Integrator plus dead time process

A pulse test is used to establish the dynamic characteristics of an integrating plus dead-time process. 

With dynamic systems that are non-self-regulating one finds, likewise, step responses with and without dead time. An integrator plus dead time process would be modeled like Eq. (77). 

Fig. 12.23. Step response of an integrator plus dead time process.

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]. 

It is important to see over what range of processes complementary feedback has an advantage over two-mode control. A single-capacity plus dead-time process will respond to a step load change under complementary feedback as shown in Fig. 4.15. Without going into the derivation of the load response curve, it turns out that the integrated area per unit load change is... 

A set of Internal Model Control (IMC) tuning rules were established by Rivera, Morari, and Skogestad for a first-order plus dead time (FOPDT) open-loop process response that simply involves the adjustment of the proportional gain in the controller, K, for tuning. The integral time constant, 7, is set equal to the first-order time constant, TFO, for PI controllers (Table 10.5). 

Proportional-plus-integral control is the most generally useful control mode and therefore the one usually applied to automated process-control. Its major limitation is in processes with large dead-time and capacitance if reset time is faster than process dead-time, the controller-response changes are faster than the process, and cycling results. In these cases, derivative control is beneficial. 

FIG 1.25. Zntegral control of dead time (abotie) is the same as proportional control of a dead-time plus integrating process (below). 

Derivative is the inverse of integral action. In theory, it is characterized by a 90 phase lead, although because of physical limitations 45° is about all that can be expected. If perfect derivative (90 lead) were available, it could halve the period of the dead-time plus capacity loop by allowing the dcnd time to contribute all 180°. Remember that perfect derivative applied to the two-capacity process provided critical damping with zero proportional band. But Fig. 1.27 indicates that perfect derivative is limited to zero damping at a period of 2t[Pg.33]

Consider a loop consisting of a dead-time plus integrating process of time constant t, hysteresis, and a proportional-plus-reset controller. Let the reset time be set for 30 phase lag and the proportional band for ( -amplitude damping at A/H of 2. Table 5.2 summarizes the effect of hysteresis. 

FIG 5.13. Minimum-time control of a dead-time plus integrating process. 

To explore what is required for minimum-time control, consider the application to a dead-time plus integrating process. In Fig. 5.13 the tracks of both the intermediate variable, i.e., the output of the integrator, and the controlled variable are plotted. Minimum time requires that m be switched from 100 percent to equal the load. 5 before the set point is reached. 

If the same program is applied to a two-capacity process, the controlled variable will be more heavily damped than necessary. Therefore this program provides the minimum-time switching only for dead-time plus integrating processes. 

Soil and related environments are both an important natural habitat of biota and a natural reservoir of biotic debris consisting of plant remains and dead animals and microorganisms. With time, dead remains are subject to continuous turnover, either mineralized or transformed to diverse organic components which are termed humus. This process is referred to as humification. Humus is composed of humic substances plus nonhumic substances that have become stabilized and are thus an integral part of soil and related environments (Table 2.1). 

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