Open-loop observers

The estimator can be regarded as formed by two parts the first (Eqs. 7a, b) is a closed-loop observer that yield the estimate sequence xs(tQ (A- = 0, 1, 2,. ..) convergent to the current sequence xs(4 and the second (Eq. 7c) is on open-loop observer driven by the Xs-estimates that yields the estimate sequence P(tQ convergent to the current sequence P(tk). ... 

Open-loop observers are used when the state variables are not observable from the available measurements. In this case, the estimation is based on the predictions of a process model. In this section, the development and use of open-loop observers is illustrated for estimation of monomer concentrations in an emulsion copolymerization reactor when only the heat of reaction is measured on-line. In this case the observability criterion is not fulfilled, which means that the estimation of the monomer concentrations cannot be carried out with feedback correction [135]. 

Equations S.2-8.8 constitute an open-loop observer/estimator that is completely dependent on model responses and whose rate of convergence is not adjustable (no feedback action) [ 132 ]. Note that if the observer error at t = 0 is equal to zero (the initial conditions are known with very high precision), it will remain equal to zero during the rest of the process, provided that the model is correct. All the model contributions to the estimator are included in D12, where the only parameters are the reactivity ratios and the heat of polymerization. These parameters can be obtained by independent experiments with a good accuracy. 

Figure 8.5 Monitoring a semibatch emulsion polymerization reaction of VAc/BA by means of an open-loop observer based on calorimetric measurements, (a) Conversion and (b) copolymer composition.

Extensions of Kalman filters and Luenberger observers [131 Solution polymerizations (conversion and molecular weight estimation) with and without on-line measurements for A4w [102, 113, 133, 134] Emulsion polymerization (monomer concentration in the particles with parameter estimation or not (n)) [45, 139[ Heat of reaction and heat transfer coefficient in polymerization reactors [135, 141, 142] Computationally fast, reiterative and constrained algorithms are more robust, multi-rate (having fast/ frequent and slow measurements can be handled)/Trial and error required for tuning the process and observation model covariance errors, model linearization required The number of industrial applications is scarce A critical article by Wilson eta/. [143] reviews the industrial implementation and shows their experiences at Ciba. Their main conclusion is that the superior performance of state estimation techniques over open-loop observers cannot be guaranteed. 

S.3 Individual Monomer Concentration Although, strictly speaking, the polymerization rate of each monomer and hence free monomer concentration and the polymer composition are not observable from the overall heat of the reaction, Q, it is possible to infer them using an open-loop observer based on a model of the process [23, 25]. The overall polymerization heat, Q, is related to the rates of reaction of the individual monomers by Equation 7.3 assuming that the heats of cross-propagation are equal to those of homopolymerization [24] ... 

The open-loop observer formed by Equations 7.23,7.27, and 7.28 has been successfully used to estimate the unreacted amounts of monomers in different monomer systems under different conditions [25-29]. Once the unreacted amounts of the monomers (Al and A ) are estimated during the course of the polymerization, the conversion and copolymer compositions can be readily estimated for batch and semibatch reactors. The accuracy of the estimation depends on the reactivity ratios and enthalpies of polymerization for each monomer that can be obtained experimentally or are available in the literature. 

Note that for homogenous polymerizations, the same open-loop observer can be applied in a more straightforward manner because monomer partitioning need not be considered. 

Vicente et al. [30] used the heat of reaction and the open-loop observers developed in Section 7.2.5.3 to determine the concentration of monomer and CTA and hence to infer the instantaneous number-average molar masses during emulsion homo- and copolymerization reactions. In addition, the authors used the inferred values for online control of the molar mass distributions of copolymers with predefined distributions. They demonstrated that polymer latexes with unimodal MMD with the minimum achievable polydispersity index in free-radical polymerization (PI = 2) and bimodal distributions could be easily produced in linear polymer systems [15, 30]. 

Open-Loop versus Closed-Loop Dynamics It is common in industry to manipulate coolant in a jacketed reacdor in order to control conditions in the reacdor itself. A simplified schematic diagram of such a reactor control system is shown in Fig. 8-2. Assume that the reacdor temperature is adjusted by a controller that increases the coolant flow in proportion to the difference between the desired reactor temperature and the temperature that is measured. The proportionality constant is K. If a small change in the temperature of the inlet stream occurs, then depending on the value or K, one might observe the reactor temperature responses shown in Fig. 8-3. The top plot shows the case for no control (K = 0), which is called the open loop, or the normal dynamic response of the process by itself. As increases, several effects can be noted. First, the reactor temperature responds faster and faster. Second, for the initial increases in K, the maximum deviation in the reactor temperature becomes smaller. Both of these effects are desirable so that disturbances from normal operation have... 

It has been observed that the value of the combined closed-loop PI control and the open-loop recycle trip control justifies its use on all turbocompressors. 

In Figure 8.8, sinee the observer dynamies will never exaetly equal the system dynamies, this open-loop arrangement means that x and x will gradually diverge. If however, an output veetor y is estimated and subtraeted from the aetual output veetor y, the differenee ean be used, in a elosed-loop sense, to modify the dynamies of the observer so that the output error (y — y) is minimized. This arrangement, sometimes ealled a Luenberger observer (1964), is shown in Figure 8.9. 

This is the idea behind the plotting of the closed-loop poles—in other words, construction of root locus plots. Of course, we need mathematical or computational tools when we have more complex systems. An important observation from Example 7.5 is that with simple first and second order systems with no open-loop zeros in the RHP, the closed-loop system is always stable. 

A periodically forced system may be considered as an open-loop control system. The intermediate and high amplitude forced responses can be used in model discrimination procedures (Bennett, 1981 Cutlip etal., 1983). Alternate choices of the forcing variable and observations of the relations and lags between various oscillating components of the response will yield information regarding intermediate steps in a reaction mechanism. Even some unstable phase plane components of the unforced system will become apparent through their role in observable effects (such as the codimension two bifurcations described above where they collide and annihilate stable, observable responses). 

As can be observed in Figure 10.1, information from the process (value of the temperature measured by the sensor) arrives at the control loop, which defines it as a closed loop. This is the case for most controllers installed in bioreactors. If the decisions of a controller do not take into account any of the monitored information of the process, it is called an open loop. An example of an open loop is a fed-batch process, where it... 

Here, max and jrm n denote, respectively, the maximum and the minimum values of the muscular activation, a determines the slope of the feedback curve, S is the displacement of the curve along the flow axis, and Fneno is a normalization value for the Henle flow. The relation between the glomerular filtration and the flow into the loop of Henle can be obtained from open-loop experiments in which a paraffin block is inserted into the proximal tubule and the rate of glomerular filtration (or, alternatively, the so-called tubular stop pressure at which the filtration ceases) is measured as a function of an externally forced rate of flow of artificial tubular fluid into the loop of Henle. Translation of the experimental results into a relation between muscular activation and Henle flow is performed by means of the model, i.e., the relation is adjusted such that it can reproduce the experimentally observed steady state relation. We have previously discussed the significance of the feedback gain a in controlling the dynamics of the system, a is one of the parameters that differ between hypertensive and normotensive rats, and a will also be one of the control parameters in our analysis of the simulation results. 

In Fig. 1(B) one can observe that the parametric controller of order p= 0 improves the response when compared with the open loop response, as expected for this class of full-scale parametric controllers. For order p = 6 (Fig. 1(C)), there is an initial strong response from the controller which makes its performance of inferior quality by comparison to cases A and B. Nevertheless, it stabilizes the states. However, controller p=3 (Fig. 1(D)) shows a poor performance as a consequent of significant model reduction. 

The control computer/DCS system consists of controllers, A/D and D/A converters, and the signal conditioifing hardware and software, i.e., filtering and validation. Each of these components requires separate evaluation. Table 15.5 lists possible problems with the controller/DCS system. One way to initially check controller tuning is to place the control loop in manual (open the control loop) and observe whether the controlled variable lines out to a steady-state or near steady-state value. Comparing the open-loop and closed-loop performance indicates whether the controller is upsetting the process. If not, disturbances to the control loop in question are the primary source of the upsets. 

The effect of the process on the closed-loop behavior can be examined directly by opening the control loop in question and observing the process behavior. Open-loop oscillatory behavior indicates a problem internal to the process. The noise level on the sensor reading can also be assessed under open-loop conditions. 

As discussed in Section 9.2.2., the introduction of feedback can lead to oscillations in a system. As pointed out by Randolph and Larson (1988) and as illustrated in Figure 9.24, a crystallizer has an inherent feedback structure because of the interrelationship between the growth rate, the nucleation rate, and the CSD. This feedback is responsible for observed instabilities of an open-loop crystallizer (i.e., a crystallizer that is not externally controlled). 

Figure 4.4 shows the open-loop response over three hours for the following scenario after 0.5 h the feed temperature increases to 308.15 K for one hour, then for 0.5 h the total flow rises to 2000 kmol/h, followed by feed reset to 1870 kmol/hr, and finally feed temperature reset to 308.15 K. It can be observed that the flash temperature follows closely the disturbances with apparently first-order dynamics. The pressure is not affected by the disturbance in temperature, and only slightly by the feed flow. 

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