Phasor Agents: Distributed Control & Learning

• Phasor Agents are distributed nodes that use phase measurements and control to coordinate operations in electrical grids, oscillator networks, and neuromorphic learning substrates.
• They implement local feedback control laws based on phasor measurements to track setpoints, optimize network performance, and ensure system stability.
• Their design leverages synchronization theory, distributed optimization, and adaptive learning to achieve robust, scalable, and real-time control in complex networks.

A Phasor Agent is a distributed computational or physical node whose fundamental design principle is the measurement, manipulation, and control of phase variables—either in the context of electrical power systems (where voltages/currents are treated as phasors), dynamical synchronization (where phase is the organizing order parameter), or neural-inspired learning substrates (where coupled oscillators encode information in relative phases). In all cases, the agent’s local control logic and inter-agent information exchange involve explicit phasor quantities, enabling robust coordination, constraint satisfaction, and adaptive learning across a network. The theoretical and practical implementations of Phasor Agents span power grid control (Moffat et al., 2022, Bolognani et al., 2013), control-theoretic agent synchronization (Wang et al., 2022), and distributed computation/learning in oscillatory graphs (Trappe, 7 Jan 2026).

1. Mathematical Foundations of Phasor Agents

Phasor Agents act on networks where the state of each node is represented by a phasor—a complex quantity encoding both magnitude and phase. In power systems, the bus voltage at node $i$ is $V_i = |V_i| e^{j\theta_i}$. In oscillator-based frameworks, the internal state is $z_i = r_i e^{j\phi_i}$, with $r_i$ amplitude and $\phi_i$ phase.

Phasor-based agent architectures may be further categorized by domain:

• Electrical Power Systems: Agents receive phasor setpoints $(|V_i|^*, \theta_i^*)$ and locally enforce them through measurement and corrective actuation, directly interacting with the AC steady-state system equations (Moffat et al., 2022, Bolognani et al., 2013).
• Oscillatory Graphs: Agents are dynamical nodes (i.e., Stuart–Landau oscillators), with states $(r_i, \phi_i)$ and learnable coupling weights $W_{ij}$; the full network dynamics encode spatiotemporal information in relative phases and amplitudes (Trappe, 7 Jan 2026).
• LTI/MIMO Agent Synchronization: Agents are viewed as LTI systems with well-defined persistent phase behavior at specific frequencies; synchronization is achieved by explicit phase-aware controller synthesis (Wang et al., 2022).

The universality of phasor-based representation arises from its ability to encode both local state and inter-agent relations (e.g., through relative phases), and its operational compatibility with feedback control, optimization, and learning protocols.

2. Phasor Feedback Control in Power Systems

Voltage Phasor Control (VPC) exemplifies the deployment of Phasor Agents in electric power distribution. Given a set of buses $\mathcal{N}$, each phasor agent implements the following local logic (Moffat et al., 2022):

• Measurement: Acquire the complex voltage $V_i^{\rm meas}$ via a Phasor Measurement Unit (PMU), operating at 30–120 samples/s, synchronized by GPS.
• Reference Tracking: Receive reference angle $\theta_{\rm ref}$ (typically substation bus); store OPF-issued setpoints $v_i^* = |V_i|^*e^{j\theta_i^*}$.
• Local Control Law: Compute actuation $(\Delta P_i, \Delta Q_i)$ by inverting the local network self-admittance and enforcing

$$(\Delta P_i + j\Delta Q_i) = -Y_{ii} (V_i^{\rm meas} - v_i^*)$$

or in real form

$\begin{pmatrix} \Delta P_i \ \Delta Q_i \end{pmatrix} = -\begin{pmatrix} \Re\{Y_{ii}\} & -\Im\{Y_{ii}\} \[6pt] \Im\{Y_{ii}\} & \Re\{Y_{ii}\} \end{pmatrix} \begin{pmatrix} |V_i|-|V_i|^* \[3pt] \theta_i - \theta_i^* \end{pmatrix}.$

• Feedback Rate: The servo loop updates at PMU acquisition rate (10–33 ms).

This phasor-targeting approach ensures immediate local counteracting of disturbances and guarantees, by construction, that $\frac{d}{dt}(V_i^{\rm meas}-v_i^*) \approx 0$ in the absence of higher-order effects.

3. Distributed Optimization and Synchronization via Phasor Agents

In distributed grid optimization, microgenerator buses are equipped as Phasor Agents, each solving for their own local reactive injection to collectively minimize active losses and enforce voltage/reactive constraints (Bolognani et al., 2013). The core features are:

• Dual Decomposition: Each agent maintains dual variables for local voltage/reactive limits.
• Update Law: Agents perform local gradient ascent (dual variables) and primal minimization, with the key local feedback depending on measured phasors and neighbor-shared phase data.
• Synchronous/Asynchronous Operation: Both lock-step and random-timer update schemes provably converge; convergence rates and conditions are precisely characterized.

The result is a plug-and-play architecture where phasor agents achieve near-centralized loss minimization and constraint satisfaction, robust to network parameter variability and requiring only neighbor-to-neighbor phasor exchange.

In general agent synchronization, phasor analysis underpins both scalable stability guarantees and controller synthesis. The key principle is the Small Phase Theorem (Wang et al., 2022), providing frequency-local synchronization conditions solely in terms of agent and communication phase margins. This allows design of agent-dependent or uniform controllers by solving local phase inequalities or LMIs, regardless of overall network size/topology.

4. Oscillatory Phasor Agents in Computational Learning

Phasor Agents can also refer to oscillatory computational substrates in which each node is a rhythm generator (Stuart–Landau oscillator), and the system as a whole implements learning, planning, and memory (Trappe, 7 Jan 2026). The main mechanisms are:

• State Variables: Each agent maintains $z_i(t) \in \mathbb{C}$; interaction among nodes is controlled by weighted adjacency $(A_{ij}, W_{ij})$.
• Encoding: Phase differences encode relational information (e.g., sequence, binding); amplitudes gate gain/activity.
• Learning: Edge weights are modified via a local three-factor plasticity rule,

$\Delta W_{ij}(t) = \eta\,g(t)\,M(t)\,e_{ij}(t)$

where $e_{ij}(t)$ is a spike-timing eligibility trace, $M(t)$ a global modulator (e.g., reward), $g(t)$ an oscillation-phase gate.

• Wake/Sleep Protocol: Learning is divided into wake "tagging" (eligibility accumulation), NREM-like deep sleep (gated consolidation), and REM-like replay (dream-based planning), inspired by biological synaptic dynamics.

Quantitative experiments demonstrate expansion of stable learning regimes (up to 67%), improved recall (4x diffusive baselines), and explicit Tolman-style latent-learning signatures arising from phase-coherent dynamics and internal replay.

5. Quantitative Performance and Comparative Analyses

Representative performance figures for Phasor Agents are available in each domain:

Power Grid VPC (Four-node Example) (Moffat et al., 2022):

Disturbance PF	$\Delta|i_{01}|_{(\text{OL})}$	$\Delta|i_{01}|_{(\text{VPC})}$	$\Delta|i_{01}|_{(\text{VMC},\,\text{APF}=0)}$	$\Delta|i_{01}|_{(\text{VMC},\,\text{APF}=1)}$
0.7 leading	+1.00	+0.50	+1.00	+1.00
unity	+1.00	+0.52	+0.20	+0.80
0.7 lagging	+1.00	+0.48	–0.10	+0.05

These results show that VPC-implemented Phasor Agents halve or better the disturbance-induced increase in upstream flow magnitude under all tested conditions, outperforming conventional VMC except when the latter is perfectly tuned to the disturbance power factor.

Oscillatory Learning Phasor Agents (Trappe, 7 Jan 2026):

• Holographic recall accuracy: 88.9% (phase-aware kernel) vs 22.2% (diffusive baseline).
• Wake+NREM regime expands stable learning region by 67% under fixed weight norm.
• REM replay improves maze success rate by +45.5 percentage points.
• REM-only agents exhibit immediate competence in latent learning tasks (39.4% success at $t=0$ vs 6.5% for wake-only).

6. Architectural Robustness, Implementation, and Generalizations

Phasor Agent-based frameworks exhibit several robust features:

• Input-to-State Stability: Local servo loops in grid applications remain stable under bounded measurement noise and communication delays (≤1–2 cycles) (Moffat et al., 2022).
• Plug-and-Play Capability: Local control laws depend only on self-admittance (power) or neighbor structure (oscillators), enabling scalability, late binding of agent addition/removal, and adaptation to topology changes (Moffat et al., 2022, Bolognani et al., 2013, Wang et al., 2022).
• Distributed Algorithms: Both synchronous and asynchronous update logics are supported, with provable convergence and explicit scaling conditions (Bolognani et al., 2013, Wang et al., 2022).
• Synchronization Theory: The small phase theorem’s applicability to arbitrary network size and heterogeneity enables design under compositional assumptions.

A plausible implication is that the phasor agent paradigm supports distributed, local, and communication-efficient implementations in both engineered and computational systems.

7. Broader Context, Applications, and Theoretical Significance

Phasor Agents constitute a unifying abstraction across grid control, dynamical system synchronization, and distributed learning:

• Electrical Grids: Phasor Agents enable constraint-satisfying, rapid, and robust real-time power flow management in increasingly dynamic and renewable-rich grids (Moffat et al., 2022, Bolognani et al., 2013).
• Network Synchronization: They provide scalable, topology-independent sufficient conditions and constructive controller synthesis for multi-agent synchronization in networks with diverse agent dynamics and communication structures (Wang et al., 2022).
• Computation and Learning Substrates: Oscillatory phasor agents serve as neuromorphic or analog substrates for learning systems capable of fast memory, robust credit assignment without backpropagation, and emergent flexible planning (Trappe, 7 Jan 2026).

Through explicit control and adaptation of phase relations, Phasor Agents realize distributed architectures that are robust to disturbances, scalable in complexity, and capable of real-time learning, positioning them as a fundamental motif in the design of future engineered and computational networks.