control//step response//first-order system

2021-10-05

A first-order system is a dynamical system with a single state whose rate of change is proportional to the gap between where it is and where its input is pushing it, and it is one of the most used models in industry: the temperature of a heated block, the speed of a motor, the level of a tank with a linear outlet, each summed up by two numbers read off one step test. Its equation is

τ x˙=−x+K u,x(t)=K u0(1−e−t/τ) after a step u0 from rest.\tau\,\dot x=-x+K\,u,\qquad x(t)=K\,u_0\left(1-e^{-t/\tau}\right)\ \text{after a step } u_0 \text{ from rest.}τx˙=−x+Ku,x(t)=Ku0(1−e−t/τ) after a step u0 from rest.

KKK is the static gain, how much output comes out per unit of input once everything has settled (degrees per percent of heater power, rpm per volt), and τ\tauτ is the time constant, how long it takes to get there. The response starts at once, with its steepest slope, and creeps towards Ku0K u_0Ku0 without ever passing it: a first-order system never overshoots, which is the quickest way to tell it from a second-order system on a trend.

Physically it is a store with a leak. Heat enters a motor housing and leaks to the air through a conductance; the time constant is the storage over the conductance (heat capacity over loss coefficient, inertia over viscous friction, a tank's section over its outlet slope), and the static gain is what the leak allows to accumulate. That is also why the same law, with the input switched off, is plain exponential decay. An integrator is the store without the leak: it ramps under a constant input instead of settling.

Two numbers from one step are enough for most loops.

A process engineer moves the valve by 5 %, records the response, fits KKK, τ\tauτ and a dead time, and has the FOPDT model that tunes the great majority of temperature, level and pressure loops in a plant (PID tuning); a forty-state model pays off only when something like a multivariable MPC needs it.

The parameters move with the operating point. Linearized around a level h∗h^*h∗, a tank emptying through an orifice behaves as a first-order system with τ=2Ah∗/q∗\tau=2Ah^*/q^*τ=2Ah∗/q∗, so the same tank is slower when full than when nearly empty (linearization). One fitted model covers the neighbourhood of the step that produced it.

Real plants are higher order, and the first-order fit captures the slowest, dominant mode. The faster modes and any transport delay end up lumped into the dead time, which is why the dead time of a fitted model is often larger than the physical pipe delay (delay and lag).

Its timing is set by τ\tauτ alone, and so is how often a controller must look at it; the canonical shape with overshoot and ringing is the second-order system, and the whole test is described in step response.