Bonus Day 9: Exponentials and Logs — Solving “Time to Target” in Model 4 скачать в хорошем качестве

Bonus Day 9: Exponentials and Logs — Solving “Time to Target” in Model 4

In this bonus lesson, we complete the core math behind PNNL Model 4 by answering one simple question: How many minutes until we’re close enough? Model 4 doesn’t guess runtime using slopes or regression. Instead, it assumes the temperature error decays by a fixed ratio each time step — a first-order response. Once you accept that assumption, the rest of the math falls out naturally. Repeated multiplication becomes an exponential. And exponentials are inverted using logarithms. In this video, we walk through that transition step-by-step and explain why logs appear in the Model 4 optimal start equation. Nothing is hand-waved. Everything is tied directly back to physical HVAC behavior and BAS logic. We cover: • How repeated decay becomes an exponential function • Why first-order systems shrink error by percentage, not degrees • What the decay factor c really means in HVAC terms • Why ln(c) is negative for a stable system • How deadbands translate into “close enough” conditions • How logs solve for time instead of error • Why Model 4 produces a closed-form time prediction We also walk through a tiny Python example using only the math library — no NumPy — to compute how many minutes it takes to reach a deadband from an initial error. This is exactly the type of computation that can run inside resource-constrained BAS hardware. By the end of this lesson, logarithms should feel like a practical engineering tool instead of abstract math. At this point, Model 4’s equation is no longer mysterious — it’s inevitable. Lesson reference https://github.com/bbartling/hvac-opt... Vibe Coding (Niagara Program Objects) https://github.com/bbartling/niagara4... HVAC Optimal Start Math Playground https://github.com/bbartling/hvac-opt... #hvac #optimalstart #buildingautomation #niagaraframework #smartbuildings #hvacmath #firstordersystems #exponentialdecayproblem #controlsystems #engineeringeducation #vibecoding