Multi-Variable Systems

We start with experiments on a multi-variable system, the bromate oxidation of ferroin, [1], also called the minimum bromate oscillator. The bistability and chemical oscillations of this system were characterized in [2]. 

The goal of these experiments is the measurement of the front propagation of one of the two stable stationary states into the other, see Chap. 5 particularly Sects. 5.1.3 and 5.1.4 and Figs. 5.2 5.7. From such measurements we can determine equistabUity conditions for the two stationary states where the front propagation velocity is zero. We thus obtain kinetic and thermodynamic conditions for the coexistence of the two stationary states. 

Other suggestions have been made concerning the measurement of relative stability. One such suggestion [3] was based on the connections of two continuous stirred tank reactors, CSTRs, each filled with one or the other stable stationary state the final state of both CSTR is predicted to be the more stable stationary state. However, the final state has been shown theoretically [4] and experimentally [5,6] to depend also on the strength and manner of mixing of the CSTRs, and therefore is not a useful, direct measure of relative stability. 

We assume that the reaction mechanism of this reaction given by Noyes et al. (NFT) [7] is adequate, although recognized to be perhaps over-simplified  

We measured bistability (1) for this reaction and the results are shown in Fig. 7.1. 

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The DMC discussed in this chapter is for a SISO system. We will say more about DMC in Chap. 17 since this methodology is fairly easily extended to multi-variable systems, which is where its real potential usefulness occurs. 

One final comment should be made about model-based control before we leave the subject. These model-based controllers depend quite strongly on the validity of the model. If we have a poor model or if the plant parameters change, the performance of a model-based controller is usually seriously affected. Model-based controllers are less robust than the more conventional PI controllers. This lack of robustness can be a problem in the single-input-single-output (SISO) loops that we have been examining. It is an even more serious problem in multi-variable systems, as we will find out in Chaps. 16 and 17. 

As shown in the above works, an optimal feedback/feedforward controller can be derived as an analytical function of the numerator and denominator polynomials of Gp(B) and Gn(B). No iteration or integration is required to generate the feedback law, as a consequence of the one step ahead criterion. Shinnar and Palmor (52) have also clearly demonstrated how dead time compensation (discrete time Smith predictor) arises naturally out of the minimum variance controller. These minimum variance techniques can also be extended to multi-variable systems, as shown by MacGregor (51). 

The reader will have noticed that we have discussed two difTerent techniques for the investigation of multi-variable systems, namely multiple correlation and the factorial experiment. 

The Israeli oil shales may be considered as a multi-variabled system, in which the main components influencing their quality are organic matter, carbonate, clay minerals and apatite. As the percentage of these components varies over the vertical section, depth also plays a significant role whenever a quality assessment of the shale is made. Compositional variations within the organic matter are responsible for changes in the relative calorific value and retorted oil yield, while fluidized bed combustion is affected by the inorganic composition. 

For a one-dimensional system, the quantity P (x, Z) (x, t) is the probability density for finding the system at position x at time t. In three dimensions, the quantity P (r, Z)T(r, t) is the probability density for finding the system at point r at time t. For a multi-variable system, the product 72,... 

In a multi-variable system, entropy production a is given by ... 

In addition to the determination of the traditional nutrients, flow methods have been reported for a number of other components using either flow spectrophotometers or other types of detectors. Flow methods can be easily combined in multi-variable systems. Despite the occasionally low accuracy, they can provide useful additional information about the analysed samples. As an example, consider conventional water sampling in an estuary using a small vessel and sample bottles on a hydrographic wire. To confirm that a sample has really been taken at the depth indicated by the cable length and sampler position on the cable, it is necessary to measure the salinity. Instead of using sample water for separate salinity determinations with a salinometer (increasing the required sampler size), a small conductivity cell in the flow analyser would provide this information. 

It is a relatively straightforward task to obtain the steady state gain matrix K for a multi-variable system from process data (e.g., see Ljung [11]), from which the degree of interaction... 

In Chap. 2 we obtained a thermodynamic state function d>, (2.13), valid for single variable non-linear systems, and (2.6), valid for single variable linear systems. We shall extend the approach used there to multi-variable systems in Chap. 4 and use the results later for comparison with experiments on relative stability. However, the generalization of the results in Chap. 2 for multi-variable linear and non-linear systems, based on the use of deterministic kinetic equations, does not yield a thermodynamic state function. In order to obtain a thermodynamic state function for multi-variable systems we need to consider fluctuations, and now turn to this analysis [1]. 

Linear Multi-Variable Systems 25 the equation satisfied by S (X) with the Hamiltonian function (not operator)... 

We note here that (3.35) and (3.37) hold for non-linear multi-variable systems as well no assumption of a linear reaction mechanism was made in their derivation. 

We turn next to consideration of a non-linear multi-variable system, for example the model... 

Here the reference state (X°,T°) replaces the starred reference state of Chap. 2 (see (2.11)). The important point is that the action and the excess work in (3.40) are state functions for single and multi-variable systems. Both X° and Y° are functions of X and Y in general, but the integrand in (3.40) is an exact differential, because p is the gradient of the action, (3.16). For the starred reference state the excess work is a state function only for single variable systems. 

This equation serves as a necessary and sufficient criterion for the existence and stability of stationary states for non-autocatal3dic and auto-catalytic stationary states in multi-variable systems. 

So far we have considered only homogeneous reaction systems in which concentrations are functions of time only. Now we turn to inhomogeneous reaction systems in which concentrations are functions of time and space. There may be concentration gradients in space and therefore diffusion will occur. We shall formulate a thermodynamic and stochastic theory for such systems [1] first we analyze one-variable systems and then two- and multi-variable systems, with two or more stable stationary states, and then apply the theory to study relative stabihty of such multiple stable stationary states. The thermodynamic and stochastic theory of diffusion and other transport processes is given in Chap. 8. 

In Chap. 5 we discussed reaction diffusion systems, obtained necessary and sufficient conditions for the existence and stability of stationary states, derived criteria of relative stability of multiple stationary states, all on the basis of deterministic kinetic equations. We began this analysis in Chap. 2 for homogeneous one-variable systems, and followed it in Chap. 3 for homogeneous multi-variable systems, but now on the basis of consideration of fluctuations. In a parallel way, we now follow the discussion of the thermod3mamics of reaction diffusion equations with deterministic kinetic equations, Chap. 5, but now based on the master equation for consideration of fluctuations. 

For multi-variable systems this approach is more difficult the determination of the stochastic potential requires sufficient measurements to determine rate coefficients and then the numerical solution of the stationary form of the master equation. Details of this procedure are described in Appendix A of [1]. 

Chang, J., and C. C. Yu, The Relative Gain for Non-Square Multi-variable Systems, Chem. Engr. Set, 45,1309 (1990). 

We will not present the formal proof for the two-variable systems which can be found in the earlier works [7]. Instead we shall generalize the notions of activator and inhibitor to multi-variable systems and show that in at least the three variable case the same conditions for the Turing instability apply. 

The first experiment on the relative stability in a bistable multi-variable system is reported in [36]. The apparatus consists of two continuously stirred tank reactors (CSTR), and a different stable stationary state of a bistable bromate-ferroin reaction is established in each CSTR with same set of influx of reactant solutions into the reactors. The reaction solution from each tank is then pumped quickly into a laminar flow reactor (LFR) where the solutions... 

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