Control Theory #32 — Set-Point Linearization (Worked Example 9)

Instructor: Dr. Süleyman Burak ÇELİK

In this worked example, we linearize a nonlinear state space system using the Jacobian method. Given x_dot = (x-1)u + 2x^2 - 2u^2 and y = 3xu, we check three candidate set-points and then compute the linearized model at the valid one. Topics covered: - Part (a): Equilibrium check — substituting each candidate into x_dot = 0, only (x=1, u=1) satisfies the condition - Part (b): Jacobian linearization — computing all four partial derivatives of f and g, evaluating at the set-point, writing the linearized state space model Key results: - Valid set-point: x_SP = 1, u_SP = 1 - Linearized state equation: delta_x_dot = 5 delta_x - 4 delta_u - Linearized output equation: delta_y = 3 delta_x + 3 delta_u - Physical interpretation: a = 5 (unstable tendency), b = -4 (control authority) Every calculation shown step by step — no steps skipped. Playlist: Control Theory - AcEdumy GitHub: https://github.com/acedumy