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npx versuz@latest install hiyenwong-ai-collection-collection-skills-kuramoto-control-theorygit clone https://github.com/hiyenwong/ai_collection.gitcp ai_collection/SKILL.MD ~/.claude/skills/hiyenwong-ai-collection-collection-skills-kuramoto-control-theory/SKILL.md--- name: kuramoto-control-theory description: "Unified control-theoretic framework for complex-valued Kuramoto networks. Based on arxiv:2604.07249 'Complex-Valued Kuramoto Networks: A Unified Control-Theoretic Framework' by Giordano et al. Use when analyzing Kuramoto network synchronization, phase locking control, switched feedforward control, sliding-mode control for oscillators, or when asked 'Kuramoto control', 'oscillator synchronization', 'phase locking design', 'complex-valued Kuramoto'." --- # Kuramoto Control Theory A unified control-theoretic framework for synchronization in complex-valued Kuramoto networks. ## Core Innovation **Problem**: Classical Kuramoto model's intrinsic nonlinearity limits analytical tractability and complicates control design. **Solution**: Complex-valued extension embeds phase dynamics into higher-dimensional linear state space, enabling modern control techniques. ## Key Concepts ### Complex-Valued Kuramoto Model **Idea**: Instead of real-valued phases θ, use complex-valued states z = r·e^(iθ) **Benefit**: - Phase dynamics → linear state space - Regulating complex-state moduli |z| to common value → recovers Kuramoto phase behavior - Modern control techniques applicable ### Control Designs #### 1. Switched Feedforward Law - Ensures exact phase correspondence at all times - No spectral gain tuning needed - Precise phase tracking #### 2. Feedforward + Sliding-Mode Law - Achieves finite-time convergence - Robust to disturbances - No spectral gain tuning #### 3. Non-autonomous MIMO Sliding-Mode Controller - Enforces phase locking at prescribed frequency - Finite-time convergence - Independent of natural frequencies and coupling strengths - Works for heterogeneous networks ## When to Use This Skill Use when: - Designing synchronization control for oscillator networks - Analyzing Kuramoto model control strategies - Need finite-time phase locking - Working with heterogeneous oscillator networks - Classical Kuramoto model fails to synchronize - Brain network phase synchronization research ## Control Design Process ### Step 1: Model Conversion Convert real-valued Kuramoto to complex-valued: ``` θ_i → z_i = r_i · e^(iθ_i) ``` ### Step 2: Choose Control Strategy **For exact phase tracking**: Switched feedforward law **For robust convergence**: Feedforward + sliding-mode **For prescribed frequency locking**: MIMO sliding-mode ### Step 3: Design Parameters - Target synchronization frequency (for MIMO) - Convergence rate (sliding-mode gain) - Robustness requirements ### Step 4: Implementation Apply chosen control law to network couplings. ## Applications ### 1. Brain Network Synchronization **Scenario**: Synchronize neural oscillators across brain regions. **Approach**: - Model brain regions as coupled oscillators - Use complex-valued Kuramoto extension - Apply MIMO sliding-mode for prescribed frequency locking **Benefits**: - Finite-time convergence (important for neural dynamics) - Handles heterogeneity (different brain regions) - Independent of natural frequencies ### 2. Power Grid Synchronization **Scenario**: Synchronize generators across power network. **Approach**: - Generators as oscillators - Complex-valued Kuramoto for stability analysis - Switched feedforward for exact phase matching ### 3. Wireless Network Clock Synchronization **Scenario**: Synchronize clocks across distributed nodes. **Approach**: - Nodes as oscillators - Feedforward + sliding-mode for robust convergence - Handles network heterogeneity ## Technical Details ### State-Space Representation Complex-valued Kuramoto in linear state space: ``` dz/dt = (natural_freq + coupling) · z ``` Control objective: Regulate |z_i| → common value, phase_i → synchronized ### Switched Control Design **Switched Feedforward**: ``` u(t) = f(phase_error, coupling_matrix) → exact correspondence ``` **Sliding-Mode**: ``` u(t) = -K · sign(sliding_surface) → finite-time convergence ``` ### MIMO Controller For n oscillators: ``` U(t) = MIMO_sliding_control(ω_target, K) ``` Ensures all phases lock to ω_target in finite time. ## Comparison: Classical vs Complex-Valued | Aspect | Classical Kuramoto | Complex-Valued | |--------|-------------------|----------------| | State space | Nonlinear (θ) | Linear (z) | | Control design | Difficult | Modern techniques applicable | | Heterogeneity | May fail | Handles easily | | Convergence | Asymptotic | Finite-time achievable | | Frequency locking | Emergent | Prescribed achievable | ## Related Skills - **kuramoto-brain-network**: Brain-specific Kuramoto applications - **brain-connectivity-analysis**: Brain network synchronization - **neural-dynamics-decision-making**: Neural oscillation dynamics - **control-systems-design**: General control theory ## Paper Reference **Full Paper**: arXiv:2604.07249 - "Complex-Valued Kuramoto Networks: A Unified Control-Theoretic Framework" by Lorenzo Giordano, Josep M. Olm, Mario di Bernardo (2026-04-08) **PDF**: papers/systems-engineering-2026-04-09/kuramoto-control.pdf **Key Quote**: "We propose two switched control designs that overcome these limitations: a switched feedforward law ensuring exact phase correspondence at all times, and a feedforward plus sliding-mode law achieving finite-time convergence without spectral gain tuning." ## Simulation Results (from Paper) - Improved transient response - Better steady-state accuracy - Enhanced robustness - Successfully synchronized heterogeneous networks (where classical Kuramoto failed) --- *Created: 2026-04-09 based on arxiv:2604.07249*