Integrated time absolute error

Graham and Lanthrop introduced the integral over time, multiplied by the absolute error ITAE (Integral Time Absolute Error), as a measure for control quality ... 

The integrated time absolute error (ITAE) criterion is the integral of the absolute value of the error multiplied by time. ITAE results in errors existing over time being penalised even though they may be small, which results in a more heavily damped response. 

Input/Output interface, 557-61 Input-output models, 81 discrete-time, 609-26 examples, 81-85, 162, 163, 166 using Laplace transforms, 159-66 Input variables, 12-14 Integral of absolute error, 302 Integral control action, 273, 277-78 advantages and drawbacks, 274-75, 307... 

Robbins recommended using a proportional gain of 0.3 UG for minimum variability in the process response. The recommended integral time constant is 1.0 times UTP for good response to load change and 2.0 times UTP for minimum lAE (integral of absolute error) in response to a step change in setpoint. 

The tables use three different tuning criteria-integral of absolute error (lAE), integral of square of error (ISE) and integral over time of the absolute error (ITAE). Not used by Smith and Murrill, another function can be defined - integral over time of the square of the error (ITSE). Each of these is a form of penalty function representing the size and duration of error. The tuning methods aim to minimise the penalty. 

Several criteria may be used to estimate the quality of control (Stephanopoulos, 1984). One of these is the integral of the time-weighted absolute error (ITAE), where... 

Integral of time-weighted absolute error (ITAE)... 

Raman spectra in the range 1545-355 cm 1 were selected for data analysis. An average of 27 (461/17) spectra were obtained for each individual with a 3 min integration time per spectrum. Each spectrum was obtained with excitation power 300 mW and integration time equivalent to 3 min. Spectra from each volunteer were analyzed using PLS with leave-one-out cross-validation, with eight factors retained for development of the regression vector. For one subject, a mean absolute error (MAE) of 7.8% (RMSECV 0.7 mM) and an R2 of 0.83 were obtained. 

The PI and PID control ler were tnned to produce a minimum integrated absolute error (lAE). Their response curves are similar in appearance to agaussian distribution curve, but with a damped cycle in the trailing edge. The peak deviation of the PID response curve is only 0.12 times the uncontrolled offset, occurring at O.l Zx the peak deviation of the PI response curve is 0.21 times the uncontrolled offset, occurring at 0.48Zx. These values can be used to predict the load response of any distributed lag whose parameters K and Zx are known or can be estimated as described below. 

We will cover later different measures of control performance but the most commonly used is integral over time of absolute error (ITAE). The higher the value of ITAE, the poorer the controller is at eliminating the error. Figure 3.19 shows the impact that switching from proportional-on-PV to proportional-on-error has on ITAE. Both algorithms have been tuned... 

Process model is known An approximate model of the process may be obtained by the step response test noted earlier (see Section 12.4.2.4) (or from first prindples). When an approximate modd of the process is known we may obtain the tuning parameters directly. We use here the example of the first-order plus dead time process, since its dynamics are so representative of the polymer equipment dynamics. Here we have chosen for the tuning criteria to minimize the integral of the time-weighted absolute error (ITAE) [Eq. (96)] [7]. 

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