= Tracking error
{title2=$e(t)=r(t)-y_m(t)$}
The <tracking error> used by the <controller> is the signed difference
$$
e(t)=r(t)-y_m(t),
$$
between the <reference signal> and the <measured output>. A positive <tracking error> means the readout is below the desired value; a negative <tracking error> means it is above. With exact measurement $y_m=y$, $r=1$ and $y=1.1$ give $e=-0.1$. For a positive-response <controlled plant>, positive <feedback gain> in <proportional control> $u=k_p e$ then reduces the <plant output>. The sign of the <controller> must also account for the sign of the <controlled plant>'s response: a positive command that reduces the <plant output> requires a different sign choice. Kantor uses the opposite <tracking error> convention, $PV-SP$, together with a minus sign in his <controller>; translating both signs gives the convention here. https://jckantor.github.io/cbe30338-book/notebooks/03.06-Proportional-Integral-Control.html[Jeffrey Kantor, negative feedback and proportional-control equations]{ref}
Distinguish the <controller>'s <tracking error> from the physical <plant output> discrepancy $\varepsilon=r-y$. If the readout has constant <sensor bias> $b$, then
$$
y_m=y+b,\qquad e=\varepsilon-b,\qquad \varepsilon=e+b.
$$
All four quantities $r,y,y_m,e$ have the same units under <dimensional analysis>; a <sensor bias> is expressed in those units too. A readout that is too high by $0.2\,{}^\circ\mathrm C$ can make $e=0$ when the physical <temperature> is $0.2\,{}^\circ\mathrm C$ below the goal. <Integral action> integrates the <tracking error> it receives; it cannot infer this missing calibration information from that <tracking error> alone. <Thermometer calibration> against an independent reference addresses a different problem from tuning a <feedback gain>.
\Image[/tracking-reference-errors-signals.png]
{title=<Reference signal>, <tracking error>, and a biased <measured output>}
{description=Original signal diagram with a synthetic <equilibrium point> for <proportional-integral control>. All output coordinates are <temperature> rises above one fixed baseline; the <control input> is in heater-command percentage points. With $r=1$ and <sensor bias> $b=0.2$, zero measured <tracking error> means $y_m=1$, while the physical <plant output> is $y=0.8$ and its discrepancy is $\varepsilon=0.2$. No plotted laboratory measurements are implied.}
A motivating apparatus/model comparison is Park, Martin, Kelly and Hedengren's temperature-control laboratory: two TIP31C heaters with attached TMP36GZ sensors and heat sinks, an Arduino, a power supply, and a USB-connected computer. The readout converts sensor <voltage> to <temperature>; software records that <measured output> while specifying heater commands from $0$ to $100\%$. For the single-heater test, one command changes in steps and the other heater stays off. Parameters are fitted to one run, then a different command sequence tests predictions without feeding measured <temperatures> into the simulation. The authors observe that their richer thermal model predicts the response better than the approximation with a single <time constant> and a delayed <control input>, especially after command changes. The real sensor has its own thermal lag. They also repeat the same command profile at different ambient <temperatures> and find changes in the response, motivating ambient-condition reporting. This experiment measures a command-to-readout response, not independently the exact heater <temperature> or delivered power. https://www.researchgate.net/publication/338614896_Benchmark_Temperature_Microcontroller_for_Process_Dynamics_and_Control[Park, Martin, Kelly and Hedengren, Benchmark Temperature Microcontroller for Process Dynamics and Control, sections 2 and 3, Figures 3, 6 and 7: author-uploaded full text]{ref}
The lesson for <tracking error> is to record three traces during a tracking test: the <reference signal>, the <measured output>, and the applied <control input>. Their first difference is the <tracking error>; a model prediction minus a readout is instead a model discrepancy. In a proposed comparison of <proportional control> and <proportional-integral control>, keep the initial <temperature>, ambient conditions, heater limits, and target changes fixed; run the second heater separately as a known <control disturbance>. Independent <thermometer calibration> is needed to assess physical accuracy. These are suggested controls, not additional experiments claimed for the paper. A sensor's lag, <measurement noise>, <actuator saturation>, and changed ambient conditions can all invalidate conclusions drawn from the ideal first-order model.
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