{title2=$r(t)$}

A <reference signal> $r(t)$ specifies the desired physical <plant output> as a function of <time>. A fixed <reference signal> is often called a setpoint; a changing <reference signal> describes a trajectory to follow. A <reference signal> is a goal, whereas the <control input> $u(t)$ is the command chosen to achieve that goal. https://jckantor.github.io/cbe30338-book/notebooks/03.06-Proportional-Integral-Control.html[Jeffrey Kantor, Practical Proportional and Proportional-Integral Control, nomenclature and sign conventions]{ref}

Asking for a <temperature> of $40\,{}^\circ\mathrm C$ does not specify a heater command of $40\%$: <dimensional analysis> distinguishes the units of these quantities. The <reference signal> and the <measured output> must describe the same quantity in the same coordinates before their difference can define a <tracking error>. An unconverted sensor readout cannot be subtracted directly from a desired <temperature>.

For example, let $y=T-T_0$ be <temperature> rise above a fixed ambient baseline $T_0$, and write the desired <temperature> as $T_d=T_0+r$. A <Heaviside step function> gives $r(t)=R$ for $t\ge0$, with $r(t)=0$ before the change: the target rise becomes $R$ at once, although a physical <controlled plant> generally cannot jump at once. A <ramp reference signal> $r(t)=vt$ instead asks the <temperature> rise to increase at a fixed rate $v$. If $R=1\,{}^\circ\mathrm C$ and $v=0.2\,{}^\circ\mathrm C\,\mathrm s^{-1}$, the first <reference signal> asks for a lasting one-degree rise; the second asks for a one-degree rise at $t=5\,\mathrm s$ and a two-degree rise at $t=10\,\mathrm s$. Here <degree Celsius> differences are also <kelvin> differences. A target that grows forever is useful for analytical tests but eventually exceeds a real heater's range.

To make the distinction between a goal and a command concrete, the examples in <steady-state tracking error> use the <first-order low-pass filter> as a new <controlled plant>, rather than the accumulating <controlled plant> $\dot x=u+w$ used in the earlier feedback demonstration:
$$
\tau\dot y+y=Ku+d,\qquad \tau>0,\quad K>0.
$$
The <time constant> $\tau$ has units of <time>, $K$ has units of <plant output> per <control input>, and the additive <control disturbance> $d$ has the units of $y$. This $d$ is an equivalent output offset, not the rate-like $w$ in $\dot x=u+w$. A heater interpretation uses $y$ in <degree Celsius> differences and $u$ in percentage points of heater command, with $K$ in degrees per percentage point; the <control input> is a command, not a direct measurement of delivered power. With exact measurement, holding $y=R$ requires $u=(R-d)/K$. Thus even a constant <reference signal> can require a persistent nonzero <control input>. The numerical model below is deliberately synthetic and has no fitted laboratory parameters.


 Reference signal (source code)