Step response method

Long time constants in the system and zone-to-zone interaction of the heaters complicated the controller design and tuning. The time available for experimental measurements was limited by the schedule of other experimental work to be performed by the extruder. The classic step response methods of tuning controllers would take on the order of hours to perform, and frequently disturbances in the polymer feed or in the ambient room conditions would invalidate the test. Consequently, a mathematical rather than an empirical approach was desirable. 

A key to the successful application of a PID control is the tuning of parameters, Xp, Tp and Tp in Equation 13.5. To tune them properly, the Ziegler-Nichols method is used, which includes an ultimate-gain method and a step-response method. 

The step response method uses the process response to the step change of input as illustrated below. 

The step response method is performed using the following steps. 

Table 13.3 Determination of parameters of PID control by step response method.

Numerical simulations have been conveniently used to describe complex fluid dynamic behavior in microstructures [21, 86]. Van der Linde et al. [87] solved the coupled diffusion equation for reacting species and compared the results with data from the oxidation of CO on alumina-supported Cr using the step-response method. Transient periodical concentration changes in microchannels have been numerically calculated by various authors [19, 58, 88]. 

Step response method The input signal is changed stepwise from the steady-state value to some specific value. This method is mainly used to discuss the dynamic characteristics of a system. 

Fig. 5.18 Electrical circuits for DC step-response methods (a) measure- ment of a current transient, (b) measurement of the time integral of current, with compensation for the large instantaneous component of the response.

The oxidation of carbon monoxide has been studied by both the usual step-response and isotopic experiments and by the TAP system (2/7). The general conclusion is that the fast response of the TAP system did not produce any additional mechanistic information to that obtained from step-response experiments. A number of the points discussed in previous paragraphs are mentioned, and it is suggested that the final pattern of multipulse response experiments be termed a pseudo-steady state. A factor not mentioned is that transient IR experiments are valuable with the step-response method but not compatible with the TAP system. 

The step-response method has been used to study NO reduction over Cu/AbOs and Cu-Cr/AbOs (236). NO is adsorbed on oxidized catalysts as nitrale/nilrite complexes, and the latter decompose only in a slightly reducing atmosphere. These experiments on the transient regime permit the estimation of the surface composition of the catalyst in contact with various gas-phase mixtures. The reaction steps were identified, but no simulation of the observed results was attempted. 

The flow characteristics of a continuous reactor have been studied using the tracer step-response method. The concentrations of tracer in the effluent at various times after the instant of addition are reported as... 

For on-line controller tuning, the relay auto-tuning and step response methods are recommended. 

Consider the experimental step response data for the heat exchanger of Exercise 12.5. Determine the PI controller settings using the step response method and two controller tuning relations ... 

All described sensor probes scan an edge of the same material to get the characteristic step response of each system. The derivation of this curve (see eq.(4) ) causes the impulse responses. The measurement frequency is 100 kHz, the distance between sensor and structure 0. Chapter 4.2.1. and 4.2.2. compare several sensors and measurement methods and show the importance of the impulse response for the comparison. 

Note that the Proeess Reaetion Method eannot be used if the open-loop step response has an overshoot, or eontains a pure integrator(s). 

Fig. 4.35 Closed-loop step response of temperature control system using PID controller tuned using Zeigler-Nichols process reaction method.

A practical method of predicting the molecular behavior within the flow system involves the RTD. A common experiment to test nonuniformities is the stimulus response experiment. A typical stimulus is a step-change in the concentration of some tracer material. The step-response is an instantaneous jump of a concentration to some new value, which is then maintained for an indefinite period. The tracer should be detectable and must not change or decompose as it passes through the mixer. Studies have shown that the flow characteristics of static mixers approach those of an ideal plug flow system. Figures 8-41 and 8-42, respectively, indicate the exit residence time distributions of the Kenics static mixer in comparison with other flow systems. 

Gal-Or and Hoelscher (G5) have recently developed a fast and simple transient-response method for the measurement of concentration and volumetric mass-transfer coefficients in gas-liquid dispersions. The method involves the use of a transient response to a step change in the composition of the feed gas. The resulting change in the composition of the liquid phase of the dispersion is measured by means of a Clark electrode, which permits the rapid and accurate analysis of oxygen or carbon dioxide concentrations in a gas, in blood, or in any liquid mixture. 

Step- and impulse-response methods. Chemisorption can conveniently be measured under flow conditions using transient techniques, in particular, step-response and impulse-response measurements. After pretreatment, pulses of probe gas are injected into a carrier gas stream passing through the reactor that contains the pre-treated. sample. The response is detected at the reactor exit. 

Differences between equations 19.3-19 and 19.3-20 are most significant if samples are collected infrequently. Ultimately, if they lead to substantially different estimates of t and of, it is necessary to verify the results using an appropriate mixing model. Example 19-2 illustrates the method for evaluating t and of from a step response, using both central and backward differencing. 

The minimization of the quadratic performance index in Equation (16.2), subject to the constraints in Equations (16.5-16.7) and the step response model such as Equation (16.1), forms a standard quadratic programming (QP) problem, described in Chapter 8. If the quadratic terms in Equation (16.2) are replaced by linear terms, a linear programming program (LP) problem results that can also be solved using standard methods. The MPC formulation for SISO control problems described earlier can easily be extended to MIMO problems and to other types of models and objective functions (Lee et al., 1994). Tuning the controller is carried out by adjusting the following parameters ... 

There is one method that is based on a time-domain model. It was developed at Shell Oil Company (C, R. Cutler and B. L. Kamaker, Dynamic Matrix Control A Computer Control Algorithm, paper presented at the 86th National AlChE Meeting, 1979) and is called dynamic matrix control (DMC). Several other methods have also been proposed ihat are quite similar. The basic idea is to use a time-domain step-response model of the process to calculate the future changes in the manipulated variable that will minimize some performance index. Much of the explanation of DMC given in this section follows the development presented by C. C. Yu in his Ph.D. thesis (Lehigh University, 1987). 

The dynamics of the system under study can, in fact, be recovered from a variety of stimulus response tests. These include impulse and step response experiments, and frequency response and cross-correlation techniques. Descriptions of these methods and the interrelationships between them are discussed in many references, see, for instance, refs. 22—25 and Sects. 3.2.1—3.2.4 of this chapter. 

Lietti and co-workers studied the kinetics of ammonia adsorption-desorption over V-Ti-O and V-W-Ti-O model catalysts in powder form by transient response methods [37, 52, 53[. Perturbations both in the ammonia concentration at constant temperature in the range 220-400 °C and in the catalyst temperature were imposed. A typical result obtained at 280 °C with a rectangular step feed of ammonia in flowing He over a V2O5-WO3/TiO2 model catalyst followed by its shut off is presented in Figure 13.5. Eventually the catalyst temperature was increased according to a linear schedule in order to complete the desorption of ammonia. 

The adsorption of NO, under lean conditions was studied by imposing a step change of NO and NO2 feed concentrations in the presence and absence of excess oxygen over the reference catalysts in a fixed-bed flow microreactor operated at 350 ° C and analyzing the transient response in the outlet concentrations of reactants and products [transient response method (TRM)[. The adsorption/desorption sequence was repeated several times in order to condition the catalytic systems fully due to the regeneration procedure adopted (either reduction with 2000 ppm H2 + He or TPD in flowing He), BaO was the most Ba-abundant species present on the catalyst surface. FT-IR spectroscopy was used as a complementary technique to investigate the nature of the stored NO species. 

A better data analysis method, as yet unused, might be to use the area under the step response curve. It can be shown that the moments are related to the step response by... 

Characterization of Data. The use of the moments for either pulse or step data would seem to be best since the data is integrated and, thus, a smoothed mean square fit is obtained. Other methods use only the data from a narrow region such as at the pulse maximum or the slope of the step response at a given location. 

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