Linear frequency response, methods

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. 

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. 

Species formed from acetylene (Ay) adsorbed in zeolite Y, mordenite, beta and ZSM-5 have been studied by IR spectroscopy. The dynamics of Ay physisorption has been characterized by the frequency response method (FR). The rate of micropore diffusion governed the transport in Na-mordenite, while sorption was the rate limiting process step for all the H-zeolites. The equilibrium constants (Ka) of Ay sorption have been determined applying the Langmuir rate equation to describe the pressure dependence of the sorption time constants. The -octane hydroconversion activity of Pt/H-zeolites was found to increase linearly with the Ka of Ay sorption on the H-zeoIites. 

If the process model is nonlinear, then advanced stability theory can be used (Khalil, 2001), or an approximate stability analysis can be performed based on a linearized transfer function model. If the transfer function model includes time delays, then an exact stability analysis can be performed using root-finding or, preferably, the frequency response methods of Chapter 14. A less desirable alternative is to approximate the terms and apply the Routh stability criterion. 

Cycled Feed. The qualitative interpretation of responses to steps and pulses is often possible, but the quantitative exploitation of the data requires the numerical integration of nonlinear differential equations incorporated into a program for the search for the best parameters. A sinusoidal variation of a feed component concentration around a steady state value can be analyzed by the well developed methods of linear analysis if the relative amplitudes of the responses are under about 0.1. The application of these ideas to a modulated molecular beam was developed by Jones et al. ( 7) in 1972. A number of simple sequences of linear steps produces frequency responses shown in Fig. 7 (7). Here e is the ratio of product to reactant amplitude, n is the sticking probability, w is the forcing frequency, and k is the desorption rate constant for the product. For the series process k- is the rate constant of the surface reaction, and for the branched process P is the fraction reacting through path 1 and desorbing with a rate constant k. This method has recently been applied to the decomposition of hydrazine on Ir(lll) by Merrill and Sawin (35). 

It can be shown(18) that this method may be applied to any system described by a linear differential equation or to a distance-velocity lag in order to obtain the relevant frequency response characteristics. 

R. Cammi and J. Tomasi, Nonequilibrium solvation theory for the polarizable continuum model - a new formulation at the SCF level with application to the case of the frequency-dependent linear electric-response function, Int. J. Quantum Chem., (1995) 465-74 B. Mennucci, R. Cammi and J. Tomasi, Excited states and solvatochromic shifts within a nonequilibrium solvation approach A new formulation of the integral equation formalism method at the self-consistent field, configuration interaction, and multiconfiguration self-consistent field level, J. Chem. Phys., 109 (1998) 2798-807 R. Cammi, L. Frediani, B. Mennucci, J. Tomasi, K. Ruud and K. V. Mikkelsen, A second-order, quadratically... 

Regarding adsorption and diffusion without reaction, Jordi and Do (49) simulated the expected results for the frequency response by completely numerical methods, with no need for linearization. In a later study, they used a linearized model coupled with analytic solutions for the diffusion inside the particles, which also took into account transport in both macropores and micropores (50). The mathematical details are clearly presented in these papers. 

The pore diameters of MFI-type zeolites are comparable to the size of many commercially important molecules, such as aromatics or linear or branched hydrocarbons [1]. Thus, the study of the difiusion of reactive molecules in the channel system of zeolite catalysts is of considerable interest for the understanding of the catalyst performance. A variety of methods has been developed and applied to the measurement of diffii-sion coefficients, amongst others gravimetric techniques [2], neutron scattering [3], NMR [4] and Frequency Response [5]. The FTIR technique offers the possibility to study sorption and sorption kinetics under conditions close to those of catalytic experiments. By the use of an IR microscope, single crystals have become accessible to the FTIR technique. 

The various experimental methods of linear viscoelasticity are summarized in Table 7.1. All information for linear viscoelastic response can, in principle, be obtained from each method. The oscillatory methods are particularly useful because they directly probe the response of the system on the time scale of the imposed frequency of oscillation 1 juj. Commercial rheometers can accomplish this with either applied stress or applied strain,... 

With nonlinear systems, however, all simplicity disappears. No general methods of solving even the simplest, nonlinear differential equations are known. Frequency response characterization is useless since sinusoidal forcing will not produce sinusoidal response. The only recourse other than arbitrary linearization of the equations is to utilize... 

In evaluating control systems for microrotorcrafts, Mettler [8] found that frequency domain identification served as a better method than time domain identification. This is because in frequency domain identification, the output measurement noise does not affect the results, it is possible to focus on a precise frequency range (which minimizes the disparity between the modes of motion), and frequency responses can completely describe the system s linear dynamics. Mettler also determined that the rigid body equations of motion needed to be expanded through use of the hybrid formulation to generate a more accurate control system. This method models the rotor motion using a tip-path plane model and expresses the rotor forces and moments in terms of the rotor states. The rotor and fuselage motions are then dynamically coupled. 

Fig. 5.5. (top) Example of frequency response functions of a resonant and a so-called broadband transducer. The frequency response function was measured using a laser-vibrometer (middle) and a face-to-face method (bottom). Amplitudes are on a linear scale. 

An interesting alternative approach is the direct calculation by polarization propagator, or linear response, methods.The poles of the propagator yield the transition frequencies, the residues yield the corresponding transition moments, and the propagator itself determines linear response properties such as the frequency-dependent (or dynamic) polarizability. 

The calculation of phonon frequencies of the crystalline structure is one of the fundamental subjects when considering the phase stability, phase transitions, and thermodynamics of crystalline materials. The approaches of ab-initio calculations fall into two classes the linear response method [670] and the direct method, see [671] and references therein. 

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