Analysis of Multivariable Systems

In Chap. 15 we reviewed a tittle matrix mathematics and notation. Now that the tools are available, we will apply them in this chapter to the analysis of multivariable processes. Our primary concern is with closedloop systems. Given a process with its matrix of openloop transfer functions, we want to be able to see the effects of using various feedback controllers. Therefore we must be able to find out if the entire closedloop multivariable system is stable. And if it is stable, we want to know how stable it is. The last question considers the robustness of the controller, i.e., the tolerance of the controller to changes in parameters. If the system becomes unstable for small changes in process gains, time constants, or deadtimes, the controller is not robust. 

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Then we address these same questions in Chap. 3 for multivariable systems, with two or more intermediates. Now our approach takes inherent fluctuations fully into account and we find a state function (analogous to AG) that satisfies the stated requirements. We also present a deterministic analysis of multivariable systems in Chap. 4 and compare the approach and the results with the fluctuational analysis. In Chap. 5 we turn to the study of reaction-diffusion systems and the issue of relative stability of multiple stationary states. The same issue is addressed in Chap. 6 on the basis of fluctuations, and in Chap. 7 we present experiments on relative stability. 

The most important method for exploratory analysis of multivariate data is reduction of the dimensionality and graphical representation of the data. The mainly applied technique is the projection of the data points onto a suitable plane, spanned by the first two principal component vectors. This type of projection preserves (in mathematical terms) a maximum of information on the data structure. This method, which is essentially a rotation of the coordinate system, is also referred to as eigenvector-projection or Karhunen-Loeve- projection (ref. 8). 

Koliopoulos, T.C. and Koliopoulou, G. 2007a. The use of input-output system analysis for sustainable development of multivariable systems. In American Institute of Physics Conference Proceedings, Todorov, M. (eds.), Vol. 946, pp. 256-60. New York American Institute of Physics Publisher. 

Hulburt and Katz (HI7) developed a framework for the analysis of particulate systems with the population balance equation for a multivariate particle number density. This number density is defined over phase space which is characterized by a vector of the least number of independent coordinates attached to a particle distribution that allow complete description of the properties of the distribution. Phase space is composed of three external particle coordinates x and m internal particle coordinates Xj. The former (Xei, x 2, A es) refer to the spatial distribution of particles. The latter coordinate properties Ocu,Xa,. . , Xt ) give a quantitative description of the state of an individual particle, such as its mass, concentration, temperature, age, etc. In the case of a homogeneous dispersion such as in a well-mixed vessel the external coordinates are unnecessary whereas for a nonideal stirred vessel or tubular configuration they may be needed. Thus (x t)d represents the number of particles per unit volume of dispersion at time t in the incremental range x, x -I- d, where x represents both coordinate sets. The number density continuity equation in particle phase space is shown to be (HI 8, R6)... 

This chapter presents fundamental elements from the process control theory necessary in assessing the controllability of a design. It prepares also the computational tools used in Chapter 13 devoted to plantwide control. Prior knowledge is welcome. If the material is too difficult, the reader should go back to specialized undergraduate books. A concise presentation of the process control essentials viewed from process engineer s perspective has been recently published by Luyben Luyben (1998). In the field of feedback control of multivariable systems with emphasis on controllability analysis we recommend the book of Skogetsad and Postletwhite (1998). 

Brocket , R.W., Mesarovic, M.D. The reproducibility of multivariable systems. Journal of Mathematical Analysis and Applications, 11 548-563, 1965. 

Multivariate analysis The analysis of a system that contains several variables (i.e., signals from several receptors)... 

Briefly, PCA models the data in terms of the significant factors, or principal components, which describe the systematic variability of the data. PCA also describes the data in terms of residuals that represent the noise in the system. PLS may be described as a method for constructing predictive models from data sets with many collinear factors. Both have received considerable attention in the analysis of multivariate data. 

The HCA method, which uses any of a variety of multivariate distance calculations to identify similar spectra, has found little use in Raman spectroscopy, although it could be of use in the growing analysis of complicated systems in which a large heterogeneous sample set is being analyzed. A study of spruce needles by Krizova et al. [54] and an investigation of cancerous skin lesions by Fendel and Schrader [55] are two examples showing the modest power of HCA. 

No quantitative applications of neural networks to quantitative analysis of linear systems have been reported where the results have been significantly better than those obtained by PLS, as would be expected since PLS (and indeed, the other multivariate methods described in this chapter) have been designed explicitly to handle linear systems. Analogous techniques, such as polynomial PLS or spline PLS, have been designed for nonlinear systems. It is interesting that with nonlinear data, neural networks have been shown to outperform any of the linear or nonlinear PLS techniques [27]. 

Correlations between structure and mass spectra were established on the basis of multivariate analysis of the spectra, database searching, or the development of knowledge-based systems, some including explicit management of chemical reactions. 

An analysis of variance can be extended to systems involving more than a single variable. For example, a two-way ANOVA can be used in a collaborative study to determine the importance to an analytical method of both the analyst and the instrumentation used. The treatment of multivariable ANOVA is beyond the scope of this text, but is covered in several of the texts listed as suggested readings at the end of the chapter. 

Scale- Up of Electrochemical Reactors. The intermediate scale of the pilot plant is frequendy used in the scale-up of an electrochemical reactor or process to full scale. Dimensional analysis (qv) has been used in chemical engineering scale-up to simplify and generalize a multivariant system, and may be appHed to electrochemical systems, but has shown limitations. It is best used in conjunction with mathematical models. Scale-up often involves seeking a few critical parameters. Eor electrochemical cells, these parameters are generally current distribution and cell resistance. The characteristics of electrolytic process scale-up have been described (63—65). 

On the other hand, when latent variables instead of the original variables are used in inverse calibration then powerful methods of multivariate calibration arise which are frequently used in multispecies analysis and single species analysis in multispecies systems. These so-called soft modeling methods are based, like the P-matrix, on the inverse calibration model by which the analytical values are regressed on the spectral data ... 

Arrays were introduced in the mid-eighties as a method to counteract the cross-selectivity of gas sensors. Their use has since become a common practice in sensor applications [1], The great advantage of this technique is that once arrays are matched with proper multivariate data analysis, the use of non-selective sensors for practical applications becomes possible. Again in the eighties, Persaud and Dodds argued that such arrays has a very close connection with mammalian olfaction systems. This conjecture opened the way to the advent of electronic noses [2], a popular name for chemical sensor arrays used for qualitative analysis of complex samples. 

Despite the broad definition of chemometrics, the most important part of it is the application of multivariate data analysis to chemistry-relevant data. Chemistry deals with compounds, their properties, and their transformations into other compounds. Major tasks of chemists are the analysis of complex mixtures, the synthesis of compounds with desired properties, and the construction and operation of chemical technological plants. However, chemical/physical systems of practical interest are often very complicated and cannot be described sufficiently by theory. Actually, a typical chemometrics approach is not based on first principles—that means scientific laws and mles of nature—but is data driven. Multivariate statistical data analysis is a powerful tool for analyzing and structuring data sets that have been obtained from such systems, and for making empirical mathematical models that are for instance capable to predict the values of important properties not directly measurable (Figure 1.1). 

In recent years a number of commercial programs have been developed that produce root locus plots (and provide other types of analysis tools). These software packages can speed up controller design. Some of the most popular include CC, CONSYD, and MATRIX-X. We will refer to these packages again later in the book since they are also useful in the frequency and z domains, as well as for handling multivariable systems. /... 

This chapter is organized in the following way. First, the general model of the CSTR process, based on first principles, is derived. A linearized approximate model of the reactor around the equilibrium points is then obtained. The analysis of this model will provide some hints about the appropriate control structures. Decentralized control as well as multivariable (MIMO) control systems can be designed according to the requirements. 

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