Integrated error

The open-loop test response fitted to a first order with dead time function GPRC can be applied to other tuning relations. One such possibility is a set of relations derived from the minimization of error integrals. Here, we just provide the basic idea behind the use of error integrals. 

As far as we are concerned, using the error integral criteria is just another empirical method. We are simply using the results of minimization obtained by other people, not to mention that the first order with dead time function is from an open-loop test. 

The controller setting is different depending on which error integral we minimize. Set point and disturbance inputs have different differential equations, and since the optimization calculation depends on the time-domain solution, the result will depend on the type of input. The closed-loop poles are the same, but the zeros, which affect the time-independent coefficients, are not. 

Time integral performance criteria (ISE, IAE, ITAE) Apply design relations derived from minimization of an error integral of the theoretical time-domain response. 

Proportional gain, integral and derivative time constants to PI and PID controllers. Different settings for load and set point changes. Different settings for different definitions of the error integral. The minimum ITAE criterion provides the least oscillatory response. 

B. INTEGRAL ACTION (RESET). Proportional action moves the control valve in direct proportion to the magnitude of the error. Integral action moves the control valve based on the time integral of the error, as sketched in Fig. 7.9b. 

A. M. Lopez, J. A. Miller, C. L. Smith and P. W. Murrill Tuning Controllers with Error-Integral Criteria Instrumentation Technology, Nov. 1969... 

The integrations with respect to Vy and are equivalent to evaluations of Graussian error integrals (see Appendix I, p. 259) the integration with respect to Vg. can, as before, be carried out by elementary means, giving... 

The numerical solution of second-order nonlinear ODEs with split boundary conditions requires trial and error integration of two coupled first-order ODEs. If one defines d p./di = Axial Grad, then the one-dimensional plug-flow mass balance with axial dispersion,... 

Table 2.5 Important areas according to the error integral.

This probability can also be derived from the distribution curve - the error integral x) - of the Gaussian distribution (Figure 2.7). An analog consideration leads to the risk a = —P. 

Important ranges for the error integral are given in Table 2.5 in connection with the related percentage and tail areas in units of the standard deviation. 

Figure 2.7 Distribution function (error integral) for the Gaussian distribution.

Di 2 = the binary diffusion coefficient for substances 1 and 2 erf = the error integral... 

Lopez, A.M., Miller, J.A., Smith, C.L. and Murrill, PW. (1967) Tuning controllers with error-integral criteria. Instrumentation Technology, 14, 57-62. 

Of these three terms the first is — 0 (0), the integral in the second term is the error integral and only imperceptibly different from... 

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