Dynamic Models, Transfer Functions & Stability

MIT’s feedback courses move from physical models to time response, frequency response, stability, and controller design ¹. For a mass-spring-damper with force input and position output,

Read the poles

The roots of the denominator are poles. Their location governs natural response. Left-half-plane continuous-time poles decay; right-half-plane poles grow. Complex poles create oscillation; poles near the imaginary axis decay slowly. Zeros shape how inputs reach outputs and can limit achievable response.

Step-response vocabulary

Define rise time, overshoot, settling time, steady-state error, and actuator effort. “Fast” is not a requirement. “Reach 95% within 0.30 s, overshoot below 5%, settle within ±0.2 mm by 0.5 s, for loads 2–6 kg” is testable.

Frequency response and robustness

Frequency response describes gain and phase versus sinusoidal frequency. Loop gain must cross unity with adequate margin before unmodeled delay, flexible modes, filtering, and sampling add enough phase lag to destabilize the system. Bandwidth trades response speed against noise, uncertainty, and actuator limits.

Models meet data

Estimate mass and geometry from design information; estimate damping, friction, or time constants from bounded experiments. Compare predicted and measured step or frequency responses. Residual patterns tell you what the model misses: dead time, backlash, saturation, Coulomb friction, flexible mode, or sensor filtering.

Model card

Document boundary, input, output, state variables, equations, parameters with sources, valid operating region, ignored effects, validation data, and decisions the model supports. A transfer function without a model card is easy to reuse where it no longer applies.