Control Theory #33 — Conical Funnel Linearization (Worked Example 10)

Instructor: Dr. Süleyman Burak ÇELİK

In this worked example, we model and linearize a conical funnel system. The 45-degree cone geometry connects to the nonlinear dynamics through area and volume relationships, and we linearize around a given set-point. Topics covered: - Part (a): Input/output identification — q_in is the input, h (water level) is the output - Part (b): State equation derivation — 45 degree cone gives A = pi h squared, V = pi h cubed / 3, mass balance leads to x_dot = u/(pi x squared) - 1/(pi x to the 3/2) - Part (c): Set-point linearization at x_SP = 1 — finding u_SP = 1 from equilibrium, computing Jacobian partial derivatives, writing the linearized model Key results: - Set-point: (x_SP, u_SP) = (1, 1) - Linearized: delta x_dot = -1/(2 pi) delta x + (1/pi) delta u - delta y = delta x - System is locally stable (a less than 0), time constant tau = 2 pi Every calculation shown step by step — no steps skipped. Playlist: Control Theory - AcEdumy GitHub: https://github.com/acedumy