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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