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Use base dimensions $\mathsf{M},\mathsf{L},\mathsf{T}$ for mass, length and time. The proposed dimensional analysis givessoEquating the independent dimensional powers requiresConsequentlyThe mass density drops out because there is no other mass-bearing quantity with which to combine it under the stated assumptions.

Dimensional analysis fixes the powers but leaves a dimensionless coefficient undetermined. For a linear deep-water gravity wave, the dispersion relation is ${\omega }^2=gk$, giving phase velocity $\omega /k=g/\omega$ and group velocity $d\omega /dk=g/(2\omega )$. Both obey the inferred scaling. The phrase wave speed alone does not distinguish these two velocities or fix the coefficient.

By dimensional analysis, power has dimensions $\mathsf{M}{\mathsf{L}}^2{\mathsf{T}}^{-3}$, mass density has dimensions $\mathsf{M}{\mathsf{L}}^{-3}$, and angular velocity has dimensions ${\mathsf{T}}^{-1}$; radians are dimensionless. ThusMatching the three independent dimensions gives $\alpha =1$, $\beta =3$ and $\gamma -3\alpha =2$, henceWith unchanged mass density, angular velocity and the same performance coefficient (a dimensionless quantity), the pump power affinity law givesThe required power therefore increases by $61.1\%$. This uses the stipulated dependence on only these three quantities, with comparable pump geometry and performance; dimensional analysis alone does not guarantee that unrelated pumps have the same performance coefficient.

The surface tension has dimensions $[\gamma ]=M{T}^{-2}$, while $[\rho ]=M{L}^{-3}$ and $[g]=L{T}^{-2}$. ConsequentlyMatching the dimension of height gives $2-n=1$, so $n=1$. Dimensional analysis fixes this exponent but cannot determine a dimensionless prefactor or a contact-angle dependence.

The physical dimension of a quantity records its powers of chosen base dimensions, independently of units. For example, velocity has dimension $L{T}^{-1}$ and force has dimension $ML{T}^{-2}$. Dimensional analysis compares these expressions in physical equations.

Physics investigates matter, motion, and interactions by building mathematical models and testing their consequences against observations. A physical law supplies a relation; a model specifies the objects, conditions, and approximations used to apply it. Classical mechanics, electromagnetism, thermodynamics, and quantum mechanics organize different aspects of these investigations. Their usefulness comes from turning measurements into predictions that can be checked and refined.

Consider a simple pendulum: how does its oscillation period change when its length changes? For small amplitude, the ideal model gives $T_0=2\pi \sqrt{\ell /g}$, where $g$ is gravitational acceleration. Taking $g=9.81m/{\mathrm{s}}^2$ as the model parameter, a one-metre pendulum has predicted period $2.0061\mathrm{s}$. Increasing its length to four metres gives $4.0121\mathrm{s}$: quadrupling the length doubles the period. The point mass cancels from the equation, so changing it alone does not change this model's answer.

The prediction identifies a mechanism and a way to challenge it. The small-angle approximation replaces $\sin \theta$ by $\theta$; the full ideal equation predicts a longer period at large amplitude. At $60^{\circ }$, its period is about $1.07318T_0$, already a $7.3\%$ change. Measurements at such amplitudes can reveal the limit of the approximation, while appreciable drag requires a damped pendulum. These distinctions are developed in simple pendulum and dimensional analysis. The small-amplitude model is also derived in Ling, Sanny and Moebs, University Physics, pendulums.

Several different experiments can become one calculation. Nondimensionalization rewrites the pendulum equation using $\tau =t\sqrt{g/\ell }$, leaving $d^2\theta /d{\tau }^2+\sin \theta =0$. All lengths share that dimensionless equation; the amplitude remains a parameter. Similarly, conservation laws and symmetry reveal features that many models share. This is one source of the subject's beauty: a compact relation can organize a large range of phenomena.

The connection to evidence remains essential when a model is elegant. In his introductory lectures, Richard Feynman emphasizes testing proposed relations by experiment and using idealizations whose accuracy depends on the conditions. ^[ref] A discrepancy can motivate a better measurement, a better approximation, or a different physical model; the mathematical calculation and the empirical assessment are both part of the explanation.

For a cloud supported by an effective speed of sound $v$ and confined by a surface pressure $P$, dimensional analysis fixes the gravitational instability mass scale to $v^4/(G^{3/2}{P}^{1/2})$. The dimensionless coefficient depends on the equilibrium model and the convention used for the instability threshold.

Exchanging a down-type squark of mass $M$ between $LQD$ and $UDD$ vertices produces a dimension-six effective operator with coefficient $C\sim {\lambda }^{\prime }{\lambda }^{\prime \prime }/M^2$. Dimensional analysis then gives a proton decay width ${\Gamma }_p=c_h|{\lambda }^{\prime }{\lambda }^{\prime \prime }{|}^2{m}_p^5/M^4$, with a dimensionless factor $c_h$ containing hadronic matrix elements and phase space. A lifetime bound ${\tau }_p>{\tau }_{\min }$ implies $|{\lambda }^{\prime }{\lambda }^{\prime \prime }|<M^2/\sqrt{c_h{\tau }_{\min }{m}_p^5}$. The bound is on the relevant flavour product and depends on the exchanged mass; dimensional analysis alone does not determine $c_h$.

If pump power depends only on fluid mass density $\rho$, impeller diameter $D$ and angular velocity $\omega$, dimensional analysis fixes $P=C_P\rho {\omega }^3{D}^5$, where $C_P$ is dimensionless. Comparing pumps with the same $C_P$ gives the displayed scaling. Geometrical or flow-regime changes can change $C_P$; the law is not a universal comparison of arbitrary pumps.

Radiogenic heating is thermal energy deposited by radioactive decay. If its specific heating rate is $Q$ in watts per kilogram, the volumetric rate is $\rho Q$ and the heat input per horizontal area to a column of depth $H$ is $\rho QH$. The distinction between specific and volumetric rates matters in dimensional analysis of a convective heat flux law.

A ratio compares two quantities $a,b$, written $a:b$ and corresponding to division $a/b$ when $b\ne 0$. Equal ratios of corresponding lengths characterize the scaling of similar triangles.

For example, $6:4$ and $3:2$ specify the same ratio because $6/4=3/2$. Multiplying both entries by the same nonzero scale preserves the quotient: $(ca)/(cb)=a/b$. This is useful when comparing mixtures or scaling a recipe; doubling both ingredient amounts preserves their ratio, while adding the same amount to both generally does not. For quantities with the same physical units, their ratio is dimensionless, connecting this elementary calculation with dimensional analysis.

Rayleigh scattering is elastic scattering of electromagnetic radiation by particles much smaller than the incident wavelength. In the stationary particle approximation, the scattered electromagnetic radiation keeps the incident frequency: its direction changes, rather than its color. The incident electric field induces an oscillating electric dipole moment, which produces electric dipole radiation. This is a useful example of how a microscopic response can explain a familiar large-scale sight, why the sky is blue. ^[ref].

The smallness condition for Rayleigh scattering by a small sphere of radius $a$ is $ka\ll 1$, where the wavenumber is $k=2\pi /\lambda$ and $\lambda$ is the surrounding medium's wavelength. For ordinary materials with modest refractive index, the incident electric field then varies little across the particle, making the electric-dipole approximation for radiation useful. For very large refractive index, also check the internal size $|m|ka\ll 1$ before applying the simple electric dipole model. Once these conditions fail, use Mie scattering for a sphere instead of extrapolating the Rayleigh scattering wavelength dependence.

Historically, the 1871 paper ^[ref] connected the color and polarization of an electromagnetic wave of scattered skylight with particles smaller than the wavelength. Its dimensional analysis anticipates scattered light intensity proportional to particle volume squared and to ${\lambda }^{-4}$. The original mechanical picture is historical; the modern explanation uses induced electric dipole radiation.

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. ^[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 $t\ge 0$, 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: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.

Construct a minimal model with one substrate and no appreciable accumulated reaction product. Only the enzyme-substrate complex $ES$ produces product, with catalytic rate constant $k_{\mathrm{cat}}$; $EI$ and $ESI$ are inactive. Write $s=[S]$ and $I=[\mathrm{I}]$ for free molar concentrations, and $E_T$ for total enzyme molar concentration. Assume a suitable quasi-steady-state approximation, rapid inhibitor binding and release relative to turnover, and negligible depletion of free inhibitor. In this construction the population relations areHere $K_M$ is the uninhibited Michaelis constant, and $K_i,K_i^{\prime }$ are the inhibition constants for the two indicated binding branches. The symbol $I$ denotes a free concentration, not an amount added to the tube. Define $V_{\max }=k_{\mathrm{cat}}{E}_T$. Conservation of enzyme gives an original short derivation:Thus fitting a Michaelis-Menten equation at a fixed $I$ would return the apparent parameters“Apparent” means conditional on that inhibitor concentration. The uninhibited Michaelis constant has not been redefined. Dimensional analysis checks the denominator: both $\alpha K_M$ and ${\alpha }^{\prime }s$ have units of molar concentration; the resulting $v$ has units of reaction rate.

The shared numerical choices make the mechanisms comparable. At $I=0$ and $s=20\mu \mathrm{M}$, all three models give $v=50\mu \mathrm{M}\mathrm{mi}{\mathrm{n}}^{-1}$. At $I=10\mu \mathrm{M}$, the respective reaction rates are $100/3$, $25$, and $30\mu \mathrm{M}\mathrm{mi}{\mathrm{n}}^{-1}$; at $I=30\mu \mathrm{M}$ they are $20$, $12.5$, and $50/3\mu \mathrm{M}\mathrm{mi}{\mathrm{n}}^{-1}$. These are independently chosen teaching models, not measured inhibition constants for a named enzyme.

A second-order reaction has overall order of reaction two: a rate law may depend on $[\mathrm{A}{]}^2$, or on $[\mathrm{A}][\mathrm{B}]$ with partial orders of reaction one and one. Doubling all relevant reactant concentrations multiplies the reaction rate by four at fixed conditions. This experimentally determined dependence does not establish molecularity or a unique reaction mechanism. For an elementary reaction obeying the law of mass action, molecularity and overall order of reaction agree; a net chemical equation alone cannot establish that the conversion is elementary.^[ref]

For the single-concentration model, explicitly choose the disappearance rate conventionHere $k_{\mathrm{obs}}$ is the coefficient for loss of A, held fixed at the specified temperature and other conditions. Its subscript labels the chosen observed disappearance rate; it does not imply a pseudo-first-order reaction. Assume a uniformly mixed closed system, fixed volume, and no appreciable reverse or competing chemical reactions. The integrated rate law isThe reciprocal-concentration plot is a straight line with slope $k_{\mathrm{obs}}$ and intercept $1/[\mathrm{A}{]}_0$. A brief derivation uses the chain rule: $d(1/[\mathrm{A}])/dt=-[\mathrm{A}{]}^{-2}d[\mathrm{A}]/dt=k_{\mathrm{obs}}$, then applies the initial condition. This is a hyperbolic decay, rather than the exponential decay of a first-order reaction.^[ref]

As an original prescribed example, choose $[\mathrm{A}{]}_0=0.020\mathrm{mol}{\mathrm{L}}^{-1}$ and $k_{\mathrm{obs}}=0.50\mathrm{L}\mathrm{mo}{\mathrm{l}}^{-1}{\mathrm{s}}^{-1}$. At $t=100\mathrm{s}$, the reciprocal-concentration calculation is $50+50=100\mathrm{L}\mathrm{mo}{\mathrm{l}}^{-1}$, giving $[\mathrm{A}]=0.010\mathrm{mol}{\mathrm{L}}^{-1}$. The initial disappearance rate is $0.50(0.020{)}^2=2.0\times 10^{-4}\mathrm{mol}{\mathrm{L}}^{-1}{\mathrm{s}}^{-1}$; after $100\mathrm{s}$ it is $5.0\times 10^{-5}\mathrm{mol}{\mathrm{L}}^{-1}{\mathrm{s}}^{-1}$. Halving the concentration quarters the disappearance rate.

Dimensional analysis checks every formula: $k_{\mathrm{obs}}[\mathrm{A}{]}^2$ has units $\mathrm{mol}{\mathrm{L}}^{-1}{\mathrm{s}}^{-1}$, $k_{\mathrm{obs}}t$ has units $\mathrm{L}\mathrm{mo}{\mathrm{l}}^{-1}$, and $k_{\mathrm{obs}}[\mathrm{A}{]}_{0}t$ is a dimensionless quantity. Changing concentration units changes the numerical rate constant: the same example is $[\mathrm{A}{]}_0=20\mathrm{mol}{\mathrm{m}}^{-3}$, $k_{\mathrm{obs}}=5.0\times 10^{-4}{\mathrm{m}}^3\mathrm{mo}{\mathrm{l}}^{-1}{\mathrm{s}}^{-1}$, giving $[\mathrm{A}](100\mathrm{s})=10\mathrm{mol}{\mathrm{m}}^{-3}$.

For an elementary reaction $2\mathrm{A}⟶\mathrm{P}$, instead define the stoichiometrically normalized reaction rate $r=k[\mathrm{A}{]}^2$. Two A entities disappear per P formed, soConsequently the reciprocal-concentration slope is $2k$, not $k$. In the prescribed example, the normalized rate constant would be $k=0.25\mathrm{L}\mathrm{mo}{\mathrm{l}}^{-1}{\mathrm{s}}^{-1}$; initially $r=1.0\times 10^{-4}\mathrm{mol}{\mathrm{L}}^{-1}{\mathrm{s}}^{-1}$ while the disappearance rate is twice that. With $[\mathrm{P}](0)=0$, stoichiometry gives $[\mathrm{P}](100\mathrm{s})=([\mathrm{A}{]}_0-[\mathrm{A}])/2=0.0050\mathrm{mol}{\mathrm{L}}^{-1}$. Both rate constants have the same dimensions, so a units check alone cannot catch their factor-of-two difference. The symbol $r$ denotes the same normalization as $v$ in reaction rate.

The half-life, meaning the time to reach half a specified starting concentration, follows by setting $[\mathrm{A}]=[\mathrm{A}{]}_0/2$:For $r=k[\mathrm{A}{]}^2$ in $2\mathrm{A}⟶\mathrm{P}$, this becomes $t_{1/2}=1/(2k[\mathrm{A}{]}_0)$. Its units are seconds because $k_{\mathrm{obs}}[\mathrm{A}{]}_0$ has units ${\mathrm{s}}^{-1}$. Doubling the starting concentration halves the first half-life; a quoted half-life needs its starting concentration. This differs from a first-order reaction, whose half-life is independent of starting concentration.^[ref]

Within the prescribed run, $[\mathrm{A}]$ reaches $0.010$, $0.0050$, and $0.0025\mathrm{mol}{\mathrm{L}}^{-1}$ at $100$, $300$, and $700\mathrm{s}$. The successive halving intervals are $100$, $200$, and $400\mathrm{s}$: each begins at a smaller concentration. More generally, reaching fraction $f$ of the initial concentration, with $0<f<1$, requires $t=(f^{-1}-1)/(k_{\mathrm{obs}}[\mathrm{A}{]}_0)$, so $90\%$ consumption takes nine initial half-lives, rather than a fixed sequence of equal waiting times.

A straight reciprocal-concentration plot tests this particular single-concentration rate equation. It does not uniquely identify a reaction mechanism, nor does curvature rule out every second-order reaction: two independently changing reactant concentrations require their own integrated rate law. At low concentration, a small measurement error $\delta [\mathrm{A}]$ becomes approximately $-\delta [\mathrm{A}]/[\mathrm{A}{]}^2$ in the reciprocal, magnifying late-time errors. Use the line as a diagnostic; a nonlinear regression of the original concentrations can retain their measurement scale and explicit error model.

For constant kinematic buoyancy flux $f_0$ from a point source, dimensional analysis gives head height and radius proportional to $f_0^{1/4}{t}^{3/4}$, head velocity proportional to $f_0^{1/4}{t}^{-1/4}$, and head reduced gravity proportional to $f_0^{1/4}{t}^{-5/4}$. Dimensionless coefficients require the chosen fluid entrainment and head closures.

A point mass constrained to a fixed-length massless arm under uniform gravitational acceleration is a simple pendulum. With no drag its angular equation is $\ddot{\theta }+(g/\ell )\sin \theta =0$. The small-angle approximation yields simple harmonic motion, whereas the full restoring force is nonlinear. Adding drag gives a damped pendulum.

For an ideal point mass $m$ swinging in a vertical plane, the torque about the pivot is $-mg\ell \sin \theta$. Dividing by its moment of inertia $m{\ell }^2$ gives the displayed equation. Both the inertia and the gravitational torque scale with $m$, explaining why the point mass cancels. This derivation and the small-amplitude period are given in Ling, Sanny and Moebs, University Physics, pendulums.

Released from rest at a small angular amplitude ${\theta }_0$, the linearized solution is $\theta (t)\simeq {\theta }_0\cos (\sqrt{g/\ell }t)$, with oscillation period $T_0=2\pi \sqrt{\ell /g}$. For $g=9.81m/{\mathrm{s}}^2$, lengths one and four metres give $T_0\approx 2.0061\mathrm{s}$ and $4.0121\mathrm{s}$. This same scaling follows from dimensional analysis; the dynamical calculation supplies the coefficient $2\pi$.

Finite angular amplitude changes the exact ideal period. Conservation of energy gives ${\dot{\theta }}^2=(2g/\ell )(\cos \theta -\cos {\theta }_0)$ for release from rest. Integrating a quarter oscillation and substituting $\sin (\theta /2)=\sin ({\theta }_0/2)\sin \varphi$ givesHere $K$ is the complete elliptic integral of the first kind. The small-angle approximation gives a ratio tending to one; at $10^{\circ },60^{\circ },90^{\circ }$, the ratios are approximately $1.00191,1.07318,1.18034$. Thus almost constant period at small amplitude has a quantitative limit. As ${\theta }_0\to \pi$, the period diverges, connecting to the pendulum separatrix approach takes infinite time.

These predictions assume the specified ideal system. Drag changes the mechanical energy; a massive support changes the moment of inertia. The distinction is useful in experiments: a change in period can test an approximation rather than merely indicate an arithmetic mistake.

For frequency-dependent transmission, the closed SIR model is a system of ordinary differential equations for the fractions $s,i,r$:Here $\beta >0$ is the coefficient of frequency-dependent transmission, and $\gamma >0$ is the recovery rate, interpreted more generally as the removal rate. Both have units of inverse time. With a constant recovery rate, the associated individual time in the infectious compartment has an exponential distribution with mean $1/\gamma$; the SIR model does not require every person to spend exactly that long there. The fractions themselves have no units.^[ref]

For counts $S=Ns$, $I=Ni$, $R=Nr$, the same SIR model equations readThe factor $1/N$ keeps the same $\beta$ in both descriptions. Dropping it while keeping these units would fail dimensional analysis. Some existing SIR model variants use the unnormalized mass-action infection convention instead, including the SIR susceptible-recovered identity; their coefficient is $b=\beta /N$ in the notation used here.

Adding the SIR model equations gives $d(s+i+r)/dt=0$, a short proof of conservation of population. Nonnegative starting fractions stay nonnegative: the outflow from either the susceptible compartment or the infectious compartment vanishes when that group is empty, while $dr/dt\ge 0$. Thus the susceptible compartment can only shrink and the removed compartment can only grow.

For a concrete initial value problem, stipulateAt the start, the flow into the infectious compartment is $0.3\times 0.99\times 0.01=0.00297$ of the total per day, and the flow into the removed compartment is $0.1\times 0.01=0.00100$ of the total per day. Hence $di/dt=0.00197$ per day. If $N=1000$, these correspond to model rates of $2.97$ and $1.00$ people per day, and net growth $1.97$ people per day. A rate need not be an integer count of events on any particular day.

For this SIR model, the basic reproduction number isIt describes one introduced member of the infectious compartment in an otherwise wholly susceptible compartment. The instantaneous reproduction number at the actual starting state is $R_0{s}_0=2.97$, obtained by freezing the current conditions and the current susceptible compartment's fraction. More generally it is $R_{0}s(t)$ in this SIR model. The basic reproduction number is a ratio of two rates, so it has no units; $r_0$ is a different quantity, the initial removed compartment's fraction.

While $i>0$, its instantaneous relative growth isInitially this equals $0.3\times 0.99-0.1=0.197{\mathrm{day}}^{-1}$. Freezing $s\approx s_0$ gives the early exponential growth approximation $i(t)\approx 0.01e^{0.197t}$ with $t$ measured in days. It must fail as the susceptible compartment shrinks. An epidemic invasion threshold occurs at $R_0{s}_0=1$; if $i_0=0$, the deterministic SIR model stays at $i=0$ even above that threshold.

For an energy-conserving strong spherical shock wave expanding into mass density $C{r}^{-\beta }$, $0\le \beta <3$, dimensional analysis gives the displayed radius law. Neglect ambient pressure, radiation losses and gravity in this blast-dominated approximation. Write $u=\dot{R}U(\xi )$, $\rho =C{R}^{-\beta }D(\xi )$ and $p=C{R}^{-\beta }{\dot{R}}^{2}P(\xi )$, with $\xi =r/R$. The radial continuity equation, momentum equation and adiabatic pressure equation reduce toThe Strong-shock Rankine-Hugoniot conditions give $U(1)=P(1)=2/(\gamma +1)$ and $D(1)=(\gamma +1)/(\gamma -1)$. The explosion energy fixes the radius coefficient through $E=4\pi C{R}^{3-\beta }{\dot{R}}^2{\int }_0^1[D{U}^2/2+P/(\gamma -1)]{\xi }^{2}d\xi$.

Use the fields of pressure similarity in a spherical blast wave, with front radius $R={\xi }_0\ell$. Since ${\rho }_0{\ell }^5/t^2=E$, changing the radial integration variable to $\xi$ givesBoth are time independent for fixed integrable similarity profiles and constant front coordinate ${\xi }_0$. With explosion energy $E$ and negligible external energy input or losses, their sum equals $E$. This normalization, together with shock matching and regularity, fixes the dimensionless blast coefficient rather than dimensional analysis alone.

The tracking error used by the controller is the signed differencebetween 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. ^[ref]

Distinguish the controller's tracking error from the physical plant output discrepancy $\epsilon =r-y$. If the readout has constant sensor bias $b$, thenAll 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.

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. ^[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.