Signals and dynamic systems

A signal is a quantity considered as it changes: screen brightness command versus time, measured motor speed, room temperature, or camera angle. A system maps input histories to output histories. Some systems respond nearly at once; others store energy or information, so the present output depends on state accumulated from the past.

A compact first-order model is τ dy/dt + y = Ku, where u(t) is input, y(t) output, K steady-state gain, and τ the time constant. For a constant step from zero to U at t = 0, y(t) = KU(1 - e^{-t/τ}). After one τ, the output reaches about 63.2% of its final change; after roughly 4τ it reaches about 98.2%. An RC voltage, a simplified thermal sensor, or a heavily damped speed response may fit this shape over a limited operating range.

State is the minimum remembered information needed to predict future behavior given future input. In a capacitor model, voltage is state; in a moving mechanism, position and velocity may both be needed. Disturbances enter outside the commanded input, while measurement noise corrupts what the controller sees. Deciding which variable is input or disturbance depends on the model boundary. This matters when choosing update rates: a sensor sampled much faster than its physical time constant may produce more data without revealing faster real dynamics.

A common misconception is that gain K alone describes response speed. Gain sets the eventual change per input; τ sets how quickly it approaches. Another is assuming any curved trace proves a first-order law. Dead time, saturation, multiple energy stores, friction, and changing gain can violate it. The optional RC transient simulation is a bounded ideal first-order model; it omits component tolerance, source resistance unless specified, leakage, and measurement loading. Use residuals and repeated tests before treating a fitted time constant as physical truth.