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Linearizing nonlinear dynamics around operating points for control design.
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For nonlinear dynamics dx/dt = f(x, u), linearize around (x_ref, u_ref):
A = ∂f/∂x |_{x_ref} (n×n matrix)
B = ∂f/∂u |_{x_ref} (n×m matrix)
For discrete-time control with timestep dt:
Euler method (simple, first-order accurate):
A_d = I + dt * A_c
B_d = dt * B_c
Matrix exponential (exact for LTI):
A_d = expm(A_c * dt)
B_d = inv(A_c) @ (A_d - I) @ B_c
The Jacobian depends on current tensions and velocities. Key partial derivatives:
∂(dT_i/dt)/∂T_i = -v_i / L
∂(dT_i/dt)/∂v_i = EA/L - T_i/L
∂(dv_i/dt)/∂T_i = -R²/J
∂(dv_i/dt)/∂u_i = R/J
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Redact text from PDF documents for blind review anonymization
中文简介正在准备中。当前显示原文说明。
Use this skill when reading sensor data from CSV files, writing simulation results to CSV, processing time-series data with pandas, or handling missing values in datasets.
中文简介正在准备中。当前显示原文说明。
DC power flow analysis for power systems. Use when computing power flows using DC approximation, building susceptance matrices, calculating line flows and loading percentages, or performing sensitivity analysis on transmission networks.
中文简介正在准备中。当前显示原文说明。
A library for building, validating, visualizing, and serializing dialogue graphs. Use this when parsing scripts or creating branching narrative structures.
中文简介正在准备中。当前显示原文说明。
Use this skill whenever the user wants to create, read, edit, or manipulate Word documents (.docx files). Triggers include: any mention of "Word doc", "word document", ".docx", or requests to produce professional documents with formatting like tables of contents, headings, page numbers, or letterheads. Also use when extracting or reorganizing content from .docx files, inserting or replacing images in documents, performing find-and-replace in Word files, working with tracked changes or comments, or converting content into a polished Word document. If the user asks for a "report", "memo", "letter", "template", or similar deliverable as a Word or .docx file, use this skill. Do NOT use for PDFs, spreadsheets, Google Docs, or general coding tasks unrelated to document generation.
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Generator economic dispatch and cost optimization for power systems. Use when minimizing generation costs, computing optimal generator setpoints, calculating operating margins, or working with generator cost functions.
中文简介正在准备中。当前显示原文说明。