L1 · Meet the machine and the question

System identification asks whether a model learned from one controlled experiment can predict observations from another. L1 uses a small machine so the boundary stays visible:

position command q_des
        -> fixed PD controller
        -> command delay
        -> actuator torque
        -> loaded pendulum under gravity
        -> encoder position q and derived velocity qd

The simulated setup has a fixed base, one rotary servo axis, a rigid arm of known length, a known point payload, and gravity. Angle q is measured from the downward vertical in radians. Positive velocity follows increasing angle.

The plant is intentionally analytical. Its gravity term is

$$ I\ddot q = \tau - g\ell\left(\frac{m_a}{2}+m_p\right)\sin q, \qquad I = \frac{m_a\ell^2}{3}+m_p\ell^2, $$

where m_a is the uniform arm mass, m_p is the point payload mass, and ell is the arm length. The arm contributes distributed inertia and a gravity load at its midpoint; the payload contributes at the tip. These values and the PD law are fixed and known. Only the effective delay needs to be estimated.

Role	What it means in the experiment
Oracle	Uses the configured non-zero command delay to generate observations.
Initial model	Uses the visible zero-delay assumption before identification.
Identified Student	Estimates delay from the fitting observations within the chosen bounds.

Torque and delayed-command traces help explain the motion in diagnostic figures. Fitting uses time, target position, recorded position, and derived velocity, named t, q_des, q, and qd in code.

Checkpoint

Before opening the report, predict where the zero-delay Initial curve should first depart from the Oracle: immediately at a command transition, or only after the delayed phase accumulates. Then inspect the machine frame and the fit plot. The answer is a timing-dependent phase difference, with its visible size depending on the excitation frequency; it is not evidence of a changed arm mass.