cortés is a UC San Diego professor whose interests include cooperative control, network science, multi-agent coordination, distributed optimization, and robotics.He is an IEEE Fellow and co-author of Distributed Control of Robotic Networks.Source: Authors Information · p. 65
• Randy Freeman is a Northwestern University professor researching nonlinear systems, distributed and robust control, multi-agent systems, optimal control, and synchronization.He has served as an associate editor of IEEE Transactions on Automatic Control and received an NSF CAREER Award.Source: Authors Information · p. 65
• Kevin Lynch is a Northwestern University professor and mechanical engineering chair whose research covers robot motion, self-organizing multi-agent systems, and functional electrical stimulation.He is affiliated with the Neuroscience and Robotics Laboratory and the Northwestern Institute on Complex Systems.Source: Authors Information · p. 66
• Sonia Martínez is a UC San Diego professor whose research includes networked and multi-agent systems, nonlinear control, and robotics.She received an NSF CAREER Award in 2007 and an outstanding paper award for “Motion coordination with Distributed Information.”Source: Authors Information · p. 66