Introduction & Context

Integral Absolute Error (IAE) quantifies the cumulative deviation between a controlled variable and its set-point during a transient response. In process engineering it is the most widely accepted scalar figure-of-merit for regulatory performance: a smaller IAE implies tighter control, less off-spec product, and reduced utility consumption. Typical applications include temperature loops in food pasteurisers, reactor concentration control, and distillation composition loops where product quality is directly linked to the deviation integral.

Methodology & Formulas

Process model
A First-Order-Plus-Dead-Time (FOPDT) representation is adopted: \[ G_p(s)=\frac{K_p\,e^{-\theta s}}{\tau s+1} \] where \(K_p\) is the process gain, \(\tau\) the dominant time constant, and \(\theta\) the dead time.
Controller structure
An ideal PI algorithm is assumed: \[ u(s)=K_c\left(1+\frac{1}{T_i s}\right)e(s) \] with controller gain \(K_c\) and integral time \(T_i\).
Closed-loop approximation
For a set-point step \(\Delta y_{sp}\) the closed-loop response is approximated by a first-order trajectory whose time constant is \[ \tau_{cl}=\frac{\tau}{1+K_c K_p \dfrac{T_i}{\tau}}. \] The resulting process variable is \[ y(t)=\Delta y_{sp}\left[1-\exp\left(-\frac{t-\theta}{\tau_{cl}}\right)\right]\,H(t-\theta) \] with \(H(\cdot)\) the Heaviside function.
Error signal
The instantaneous error is \[ e(t)=\Delta y_{sp}-y(t). \]
IAE evaluation
The integral of the absolute error over the horizon \(T_{eval}\) is computed with the trapezoidal rule: \[ \text{IAE}=\sum_{k=0}^{N-1}|e_k|\,\Delta t,\qquad \Delta t=\frac{T_{eval}}{N},\qquad T_{eval}=\alpha(\tau+\theta), \] where \(\alpha\) is a user-selected multiplier.
Overshoot estimate
Percentage overshoot is obtained from the peak value: \[ \text{Overshoot}=\frac{\max_t y(t)-\Delta y_{sp}}{\Delta y_{sp}}\times100\%. \]

Recommended evaluation parameters
Parameter	Symbol	Recommended range
Evaluation horizon multiplier	\(\alpha\)	4 – 6
Sampling time	\(\Delta t\)	\(\leq \tau/5\)
Max acceptable overshoot		0.5 °C (food-grade)

IAE is the time-weighted sum of the absolute value of the difference between setpoint and measured process variable. A smaller IAE indicates tighter control, so engineers use it to compare tuning settings, judge loop performance, and justify control improvement projects.

Export the setpoint (SP) and process variable (PV) traces at the fastest sample rate available.
Compute the error vector: |SP − PV| for every sample.
Multiply each error by the sample interval Δt to get the area of each trapezoid.
Sum the areas over the evaluation window to obtain total IAE in engineering units·time (e.g., °C·min or bar·s).
Normalize by the time span if you need a dimensionless metric for benchmarking.

Stop when the PV has settled inside ±2 % of the setpoint band and the controller output has ceased cycling; alternatively integrate for one full process time-constant after the last visible overshoot. Extending too far inflates IAE with noise, while cutting too early hides long-tail behavior.

Yes, but because IAE weights all errors equally it may rank an oscillatory loop better than it deserves. ISE squares the error so large deviations dominate, while ITAE time-weights the error so later deviations count more. Use IAE when you want a simple, easily explained metric; switch to ITAE if overshoots late in the response are critical.

Worked Example – IAE for a Temperature Loop

A refinery pre-heater uses a PI controller to keep the crude outlet temperature at 150 °C. After a 5 °C set-point change, the engineer wants the Integral Absolute Error (IAE) to judge how tightly the loop was tuned.

Knowns

Process gain \(K_p = 1.2\) °C/%
Dominant time constant \(\tau_{dom} = 45\) s
Dead time \(L = 8\) s
Set-point step \(\Delta SP = 5\) °C
Controller gain \(K_c = 2.2\) %/°C
Integral time \(T_i = 31\) s
Evaluation horizon \(T_{eval} = 265\) s
Sampling interval \(dt = 1\) s

Step-by-step IAE calculation

Record the closed-loop error \(e(t) = SP(t) - PV(t)\) every second for 265 s.
Take the absolute value \(|e(t)|\) at each sampling instant.
Integrate numerically with the rectangle rule: \[ IAE = \sum_{k=0}^{N-1} |e_k|\,dt \] where \(N = T_{eval}/dt = 265\) and \(dt = 1\) s.
Insert the recorded absolute errors; the sum evaluates to 122.351 °C·s.

Final Answer

IAE = 122.351 °C·s

"Un projet n'est jamais trop grand s'il est bien conçu." — André Citroën

"La difficulté attire l'homme de caractère, car c'est en l'étreignant qu'il se réalise." — Charles de Gaulle