Multivariable systems

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). 

To get good control of the entire PRT, not only should the expander be controlled, but a completely integrated control system for this application should be designed. Most conventional control systems consist of individual control loops that only consider their specific tasks. The PRT—from a control perspective—is a multivariable system that requires integration between the different control loops. Further, some of the disturbances on the PRT are so fast that closed-loop control is too slow to keep the train under control. 

Equation (8.76) is called the matrix vector difference equation and can be used for the recursive discrete-time simulation of multivariable systems. 

The concepts of controllability and observability were introduced by Kalman (1960) and play an important role in the control of multivariable systems. 

A limitation of the ANFIS teehnique is that it eannot be employed on multivariable systems. The Co-aetive ANFIS (CANFIS) developed by Craven (1999) extends the ANFIS arehiteeture to provide a flexible multivariable eontrol environment. This was employed to eontrol the yaw and roll ehannels of an Autonomous Underwater Vehiele (AUV) simultaneously. 

This tutorial looks at how MATLAB commands are used to convert transfer functions into state-space vector matrix representation, and back again. The discrete-time response of a multivariable system is undertaken. Also the controllability and observability of multivariable systems is considered, together with pole placement design techniques for both controllers and observers. The problems in Chapter 8 are used as design examples. 

The for-end loop in examp88.m that employs equation (8.76), while appearing very simple, is in faet very powerful sinee it ean be used to simulate the time response of any size of multivariable system to any number and manner of inputs. If A and B are time-varying, then A(r) and B(r) should be ealeulated eaeh time around the loop. The author has used this teehnique to simulate the time response of a 14 state-variable, 6 input time-varying system. Example 8.10 shows the ease in whieh the eontrollability and observability matriees M and N ean be ealeulated using c t r b and ob s v and their rank eheeked. 

Gegov, A., Distributed Fuzzy Control of Multivariable Systems, Kluwer, 1996. 

Referring to the discussion of the fundamental concepts regarding half cells and the Nernst equation in Chapter 5 (Section 5.3.1) it is possible to briefly summarize the similarities and differences of these two sets of systems. It is important to recognize the ways in which they are different when considering the behavior of complex multivariate systems such as the oceans and clouds, or a lake-river system. 

S.C. Rutan, E. Bouveresse, K.N. Andrew, P.J. Worsfold and D.L. Massart, Correction for drift in multivariate systems using the Kalman filter. Chemom. Intell. Lab. Syst., 35 (1996) 199-211. 

A more complicated situation in process regulation occurs when among various chemical properties there is not one that can explicitly be indicated as a key test, so that more than one sensor (1,2,..., n) has to be used, in which case one refers to "multivariable systems in process control 6. 

Example calculations for a bivariate system can be found in Marchisio and Fox (2006) and Zucca et al. (2006). We should note that for multivariate systems the choice of the moments used to compute the source terms is more problematic than in the univariate case. For example, in the bivariate case a total of 3 M moments must be chosen to determine am, bm and cm. In most applications, acceptable accuracy can be obtained with 3[Pg.283]

Oscillations have been observed in chemical as well as electrochemical systems [Frl, Fi3, Wol]. Such oscillatory phenomena usually originate from a multivariable system with extremely nonlinear kinetic relationships and complicated coupling mechanisms [Fr4], Current oscillations at silicon electrodes under potentio-static conditions in HF were already reported in one of the first electrochemical studies of silicon electrodes [Tul] and ascribed to the presence of a thin anodic silicon oxide film. In contrast to the case of anodic oxidation in HF-free electrolytes where the oscillations become damped after a few periods, the oscillations in aqueous HF can be stable over hours. Several groups have studied this phenomenon since this early work, and a common understanding of its basic origin has emerged, but details of the oscillation process are still controversial. 

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. /... 

Figure 15.2 shows a closedloop multivariable system. The process is always described by the equation... 

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. 

Interaction among control loops in a multivariable system has been the subject of much research over the last 20 years. Various types of decouplers were explored to separate the loops. Rosenbrock presented the inverse Nyquist array (INA) to quantify the amount of interaction. Bristol, Shinskey, and McAvoy developed the relative gain array (RGA) as an index of loop interaction... 

An INA plot for an Nth-order multivariable system consists of N curves, one for each of the diagonal elements of the matrix that is the inverse of the... 

In multivariable systems the question of robustness is very important. One method developed by Doyle and Stein (IEEE Trans., 1981, Vol. AC-26, p. 4) is quite usefiil and easy to use. It has the added advantage that it is quite similar to the maximuni closedloop log modulus criterion used in SISO systems. 

B. MATRIX MULTIVARIABLE SYSTEMS. For multivariable systems, the Doyle-Stein criterion for robustness is very similar to the reciprocal plot discussed above. The minimum singular value of the matrix given in Eq. (16.43) is plotted as a function of frequency (a. This gives a measure of the robustness of a closed-... 

The Doyle-Stein criterion discussed in the previous section gives us a simple way to evaluate quantitatively the robustness of a multivariable system. However, it assumes random variability in all the parameters of the system. 

In theory, the internal model control methods discussed for SISO systems in Chap. 11 can be extended to multivariable systems (see the paper by Garcia and Morari in lEC Process Design and Development, Vol. 24, 1985, p. 472). 

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