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Feedback control selects a control input using observations of the actual controlled plant. In a tracking loop, compare the reference signal $r$ with the measured output $y_m$ to form the tracking error $e=r-y_m$, then let the controller change $u$. The control input changes the plant output, which is measured again: that return path closes the loop. A controller can use earlier observations as well as the latest one. ^[ref]

For the ideal controlled plant, proportional control $u=2(1-x)$ initially commands $u(0)=2$. As $x$ approaches $1$, the command shrinks. When a positive control disturbance pushes $x$ above $1$, the tracking error becomes negative and the control input pushes back. The result is a bounded offset rather than the continuing drift of open-loop control. Integral action can remove that offset by remembering past tracking errors.

A control selected from the observed current state, rather than specified only as a function of the initial state and time. For $x_{t+1}=x_t+u_t$ and stage cost $x_t^2+2x_t{u}_t+7u_t^2$, the infinite-horizon feedback $u_t=-x_t/3$ gives $x_t=(2/3{)}^t{x}_0$ and total cost $2x_0^2$. The finite-horizon Bellman coefficients satisfy ${\Pi }_0=0$, ${\Pi }_s=6(1+{\Pi }_{s-1})/(7+{\Pi }_{s-1})$ and increase to two, proving optimality.

A controller is the rule or mechanism that chooses a control signal. An open-loop control can depend on a precomputed schedule; a feedback control also uses a measured output and can have its own memory, as in integral action.

Control theory studies how to choose control inputs for a dynamical system so that its plant output follows a reference signal, despite control disturbances and imperfect knowledge of the controlled plant. A controller turns the available information into a control signal. The design must consider both the resulting response and stability theory: a rule that corrects a small mismatch can still produce growing deviations if the interconnection is badly chosen. ^[ref]

The central distinction is informational. Open-loop control executes planned commands; feedback control uses the measured output of the actual controlled plant to update those commands. A reference signal can describe a destination, while control disturbances describe influences that the controller did not command. A model can predict where the controlled plant should be; measurement tells the controller what actually happened. The authors provide a free full-book PDF through ^[ref].

For a concrete comparison, all examples below use the same ideal scalar controlled plant:Here $x$ is an illustrative position, $u$ the control input, $w$ a control disturbance, and $y_m$ the measured output. Both $x$ and $t$ are dimensionless quantities; the numbers are chosen to expose the mechanism, with no empirical interpretation. The ordinary differential equation says that the present control input and control disturbance accumulate into $x$. The measured output equals the entire state only in this ideal example; a general plant output need not reveal the entire state.

A feedback delay means a controller reacts to an earlier plant output. An accurate measurement can still be stale: when the plant output has already crossed the reference signal, a controller using the earlier value may continue pushing it away from the target. In a pure constant-delay model, $y_m(t)=y(t-\tau )$ with $\tau \ge 0$. This isolates timing from measurement noise; it does not describe every sampling or communication mechanism. ^[ref]

For the ideal scalar controlled plant $\dot{x}=u$, a zero reference signal, no control disturbance, and proportional control with constant feedback gain $k>0$, the equations areWith $\tau =0$, this is an ordinary differential equation with exponential decay $x(t)=x(0)e^{-kt}$. With $\tau >0$, it is a delay differential equation, requiring an initial history of delayed feedback. The delay stability of proportional integrator control depends on the product $k\tau$: even this simple controlled plant loses asymptotic stability at $k\tau =\pi /2$.

For positive feedback delay $\tau$, an initial condition at a single instant is insufficient. Prescribe a continuous function $\varphi$ throughout $[-\tau ,0]$: for $0\le t\le \tau$, the derivative depends on $x(t-\tau )=\varphi (t-\tau )$. A right derivative at $t=0$ is understood; the earlier history need not itself obey the subsequent control law. Once this first interval is computed, it supplies the delayed values for the next interval. This successive integration is the method of steps for a delay differential equation. ^[ref]

For the constant initial history of delayed feedback $\varphi (t)=1$, direct integration givesFor $k=1$, $\tau =1.2$, the plant output crosses zero at $t=1$, while its measured output remains positive until $t=2.2$. At $t=1.4$, the delayed command therefore increases the magnitude of the actual deviation. This short exact calculation explains the first diagram without relying on a numerical approximation.

The measured output is the information about the plant output available to the controller. It may differ from the actual plant output because of measurement noise or feedback delay. In the ideal examples $y_m=y=x$, so the controller observes the actual current state exactly.

Measurement noise is a fluctuating contribution to measurement error. An additive model for a measured output is $y_m(t)=y(t)+n(t)$, where $y$ is the actual plant output. A controller using feedback control can react to $n$ as though it were a change in the plant output. This effect is absent from the ideal noiseless examples under control theory; an errors-in-variables model represents a related distinction between latent quantities and their noisy observations.

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.

A sensor bias is a systematic offset between the measured output and the physical plant output, modeled here as $y_m=y+b$ with constant $b$. Unlike zero-mean measurement noise, a sensor bias persists when repeated readouts are averaged. A sensor bias affects the physical meaning of a zero tracking error; for a temperature readout, independent thermometer calibration can detect it.

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.