Identification of linear systems

Transforming (5.65) to a system of m first - order differential equations it can be solved numerically, and fitting models of different order we can also estimate its parameters. There exists, however, a special family of methods based on the use of the convolution integral 

Consider now the problem of identifying a linear system in the form of its weighting function h(t), using the relationship (5.66). This problem is called deconvolution. Discrete Fourier transformation offers a standard technique performing numerical deconvolution as mentioned in Section 4.3.3. It 

As shown in Fig. 5.4, the area under the curve of the input remains unchanged in this approximation. 

Similar stepwise approximation of the weighting function h(t) with the discrete values h, . .., hy, and replacement of ytt ) by the observed values 

While the point - area method is very convenient in terms of computational efforts, it has a serious drawback. The matrix of the linear system (5.6B) is inherently ill - conditioned (ref. 25), and the result is very sensitive to the errors in the observations. 

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The data are usually given a priori. Even when experimentation is tolerated, there exist very few cases where it is known how to construct good experiments to produce useful knowledge suitable for particular model forms. Such a case is the identification of linear systems and the related issue on data quality is known under the term persistency of excitation (Ljung, 1987). 

B.R. Hunt, Biased estimation for nonparametric identification of linear systems, Math. Biosciences, 10 (1971) 215-237. 

Schoukens, J. and Pintelon, R. Identification of Linear Systems A Practical Guideline for Accurate Modeling. Pergamon Press, London, 1991. 

Van Overschee P., DeMoor B., 1996. Subspace Identification of Linear Systems Theory, Implementation, Applications, Kluwer Academic Publishers. 

Although many techniques have been proposed, we limit our discussion to the methods that are widely used in the chemical and petroleum industries. Only the identification of linear transfer function models is discussed. We illustrate the use of the MATLAB System Identification Toolbox. 

Basu B, Nagarajaiah S, Oiakraborty A (2008) Online identification of linear time-varying stiffness of structural systems by wavelet analysis. Struct Health Monit IntJ7(l) 21-36... 

Figure 3-1 shows calculated plots of Eq. (3-16) for hypothetical systems in which kilk2 has the values 1 and 5. It is evident from the example in which k, = 5 2 that the curvature persists well into the reaction and that unambiguous identification of the terminal linear portion may be difficult. The long extrapolation to find eg is also uncertain. The accuracy of this procedure depends upon the ratios kilk2 and... 

The PBL reactor considered in the present study is a typical batch process and the open-loop test is inadequate to identify the process. We employed a closed-loop subspace identification method. This method identifies the linear state-space model using high order ARX model. To apply the linear system identification method to the PBL reactor, we first divide a single batch into several sections according to the injection time of initiators, changes of the reactant temperature and changes of the setpoint profile, etc. Each section is assumed to be linear. The initial state values for each section should be computed in advance. The linear state models obtained for each section were evaluated through numerical simulations. 

Linear absorption measurements can therefore give the first indication of possible alloy formation. Nevertheless, in systems containing transition metals (Pd-Ag, Co-Ni,. ..) such a simple technique is no longer effective as interband transitions completely mask the SPR peak, resulting in a structurless absorption, which hinders any unambiguous identification of the alloy. In such cases, one has to rely on structural techniques like TEM (selected-area electron diffraction, SAED and energy-dispersive X-ray spectroscopy, EDS) or EXAFS (extended X-ray absorption fine structure) to establish alloy formation. 

Cyclic oligomers of PA6 can be separated by PC [385,386] also PET and linear PET oligomers were separated by this technique [387]. Similarly, PC has been used for the determination of PEGs, but was limited by its insensitivity and low repeatability [388]. PC was also used in the determination of Cd, Pb and Zn salts of fatty acids [389]. ATR-IR has been used to identify the plasticisers DEHP and TEHTM separated by PC [390]. Although this combined method is inferior in sensitivity and resolution to modem hyphenated separation systems it is simple, cheap and suitable for routine analysis of components like polymer additives. However, the applicability of ATR-IR for in situ identification of components separated by PC is severely restricted by background interference. 

With the identification of the TS trajectory, we have taken the crucial step that enables us to carry over the constructions of the geometric TST into time-dependent settings. We now have at our disposal an invariant object that is analogous to the fixed point in an autonomous system in that it never leaves the barrier region. However, although this dynamical boundedness is characteristic of the saddle point and the NHIMs, what makes them important for TST are the invariant manifolds that are attached to them. It remains to be shown that the TS trajectory can take over their role in this respect. In doing so, we follow the two main steps of time-independent TST first describe the dynamics in the linear approximation, then verify that important features remain qualitatively intact in the full nonlinear system. 

The first case study consists of a section of an olefin plant located at the Orica Botany Site in Sydney, Australia. In this example, all the theoretical results discussed in Chapters 4,5,6, and 7 for linear systems are fully exploited for variable classification, system decomposition, and data reconciliation, as well as gross error detection and identification. 

We have shown thus far how the system of equations representing a process can be related to a linear diagraph and its associated Boolean adjacency matrix. In the Section IV, we show how the location of the maximal loops in this adjacency matrix leads to identification of the subsystems of equations that must be solved simultaneously. 

System Identification Techniques. In system identification, the (nonlinear) resi pnses of the outputs of a system to the input signals are approximated by a linear model. The parameters in this linear model are determined by minimizing a criterion function that is based on some difference between the input-output data and the responses predictedv by the model. Several model structures can be chosen and depending on this structure, different criteria can be used (l ,IX) System identification is mainly used as a technique to determine models from measured input-output data of processes, but can also be used to determine compact models for complex physical models The input-output data is then obtained from simulations of the physical model. 

The retention time tj is a non-linear function of the ring size n and thus allows the identification of new species of the homologous series S by interpolation. Furthermore, the logarithm of the capacity factor k = (t — (with = death time of the chromatographic system) is a linear function of the ring size, n, although two such functions are obtained see Fig. 7... 

For continuous process systems, empirical models are used most often for control system development and implementation. Model predictive control strategies often make use of linear input-output models, developed through empirical identification steps conducted on the actual plant. Linear input-output models are obtained from a fit to input-output data from this plant. For batch processes such as autoclave curing, however, the time-dependent nature of these processes—and the extreme state variations that occur during them—prevent use of these models. Hence, one must use a nonlinear process model, obtained through a nonlinear regression technique for fitting data from many batch runs. 

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