Multivariable process

Bristol, E.H., 1966. On a new measure of interactions for multivariable process control, IEEE Transactions of Automatic Control, AC-11, 133. 

Multivariate process monitoring and diagnosis of a full-scale industrial wastewater treatment plant... 

Multiway and particularly three-way analysis of data has become an important subject in chemometrics. This is the result of the development of hyphenated detection methods (such as in combined chromatography-spectrometry) and yields three-way data structures the ways of which are defined by samples, retention times and wavelengths. In multivariate process analysis, three-way data are obtained from various batches, quality measures and times of observation [55]. In image analysis, the three modes are formed by the horizontal and vertical coordinates of the pixels within a frame and the successive frames that have been recorded. In this rapidly developing field one already finds an extensive body of literature and only a brief outline can be given here. For a more comprehensive reading and a discussion of practical applications we refer to the reviews by Geladi [56], Smilde [57] and Henrion [58]. 

MacGregor, J. F.. and Kourti, T., Statistical process control of multivariate processes, Coni. Eng. Prac. 3, 404-414 (1995). 

A data matrix is the structure most commonly found in environmental monitoring studies. In these data tables or matrices, the different analyzed samples are placed in the rows of the data matrix, and the measured variables (chemical compound concentrations, physicochemical parameters, etc.) are placed in the columns of the data matrix. The statistical techniques necessary for the multivariate processing of these data are grouped in a table or matrix, or use tools, formulations, and notations of the lineal algebra. 

However, these statements are generalizations, and it is not necessarily true to say that all biotransformations will be greener than the chemical alternative. Therefore, it is important to analyse each comparison objectively on a case-by-case basis using a multivariate process to take into account the complexity of the analysis. Designing greener processes involves, for example ... 

Chapter 16 covers the analysis of multivariable processes stability, robustness, performance. Chapter 17 presents a practical procedure for designing conventional multiloop SISO controllers (the diagonal control structure) and briefly discusses some of the full-blown multivariable controller structures that have been developed in recent years. 

We will use this matrix of transfer function representation extensively in the rest of our work with multivariable processes. 

Table 15.1 gives some simple FORTRAN subroutines for handling matrices of complex numbers. These will be used in later chapters in programs to examine the stability, performance, and robustness of multivariable processes. 

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. 

The Nyquist stability criterion that we developed in Chap. 13 can be directly applied to multivariable processes. As you should recall, the procedure is based on a complex variable theorem which says that the dilTerence between the number of zeros and poles that a function has inside a dosed contour can be found by plotting the function and looking at the number of times it endrdes the origin. 

Unfortunately much of Ais interaction analysis work has clouded the issue of how to design an effective control system for a multivariable process. In most process control applications the problem is not setpoint responses but load responses. We want a sy stem that holds the process at the desired values in the face of load disturbances. Interaction is therefore not necessarily bad, and in fact in some systems it helps in rejecting the effects of load disturbances. Niederlinski [AIChE J 1971, Vol 17, p. 1261) showed in an early paper that the use of decouplers made the load rejection worse. 

A 2 x 2 multivariable process has the following openloop plant transfer function matrix ... 

The closedloop relationships for a multivariable process were derived in Chap. 15 [Eq. (15.64)]... 

As a result, there has been a lot of research activity in multivariable control, both in academia and in industry. Some practical, useful tools have been developed to design control systems for these multivariable processes. The second edition includes a fairly comprehensive discussion of what 1 feel are the useful techniques for controlling multivariable processes. 

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