Linear time domain analysis

From the last example, we may see why the primary mathematical tools in modem control are based on linear system theories and time domain analysis. Part of the confusion in learning these more advanced techniques is that the umbilical cord to Laplace transform is not entirely severed, and we need to appreciate the link between the two approaches. On the bright side, if we can convert a state space model to transfer function form, we can still make use of classical control techniques. A couple of examples in Chapter 9 will illustrate how classical and state space techniques can work together. 

In the next chapter we take a quantitative look at the dynamics of these CSTR systems using primarily rigorous nonlinear dynamic simulations (time-domain analysis). However, some of the powerful linear Laplace and frequency-domain techniques will be used to gain insight into the dynamics of these systems. 

Time-domain plots must be used for all linear and reciprocating motion machinery. They are useful in the overall analysis of machine-trains to study changes in operating conditions. However, time-domain data are difficult to use. Because all the vibration data in this type of plot are added together to represent the total displacement at any given time, it is difficult to directly see the contribution of any particular vibration source. 

Let us first state a few important points about the application of Laplace transform in solving differential equations (Fig. 2.1). After we have formulated a model in terms of a linear or linearized differential equation, dy/dt = f(y), we can solve for y(t). Alternatively, we can transform the equation into an algebraic problem as represented by the function G(s) in the Laplace domain and solve for Y(s). The time domain solution y(t) can be obtained with an inverse transform, but we rarely do so in control analysis. 

This circuit is a bridge rectifier followed by a filter capacitor to produce a DC voltage with ripple at Vin. Connected to Vin is a linear regulator made from a Zener voltage reference and an NPN pass transistor. We will first run a Transient Analysis to see the operation of the circuit at room temperature (27°C). To set up a Transient Analysis, select PSpice and then New Simulation Profile from the Capture menus, enter a name for the profile and then click the Create button. By default the Time Domain (Transient) Analysis type is selected. Fill in the parameters as shown in the Time Domain dialog box below ... 

The original linear prediction and state-space methods are known in the nuclear magnetic resonance literature as LPSVD and Hankel singular value decomposition (HSVD), respectively, and many variants of them exist. Not only do these methods model the data, but also the fitted model parameters relate directly to actual physical parameters, thus making modelling and quantification a one-step process. The analysis is carried out in the time domain, although it is usually more convenient to display the results in the frequency domain by Fourier transformation of the fitted function. 

The analysis of outlet peaks is based on the model of processes in the column. Today the Kubi n - Kucera model [14,15], which accounts for all the above-mentioned processes, as long as they can be described by linear (differential) equations, is used nearly exclusively. Several possibilities exist for obtaining rate parameters of intracolumn processes (axial dispersion coefficient, external mass transfer coefficient, effective diffusion coefficient, adsorption/desorption rate or equilibrium constants) from the column response peaks. The moment approach in which moments of the outlet peaks are matched to theoretical expressions developed for the system of model (partial) differential equations is widespread because of its simplicity [16]. The today s availability of computers makes matching of column response peaks to model equations the preferred analysis method. Such matching can be performed in the Laplace- [17] or Fourier-domain [18], or, preferably in the time-domain [19,20]. 

PCA, principal component analysis NN, neural net LOA, linear discriminant analysis LOO, leavc-one-out HLSVD, filtering signals in the time domain ANN, artificial neural networks. 

Non-linear least-squares fitting by the Marquardt method [19,20] appears to be the most commonly used technique for hiexponential fluorescence decay analysis, at least for a time-domain measurement such as used here [21,22]. Fitting by this method requires evaluation of the derivatives of the model equation (Equation... 

However, a solution in the Laplace domain has been derived by Kucera [30] and Kubin [31]. The solution cannot be transformed back into the time domain, but from that solution, these authors have derived the expressions for the first five statistical moments (see Section 6.4.1). For a linear isotherm, this model has been studied extensively in the literature. The solution of an extension of this model, using a macro-micropore diffusion model with external film mass transfer resistance, has also been discussed [32]. All these studies use the Laplace domain solution and moment analysis. 

A similar zone folding also occurs at Cs/Pt(lll) and the phonon mode appears as a small dip in the Fourier spectrum in Figure 19.4. A detailed analysis of the time domain data by linear prediction singular value decomposition has been performed and a decomposition of the time-domain data to phonon modes and alkali-substrate stretching modes has been carried out. Coherent nuclear motions have been observed on substrates other than Pt. Figure 19.5a shows time-resolved SHG traces... 

While there many examples of multivariate analysis involving NMR time domain data,24-36 Yvi II be assumed in this review that the data matrix of interest is (or is derived from) the frequency domain data /y(otherwise specified. In most situations the NMR spectrum of the /th sample (different mixture, time, etc.) can be described as the linear combination of comp pure component spectrum, each with a concentration... 

PSOLA, which operates in the time domain. It separates the original speech into frames pitch-synchronously and performs modification ly overlapping and adding these frames onto a new set of epochs, created to match the synthesis specification. Residual-excited linear prediction performs LP analysis, but uses the whole residual in resynthesis rather than an impulse. The residual is modified in a manner very similar to that of PSOLA. 

MATLAB and SIMULINK are invaluable tools for the finequency- and time-domain calculations required for C R analysis. In this section, several examples are carried out using MATLAB, it being assumed that the reader is familiar with the MATLAB syntax. The reader is referred to Bequette (1998) for details of MATLAB usage in dynamical analysis and control, and to the multimedia CD-ROM that accompanies this text for sources of these and other useful MATLAB functions and scripts for C R analysis. In particular, the interactive C R Tutorial CRGUI can be used to test three example linear processes for controllability and resiliency and simulate their closed-loop response under single-loop PI control. 

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