[Paper Review] Controlling of clock synchronization in WSNs: structure of optimal solutions

Larisa Manita|arXiv (Cornell University)|Aug 6, 2014

Network Time Synchronization Technologies5 references3 citations

TL;DR

This paper formulates and solves an optimal control problem for clock synchronization in wireless sensor networks (WSNs), minimizing a trade-off between synchronization error and energy consumption. Using Pontryagin's maximum principle, it derives the structure of optimal control, showing that for sufficiently high transmission power, optimal solutions include singular arcs where the control remains constant at an intermediate value, significantly improving energy efficiency while maintaining synchronization accuracy.

ABSTRACT

Energy-saving optimization is very important for various engineering problems related to modern distributed systems. We consider here a control problem for a wireless sensor network with a single time server node and a large number of client nodes. The problem is to minimize a functional which accumulates clock synchronization errors in the clients nodes and the energy consumption of the server over some time interval $[0,T]$. The control function $u=u(t)$, $0\leq u(t)\leq u_{1}$, corresponds to the power of the server node transmitting synchronization signals to the clients. For all possible parameter values we find the structure of optimal trajectories. We show that for sufficiently large $u_{1}$ the solutions contain singular arcs.

Motivation & Objective

• To model and solve an optimal control problem for clock synchronization in large-scale WSNs with a single time server and multiple client nodes.
• To minimize a cost functional that balances mean-square clock desynchronization in clients and energy consumption of the server.
• To characterize the structure of optimal control inputs (transmission power) across all parameter regimes.
• To identify conditions under which singular control arcs emerge, enabling energy-efficient synchronization.

Proposed method

• Formulates a bilinear control system where the state R(t) represents the expected squared clock deviation between client and server nodes.
• Derives the dynamics Ṙ(t) = -u(t)R(t) + Nσ², modeling synchronization error decay under control input u(t).
• Applies Pontryagin’s maximum principle to the optimal control problem with bounded control u(t) ∈ [0, u₁].
• Analyzes the Hamiltonian and adjoint equations to classify optimal trajectories into nonsingular and singular cases.
• Uses weak-* convergence and compactness arguments to prove existence of optimal solutions in L∞[0,T].
• Classifies optimal control structures based on parameter thresholds, particularly the critical value √(αNσ²/β) relative to u₁.

Experimental results

Research questions

• RQ1 Under what conditions does the optimal control strategy for WSN clock synchronization include singular arcs rather than bang-bang switching?
• RQ2 How does the optimal control structure depend on initial desynchronization R(0), time horizon T, and system parameters α, β, N, σ, u₁?
• RQ3 What is the precise structure of optimal control trajectories when the maximum transmission power u₁ is sufficiently large?
• RQ4 How does the optimal control policy ensure that no synchronization messages are sent near the final time T?
• RQ5 What is the role of the singular control value us in minimizing the combined cost of energy and synchronization error?

Key findings

• For all parameter values, the optimal control policy includes a final interval [τ, T] where u(t) = 0, indicating that sending messages near time T is suboptimal.
• When u₁ > √(αNσ²/β), the optimal control may contain singular arcs where u(t) = us ∈ (0, u₁), leading to constant R(t) = RS over time intervals.
• The optimal control structure is piecewise constant with at most two switches between u=0, u=u₁, and u=us, depending on initial desynchronization R(0) and time horizon T.
• The existence of singular arcs is contingent on u₁ being sufficiently large relative to the cost trade-off parameters α, β, and σ.
• The optimal solution is unique and can be fully characterized by partitioning the (T, R(0)) plane into regions corresponding to distinct control structures.
• Numerical plots confirm that singular control regions emerge in the (T, R(0)) plane, labeled as (us,0) or (a,us,0), where a ∈ {0, u₁}.