Step response analysis

It is possible to identify the mathematieal model of an underdamped seeond-order system from its step response funetion. 

Consider a unity-gain K = ) seeond-order underdamped system responding to an input of the form 

Method (h) Logarithmic decrement. Consider the ratio of successive peaks a and 02 

Equation (3.71) can only be used if the damping is light and there is more than one overshoot. Equation (3.67) can now be employed to calculate the undamped natural frequency 

---

Fukushima, T. (1994) Predictive Method for the Progress of Neutralization (Carbonation) of Concrete by Step Response Analysis, Proceedings of the 48th Meeting on Cement Technology, Japan Cement Association, Tokyo (in Japanese) pp. 614—619. 

Choudhury SR, Choudhury SR, Rangarajan J et al (2005) Step response analysis of phosphoric acid fuel cell (PARC) cathode through a transient model. J Power Sources 140 274—279... 

The mechanism of CO oxidation over Lao.8Ceo.2Mn03 has been recently investigated by step response analysis [51]. It was proposed that CO is adsorbed on the surfece in the form of stable carbonates, whereas oxygen is dissociatively adsorbed on reduced active sites. A rapid exchange between surface and sub-surfece oxygen was revealed [51]. 

Merzlyakov M, Schick C. Step response analysis in DSC - a fast way to generate heat capacity spectra. Thermochim Acta 2001 380(1) 5-12. 

When we use pade () without the left-hand argument [q, p], the function automatically plots the step and phase responses and compares them with the exact responses of the time delay. Pade approximation has unit gain at all frequencies. These points will not make sense until we get to frequency response analysis in Chapter 8. So for now, keep the [q, p] on the left hand side of the command. 

A dose-response analysis is generally developed from each smdy that reports quantitative data on dose and response. A two-step approach distinguishes the analysis of the dose-response data. The first step is an analysis of dose and response in the range of observation of the experimental or epidemiological smdies to yield a Point of Departure (POD). The second step is extrapolation to lower doses. 

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... 

For each filter, an AC analysis was run for comparison between the different software packages. These results are displayed along with the step response from each of the filters. The results of the IsSpice model are displayed in Figs. 3.38 and 3.39. The PSpice results are shown in Figs. 3.40 and 3.41. The results from the Micro-Cap model are shown in Figs. 3.42 and 3.43. 

A cyclic input of tracer will give rise to a cyclic output. This type of input is more troublesome to apply than a pulse or step input but has some advantages for frequency response analysis. A truly instantaneous pulse input is also difficult to apply in practice, but an idealised input pulse is not essential 2,3 for obtaining the desired information (See Section 2.3.5 (b)). 

The plug flow reactor is increasingly being used under transient conditions to obtain kinetic data by analysing the combined reactor and catalyst response upon a stimulus. Mostly used are a small reactant pulse (e.g. in temporal analysis of products (TAP) [16] and positron emission profiling (PEP) [17, 18]) or a concentration step change (in step-response measurements (SRE) [19]). Isotopically labeled compounds are used which allow operation under overall steady state conditions, but under transient conditions with respect to the labeled compound [18, 20-23]. In this type of experiments both time- and position-dependent concentration profiles will develop which are described by sets of coupled partial differential equations (PDEs). These include the concentrations of proposed intermediates at the catalyst. The mathematical treatment is more complex and more parameters are to be estimated [17]. Basically, kinetic studies consist of ... 

In the concept proposed in 1983 in the US, risk assessment comprised of four steps, namely, hazard identification, dose-response analysis, exposure analysis, and risk characterisation. In a simplified procedure of risk assessment, only three types of information is needed, namely, physico-chemical characteristics, toxicology, the behavior of the chemical at the use situation. The physicochemical data is supposed to show some sense of toxicity and behaviour of the chemical. The toxicology data shows the kind of symptoms to be elucidated, the target organism, and the amount of chemicals needed for showing the symptoms. Behaviour data would show the extent the receptor - here, humans or other natural organisms - is contacted by the chemical at the use situation. The risk assessment is simply to compare the extent the receptor is contacted and the amount of the chemicals needed to show the symptom. 

The overall process and critical activities for robotic implementation of automated HTS from intake of the researcher s benchtop assay through to the identification of a validated, optimized lead series of compounds around a bona fide chemical scaffold are shown in Fig. 2. The downstream steps of hit confirmation/ validation and structure-activity relationship (SAR) elucidation and hit to lead (HTL) have been included to emphasize that the same robotic assay used for production HTS can and should be used by the HTS team to perform high-volume hit confirmation and first-line automated dose-response analysis of confirmed hits to obtain quantitative potency (IC50) rankings to elucidate the nascent SAR of emergent hit series... 

In order to gain perspective in discussing the specific problems encountered in measuring and characterizing heterogeneous surfaces it is useful to take a broad look at dynamic response analysis. A brief general introduction to impulse response, step response, and frequency response together with other reference sources may be found in Forman (1). 

For systems with transfer functions that are very difficult to factor and consequently very hard to complete the frequency response analysis, Luyben [Ref. 7] discusses various numerical solution techniques. He has also included a computer program in FORTRAN which uses the stepping technique to develop the Bode and Nyquist plots for a distillation column. More details on the philosophy of the Ziegler-Nichols tuning method can be found in the original work ... 

The apparatus for the step-response experiments can consist of gas inlet, reactor and gas analysis sections (Figure 8.4). The catalyst is placed in a cylindrical glass tube, through which the reactant gases flow. A thermoelement could be installed at the exit of the catalyst bed. The gas flows are regulated with mass flow controllers. A small portion of the product flow can be taken through a capillary into a quadrupole mass spectrometer, and the mass numbers of interest can be monitored. 

In these equations the reaction rate is denoted by r concentrations by c, the rate constant by kj and adsorbed species by. The concentration of vacant sites is expressed as c. The kinetic analysis of a reaction can be performed with the use of model fitting and non-linear regression analysis. The transient step-responses could be described quantitatively with a dynamic plug flow model as discussed above. A comparison between the experimental data and the simulations is given in Figure 8.10. 

The process gain is a steady-state characteristic of the process and is simply the ratio, Ay/Ap. The time delay, 9, is the time elapsed before Ay deviates from zero. The time constant is indicative of the speed of response the time to reach 63% of the final response is equal to 6 + t. Graphical analysis of the step response can be employed to compute good estimates of 9 and t when the response deviates from the simplified model. Table 9.4 lists one popular correlation of P, PI, and PID controllers (Stephanopoulos 1984), based on the 1953 work of Cohen and Coon using the 1/4 decay ratio. 

Step response. Although we can in principle use any feed concentration time function to determine the RTD, some choices are convenient for ease of data analysis. One of these is the step response. In the step-response experiment, at time zero we abruptly change the feed tracer concentration from steady value co to steady value c/. For convenience we assume cq =. 0. Exercise. 8.4 shows, that, we can easily remove this... 

Pulse-response and step-funetion-response experiments are perhaps the easiest to carry out and analyze however, any perturbation-response technique can be used to determine age distributions. Kramers and Alberda [H. Kramers and G. Alberda, Chem. Eng. Sci., 2, 173 (1953)) describe a frequency-response analysis, and a general treatment of arbitrary input functions. Errors associated with input and... 

The compound responsible for the purple color, or at least one of the compounds responsible, has been isolated. A larger number of compounds, which is believed to be responsible for the yellow reaction product of the first step of analysis, have also been isolated. But still, it is evident that a very complex chain of chemical reactions takes place during the procedure. The core of the reaction is, apparently, that the compound in question, through a number of steps, is oxidized into the intermediate alloxane that ultimately condenses to murexoin. The ammonium salt of murexoin is the principal contributor to the purple color of the final test solution. The murexide reaction of caffeine can therefore, extremely simplified, be written as above (Figure 3.34.6). 

---