Model system frequency response

To model the system frequency response, we rewrite the PDEs, represented by equation (8.43), by replacing u(x,t) with u<,(x)+u (x,f), the steady state solution plus a small time varying perturbation u. The steady state and the perturbation were set by the input current density j t)= Jo + / (0- Note we could have perturbed the other inputs and Xq, but this is not experimentally practical. We substitute u +u into equation (8.43) and express the c matrix as C1U+C2. This is to cover the nonlinear situation in the migration terms containing y j d /dx and y dX/dx in equation (8.45). 

An important difference between analysis of stability in the. v-plane and stability in the frequency domain is that, in the former, system models in the form of transfer functions need to be known. In the latter, however, either models or a set of input-output measured open-loop frequency response data from an unknown system may be employed. 

There are a number of ways to obtain the frequency response of a process. Experimental methods, discussed in Chap. 14, are used when a mathematical model of tbe system is not available. If equations can be developed that adequately describe the system, the frequency response can be obtained directly from the system transfer function. 

Instead of converting the step or pulse responses of a system into frequency response curves, it is fairly easy to use classical least-squares methods to solve for the best values of parameters of a model that fit the time-domain data. 

Sections 3.2.1—3.2.3 have referred specifically to the system illustrated in Fig. 6. However, the approach in these sections is quite general and can therefore be used in situations where the system transfer function G(s) is other than that given by eqn. (7). For the case of the ideal PFR responses, G(s) is exp(— st) and impulse, step and frequency responses are simply these respective input functions delayed by a length of time equal to r. The non-ideal transfer function models of Sect. 5 may be used to produce families of predicted responses which depend on chosen model parameters. 

Treffer s model system consists of signal power P (in watts) falling on the input aperture of the modulator (i.e., the spectrophotometer) that modulates the input power as a function of time by a factor M(t) such that 0 < M(t) < 1. The modulation function M(t) is not the modulation of the signal due to chopping but modulation of the signal due to scanning. The modulated signal falls on a detector with responsivity R (in volts per watt) (Kruse et al., 1962 Stewart, 1970) and flat frequency response. The idealized instantaneous... 

The results show the transient effects due to a 15 kW step decrease to the generator (from 60 to 45 kW), which occurred at time zero. Figure 8.13 compares the dynamics of the shaft speed and shows that both models are in good agreement with the experimental data with regards to frequency. This is not surprising since the system volume will have the primary influence on the frequency response, and the mod-... 

Applying harmonic filters requires careful consideration. Series-tuned filters appear to be of low impedance to harmonic currents but they also form a parallel resonance circuit with the source impedance. In some instances, a situation can be created that is worse than the condition being corrected. It is imperative that computer simulations of the entire power system be performed prior to applying harmonic filters. As a first step in the computer simulation, the power system is modeled to indicate the locations of the harmonic sources, then hypothetical harmonic filters are placed in the model and the response of the power system to the filter is examined. If unacceptable results are obtained, the location and values of the filter parameters are changed until the results are satisfactory. When applying harmonic filters, the units are almost never tuned to the exact harmonic frequency. For example, the 5th harmonic frequency may be designed for resonance at the 4.7th harmonic frequency. 

The basic features of ET energetics are summarized here for the case of an ET system (solute) linearly coupled to a bath (nuclear modes of the solute and medium) [11,30]. We further assume that the individual modes of the bath (whether localized or extended collective modes) are separable, harmonic, and classical (i.e., hv < kBT for each mode, where v is the harmonic frequency and kB is the Boltzmann constant). Consistent with the overall linear model, the frequencies are taken as the same for initial and final ET states. According to the FC control discussed above, the nuclear modes are frozen on the timescale of the actual ET event, while the medium electrons respond instantaneously (further aspects of this response are dealt with in Section 3.5.4, Reaction Field Hamiltonian). The energetics introduced below correspond to free energies. Solvation free energies may have entropic contributions, as discussed elsewhere [19], Before turning to the DC representation of solvent energetics, we first display the somewhat more transparent expressions for a discrete set of modes. 

Systems approach borrowed from the optimization and control communities can be used to achieve various other tasks of interest in multiscale simulation. For example, Hurst and Wen (2005) have recently considered shear viscosity as a scalar input/output map from shear stress to shear strain rate, and estimated the viscosity from the frequency response of the system by performing short, non-equilibrium MD. Multiscale model reduction, along with optimal control and design strategies, offers substantial promise for engineering systems. Intensive work on this topic is therefore expected in the near future. 

Bode diagram, 330-31, 334-37 frequency response, 323-24 interacting capacities, 197-200 noninteracting capacities, 194-96 pulse transfer function, 619 Multiple-input multiple-output system, 20 discrete-time model, 586 discrete transfer function, 612 input-output model, 83-85, 163-68 linearization, 121-26 transfer-function matrix, 164, 166 Multiple loop control systems, 394-409 Multiplexer, 560, 564 Multivariable control systems, 461-62 alternative configurations, 467-84 decoupling of loops, 503-8 design questions, 461-62 interaction of loops, 487-94 selection of loops, 494-503 Multivariable process (see Multiple-input multiple-output system)... 

There are several approaches that can be used to tune PID controllers, including model-based correlations, response specifications, and frequency response (Smith and Corripio 1985 Stephanopoulos 1984). An approach that has received much attention recently is model-based controller design. Model-based control requires a dynamic model of the process the dynamic model can be empirical, such as the popular first-order plus time delay model, or it can be a physical model. The selection of the controller parameters Kc, ti, to) is based on optimizing the dynamic performance of the system while maintaining closed-loop stability. 

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