Frequency response methods

At this point, we should disclose a little secret. Just from the terminology, we may gather that control analysis involves quite a bit of mathematics, especially when we go over stability and frequency response methods. That is one reason why we delay introducing these topics. Nonetheless, we have to accept the prospect of working with mathematics. We would be lying if we say that one can be good in process control without sound mathematical skills. 

If we want to design a PI controller, how should we proceed with frequency response methods Let s presume that the unit of the time constants is in minutes. 

A final word Can frequency response methods replace root locus ... 

As you will see, several different approaches are used in this book to analyze the dynamics of systems. Direct solution of the differential equations to give functions of time is a time domain teehnique. Use of Laplace transforms to characterize the dynamics of systems is a Laplace domain technique. Frequency response methods provide another approaeh to the problem. 

First we will look at the question of stability in the z plane. Then root locus and frequency response methods will be used to analyze sampled-data systems. Various types of processes and controllers will be studied. 

Yasuda, Y. Mizusawa, H. Kamimura, T. Frequency Response Method for Investigation of Kinetic Details of a Heterogeneous Catalyzed Reaction of Gases. J. Phys. Chem. B 2002, 106, 6706-6712. 

The increase in time resolution of advanced sorption uptake methods and the joint use of sorption and radio-spectroscopic techniques allow for a more detailed analysis of the so-called "non-Fickian" behaviour of sorbing species in the intracrystalline bulk phase [18,28,29,76]. Correspondingly, information on molecular dynamics has been obtained for n-butane and 2-but ne in NFI zeolites by means of the single step frequency response method and C n.m.r. line-shape analysis [29]. As can be seen from Figures 4 and 5, the ad- / desorption for both sorbates proceeds very quickly, but with a... 

Forced oscillation is a well-known technique for the characterization of linear systems and is referred to as a frequency response method in the process control field. By contrast, the response of nonlinear systems to forcing is much more diverse and not yet fully understood. In nonlinear systems, the forced response can be periodic with a period that is some integer multiple of the forcing period (a subharmonic response), or quasi-periodic (characterized by more than one frequency) or even chaotic, when the time series of the response appears to be random. In addition, abrupt transitions or bifurcations can occur between any of these responses as one or more of the parameters is varied and there can be more than one possible response for a given set of parameters depending on the initial conditions or recent history of the system. 

Kochenburger, R. J. Trans. AIEE 69 (1950) 270. A frequency response method for analysing and synthesising contactor servomechanisms. 

Frequency response method The input signal is changed in the form of a trigonometrically function. 

Other frequency domain techniques which have been proposed include the commutative controller (31), sequential return difference (32), and the direct Nyquist array (33). In chemical pro-ess control, a number of recent applications of multivariable frequency response methods include distillation columns (34), (35), and reactors (36). 

The primary interest in the pole placement literature recently has been in finding an analytical solution for the feedback matrix so that the closed loop system has a set of prescribed eigenvalues. In this context pole placement is often regarded as a simpler alternative than optimal control or frequency response methods. For a single control (r=l), the pole placement problem yields an analytical solution for full state feedback (e.g., (38), (39)). The more difficult case of output feedback pole placement for MIMO systems has not yet been fully solved(40). 

Physical state space models are more attractive for use with the LQP (especially when state variables are directly measurable), while multivariable black box models are probably better treated by frequency response methods (22) or minimum variance control (discussed later in this section). 

A fundamental advantage of the frequency response method is its ability to yield information concerning the distribution of molecular mobilities. For example, a bimodal distribution of diffusivites, which is difficult to detect by conventional sorption measurements, leads to two different resonances [49], Moreover, from an analysis of the frequency response spectrum it is even possible to monitor molecular diffusion in combination with chemical reactions [45]. As in conventional sorption experiments, however, the intrusion of heat effects limits the information provided by this technique for fast adsorption-desorption processes [50]. 

Clements also estimated the frequency response from the pulse response. A frequency-response method is also suggested by Kramers and Alberda.52 Schwartz et al.9S suggested a new, two-tracer technique for simultaneous determination of liquid holdup and Peclet number. 

Yasuda and coworkers (46, 47) extended the use of the frequency-response method to heterogeneous catalytic reactions. The input remains the sinusoidal variation of the volume of the reactor, but with a continuous flow of reactants and measurement by mass spectrometer of the response of the concentrations of the products. Yasuda recently reviewed all his work (48). 

Other tracer techniques exist, such as negative step (i.e., elution), frequency-response methods, and methods that use inputs other than steps or pulses. These methods are usually much more difficult to carry out than the ones presented and are not encountered as often. For this reason they will not be treated here, and the literature should be consulted for their virtues, defects, and the details of implementing them and analyzing the results. A good source for this information is Wen and Fan. ... 

The theoretical solutions of the frequency response method have been comprehensive developed for the kinetic behaviour of a gas-surfrce stem [1,6]. The frequency-response parameters (phase lag and anplitude) are derived from the equivalent fundamental sine-wave perturbations by a Fourier transformations of the volume and pressure square waves. 

The bimetallic catalysts exhibit higher catalytic activity at lower temperatures in comparison with pure Pt and Rh ones, as shown in Fig. 2. Otherf workers have also observed this synergism for Pt-Rl bimetallic catalysts.s The rate constants found by the RF-GC technique such as those in Table 1, are very close to thosm determined experimentally by the frequency response method " for the adsorption of CO on Pt/Si02. The values of the estimated activation energies for CC dissociative adsorption, given in Table 2, are low. ... 

Another contemporary and noteworthy review article by Koper follows yet another concept. Koper first stresses the importance of the electric circuit by evaluating, in a rigorous way, the stability of electrochemical systems by frequency response methods. He then thoroughly discusses the dynamics of selected examples, including some semiconductor systems, which are not included in this chapter, with special emphasis on how they relate to the frequency response theory. 

A further use for the control engineer is that the linear analysis allows him to use frequency response methods for control system design. The fact that the system has been broken down into subsystems also means that any nonlinearities such as deadzone that are known to exist may be introduced easily into the block diagram layout and represented by describing functions. 

Frequency response methods have been found useful in both theoretical and experimental analysis of gas mixing in fluidized beds. Experiments in a fluidized-bed reactor related to mixing theory were made by Bamstone and Harriott 24). Testin and Stuart have measured diffusion coefficients in gas-solid adsorption studies 25). 

In Chapter 14 we define mathematically the sampling process, derive the z transforms of common functions (learn our German vocabulary), develop transfer functions in the z domain, and discuss stability. Design of digital controllers is studied in Chapter 15 using root locus and frequency response methods in the z plane. We use practically all the stability analysis and controller design techniques that we introduced in the Laplace and frequency domains, now applying them in the z domain for sampled-data systems. 

The first two sampled-data controller design methods use conventional root locus and frequency response methods, which are completely analogous to the techniques in continuous systems. Instead of looking at the s plane, however, we look at the z plane. The third sampled-data controller design method is similar to the direct synthesis method discussed in Chapter 9. 

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