Integrated absolute error, control

The integral of the absolute value of the deviation lAE (Integral Absolute Error) was also proposed as a measure for control quality ... 

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. 

IAE = Integral of the Absolute Error, cost functional for determining quality of control (the higher the value, the worse the control)... 

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... 

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... 

A comparative study of the energy requirements and control properties of three thermally coupled distillation schemes and two conventional distillation sequences for the separation of ternary mixtures is presented. The responses to set point changes under closed loop operation with proportional-integral (PI) controllers were obtained. Three composition control loops were used, and for each separation scheme, the parameters of the PI controllers were optimized using the integral of the absolute error criterion. The effects of feed composition and of the ease of separability index were considered. The results indicate that there exist cases in which integrated systems may exhibit better control properties than sequences based on conventional distillation columns. 

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... 

Figures 20.13 and 20.14 compare the performance of the MPC and multiloop control systems for a +1% set-point change in X at r = 0, followed by two feed flow rate disturbances a +30% increase at r = 50 min and a return to the original value at r = 100 min. The input and output variables are displayed as deviation variables. The numerical values of the integral of the absolute error (lAE) performance index (Chapter 12) are included for each output.

For fast computation the determination of the best step-size (interval) is crucial steps that are too small result in correct concentrations at the expense of long computation times. On the other hand, steps that are too long save computation time but result in poor approximations. The best intervals lead to the fastest computation of concentration profiles within some pre-defined error limits. This of course requires knowledge about the required accuracy. The ideal step-size is not constant during the reaction and so needs to be adjusted continuously. If more complex mechanisms and thus systems of differential equations are to be integrated, adaptive step size control is absolutely essential. 

---