Slow time scales

In sharp contrast to the large number of experimental and computer simulation studies reported in literature, there have been relatively few analytical or model dependent studies on the dynamics of protein hydration layer. A simple phenomenological model, proposed earlier by Nandi and Bagchi [4] explains the observed slow relaxation in the hydration layer in terms of a dynamic equilibrium between the bound and the free states of water molecules within the layer. The slow time scale is the inverse of the rate of bound to free transition. In this model, the transition between the free and bound states occurs by rotation. Recently Mukherjee and Bagchi [14] have numerically solved the space dependent reaction-diffusion model to obtain the probability distribution and the time dependent mean-square displacement (MSD). The model predicts a transition from sub-diffusive to super-diffusive translational behaviour, before it attains a diffusive nature in the long time. However, a microscopic theory of hydration layer dynamics is yet to be fully developed. 

Fig. 2.9 Two-dimensional bifurcation diagram outlining the main bifurcation structure for the /3-cell model in the (Vs, ks) parameter plane. Notice the squid-shaped black region with chaotic dynamics. Vs characterizes the voltage dependence of the gating variable for the calcium conductance, and k is a measure of the ratio of the fast and the slow time scales.

However, one of the interesting aspects of NMR structural results is that they often suggest no discernible solution structure for small systems, such as a tri- or tetra-peptides. This is due to the flexibility and structural variance displayed by these sample systems. However, the lack of interactions between parts of the molecule, which normally are detected via Nuclear Overhauser enhancements and refined into molecular structures during NMR structural determinations, should not be interpreted as a lack of a solution structure. It is the slow time scale of NMR, coupled with the rapidly interconverting conformations, which weakens these effects to the point where they can no longer be detected with certainty, and structural techniques which operate on a much faster time scale (e.g., UV-CD spectroscopy, or forms of vibrational spectroscopy) demonstrate that there is a preferred class of solution conformers even in small peptide systems. 

Here, x2 is the slow transient of X2 and x2 is the fast transient. Note that, for the corrected approximation in Equation (2.18) to converge rapidly to the slow approximative solution (2.15), the term x2 must decay as t —> oo to an 0(e) quantity. In the slow time scale t, this decay is fast, since... 

As mentioned before, obtaining an explicit variable separation for the system in Equation (2.36) requires a nonlinear coordinate transformation. The fact that k(x) = 0 in the slow time scale t and k(x) 0 in the fast time scale r indicates that the functions fcj(x) should be used in such a coordinate transformation as fast variables. Then, it can be shown (see, e.g., Kumar and Daoutidis 1999a) that a coordinate change of the form... 

The representation (2.74) of the slow dynamics provides yet another valuable insight while the temperatures of Bi and B2 exhibit a two-time-scale behavior, the total enthalpy of the system (captured by the variable Q is a true slow variable, evolving only in the slow time scale. 

Figure 2.8 The system temperatures exhibit both fast and slow transients (top), while the total enthalpy evolves only in the slow time scale (bottom).

The above time-scale decomposition provides a transparent framework for the selection of manipulated inputs that can be used for control in the two time scales. Specifically, it establishes that output variables y1 need to be controlled in the fast time scale, using the large flow rates u1, while the control of the variables ys is to be considered in the slow time scale, using the variables us. Moreover, the reduced-order approximate models for the fast (Equation (3.11)) and slow (the state-space realization of Equation (3.16)) dynamics can serve as a basis for the synthesis of well-conditioned nonlinear controllers in each time scale. 

Using the methods presented in Chapter 2, the above formulation can be used to derive a state-space realization of the slow dynamics of the type in Equation (2.48). The resulting low-dimensional model should subsequently form the basis for formulating and solving the control problems associated with the slow time scale, i.e., stabilization, output tracking, and disturbance rejection at the process level. 

Practical considerations in implementing the hierarchical control framework developed above concern the availability of manipulated inputs to address the control objectives in the slow time scale (it is possible that dim(us) < dim(ys)), as well as achieving a tighter coordination between the distributed and supervisory control layers. Both issues are effectively addressed by using a cascaded control configuration, which extends the choice of controlled variables in the slow time scale to include the setpoints y)p of the distributed controllers. 

It can be shown (Kumar and Daoutidis 1996, Contou-Carrere el al. 2004) that, under some mild assumptions (including Assumption 3.1), the models of the process systems under consideration can be transformed into regular DAE systems by introducing an additional set of appropriately defined differential variables, i.e., by constructing a dynamic extension of the process model. Within this framework, considering that a subset y C ygp of the setpoints of the fast controllers are used as manipulated inputs in the slow time scale, the dynamic extension... 

The supervisory control objective to be addressed in the slow time scale is the regulation of the product purity. Additionally, we must consider the control of the total material holdup Mt = Mr + Me + Mb, which is not affected by changes in the flow rates of the large internal material streams F, D, and V, as can easily be verified from the corresponding mass-balance equations. 

In order to address these objectives, we follow the procedure outlined in Section 3.4.3 to obtain a reduced-order model of the dynamics in the slow time scale. Specifically we consider the limit of an infinitely high recycle how rate... 

A potential choice of manipulated inputs to address the control objectives in the slow time scale is [ 3 Mrsp]t, i.e., the product flow rate from the column reboiler, and the setpoint for the reactor holdup used in the proportional feedback controller of Equation (3.35). This cascade control configuration is physically meaningful as well intuitively, the regulation of the product purity 23 is associated with the conversion and selectivity achieved by the reactor, which in turn are affected by the reactor residence time. 

Let us consider again the limit of an infinitely small purge number (e —> 0), this time in the newly defined slow time scale. This yields the algebraic equations... 

The model of the slow dynamics of the system consists therefore of a set of coupled DAEs of nontrivial index, since the variables z (that physically correspond to the net material flows of the system in the slow time scale) are implicitly fixed by the quasi-steady-state constraints, rather than explicitly specified in the dynamic model. Also, note that the DAE model (4.27) has a well-defined index only if the flow rates u1 which appear in the algebraic constraints are specified as functions of the state variables x. This is typically accomplished via a control law u (x). 

Turning now to the slow dynamics, we define the slow time scale r = te and consider the limit e —> 0, obtaining a description of the slow dynamics of the form (4.27) ... 

The control of the impurity levels in the process should be undertaken in the slow time scale, and any control strategy should account for the long time horizon that the respective variables evolve in. One could, for example, employ a description of the slow dynamics (Equation (4.35)) for synthesizing a model-based controller. 

It is important to note that, in typical practical situations in which cost constraints play an important role, impurity-concentration measurements are available for only a few units (and, more often than not, just for a single unit). Thus, a model of the evolution of the total impurity inventory (such as those developed in the examples above, i.e., Equations (4.43) and (4.45)) is not well suited for controller design. Rather, an appropriate coordinate change of the type in Equation (4.33) should be used to obtain a model of the evolution of the measured concentration variable in the slow time scale. An example of this approach is presented in the case study following this section. 

Figure 4.7 shows the evolution of the total impurity holdup for the same simulation note that this variable exhibits dynamics only in the slow time scale, which is - again - consistent with our previous findings. 

Drawing again on our theoretical analysis, the control of y tr should be addressed in the slow time scale using the purge stream as a manipulated input. 

The concentration of impurities (present in the feed) in the process evolves over a very slow horizon (days or, possibly, weeks). Moreover, the presence of impurities in the feed stream, together with significant material recycling, can lead to the accumulation of impurities in the recycle loop, with detrimental effects on the operation of the process and on its economics (Baldea et al. 2006). Therefore, as was shown in Chapter 4, the control of the impurity levels in the process is an important operational objective, and, according to the analysis presented above, it should be addressed in the slow time scale, using the flow rate of the purge stream, up, as a manipulated input. 

The terms lime, o(l/ )r 1(x, 0)lo1 (which, being based on Equation (6.7), represent differences between large internal energy flows), become indeterminate in the slow time scale. These terms do, however, remain finite, and constitute an additional set of algebraic (rather than differential) variables in the model of the slow dynamics. On defining z = lime, o(l/ )f (x, 0)u> the reduced-order representation of the slow dynamics becomes... 

Remark 6.3. The vector function <5(x, 0) can be arbitrarily chosen (as long as the invertibility of T(x, 0) is preserved), which allows us to describe the slow component of the energy dynamics in terms of the enthalpy/temperature of any one of the units. Furthermore, <5(x, 0) may be chosen in such a way that (0d/50)B(x, 0) = 0. In this case, the model (6.18) will be independent of z and the corresponding Q represents a true slow variable in the system (whereas the original state variables evolve both in the fast and in the slow time scales). For example, on choosing <5(x, 0) as the sum of all the unit enthalpies (Equation (6.13)), it can be shown that indeed (88/89)B(x, 0) = 0. Thus, the total enthalpy of the process evolves only over a slow time scale. 

Subsequently, we considered the control implications of our findings, and showed that control objectives related to the energy dynamics of the individual units (e.g., temperature control) should typically be addressed in the fast time scale. On the other hand, control objectives related to the energy dynamic at the process level (such as managing energy use) should be addressed in the slow time scale. These concepts were illustrated through several examples and a simulation case study. 

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