Models unstable process

In the last chapter we used Laplace-domain techniques to study the dynamics and stability of simple closedloop control systems. In this chapter we want to apply these same methods to more complex systems cascade control, feedforward control, openloop unstable processes, and processes with inverse response. Finally we will discuss an alternative way to look at controller design that is called model-based control. 

This second-level modeling of the feedback mechanisms leads to nonlinear models for processes, which, under some experimental conditions, may exhibit chaotic behavior. The previous equation is termed bilinear because of the presence of the b [y (/,)] r (I,) term and it is the general formalism for models in biology, ecology, industrial applications, and socioeconomic processes [601]. Bilinear mathematical models are useful to real-world dynamic behavior because of their variable structure. It has been shown that processes described by bilinear models are generally more controllable and offer better performance in control than linear systems. We emphasize that the unstable inherent character of chaotic systems fits exactly within the complete controllability principle discussed for bilinear mathematical models [601] additive control may be used to steer the system to new equilibrium points, and multiplicative control, either to stabilize a chaotic behavior or to enlarge the attainable space. Then, bilinear systems are of extreme importance in the design and use of optimal control for chaotic behaviors. We can now understand the butterfly effect, i.e., the extreme sensitivity of chaotic systems to tiny perturbations described in Chapter 3. 

Unstable process model. In that case, an FIR process model cannot be used. A state-space model such as... 

University of Saarland (Germany), 290-291 University of Southern CaMfomia Information Sciences Institute, 1112 University of St. Gallen, 217-218 University of Utah, 238 Uimecessary work, 1459 Unpaced fines models of, 1639 Umelated events, 2147 Unstable processes, 1830 Unsupervised neural networks, 1779-1780 UnsustainabiMty, 997 Upper-Extremity CheckHst, 1143-1144 Upper-extremity cumulative trauma disorders, 1365... 

Certain nucleophilic species add to carbonyl groups to give tetrahedral intermediates that are unstable and break down to form a new double bond. An important group of such reactions is that between compounds containing primary amino groups and ketones or aldehydes. Scheme 8.2 lists some of the more familiar classes of such reactions. At one time, a principal interest in these reactions was for the preparation of crystalline derivatives of ketones and aldehydes for characterization, but more recently, these types of reactions have been studied in detail because they are models of processes that are of importance in biological reactions. In... 

Dynamic models are essential for discovering intrinsic instability and for choosing the proper control action to stabilize the unstable process. With regard to the classical control objective, although it may be achieved without the need of a model for the process, this, in fact, can be quite dangerous because the closed-loop dynamics of a process can be unstable, even when the process is intrinsically stable. Also, the model-based control is almost always more efficient and robust than control not based on reliable dynamic models. 

The c — c two-equation model equations for adsorption are similar to those of absorption except that adsorption is an unstable process and the time parameter should be involved. On the other hand, the gas adsorption process consists of gas and solid phases, and the corresponding equations should established for each... 

In reahty the chemistry of breakpoint chlorination is much more complex and has been modeled by computer (21). Conversion of NH/ to monochloramine is rapid and causes an essentially linear increase in CAC with chlorine dosage. Further addition of chlorine results in formation of unstable dichloramine which decomposes to N2 thereby causing a reduction in CAC (22). At breakpoint, the process is essentially complete, and further addition of chlorine causes an equivalent linear increase in free available chlorine. Small concentrations of combined chlorine remaining beyond breakpoint are due primarily to organic chloramines. Breakpoint occurs slightly above the theoretical C1 N ratio (1.75 vs 1.5) because of competitive oxidation of NH/ to nitrate ion. Organic matter consumes chlorine and its oxidation also increases the breakpoint chlorine demand. Cyanuric acid does not interfere with breakpoint chlorination (23). 

The onset of flow instability in a heated capillary with vaporizing meniscus is considered in Chap 11. The behavior of a vapor/liquid system undergoing small perturbations is analyzed by linear approximation, in the frame work of a onedimensional model of capillary flow with a distinct interface. The effect of the physical properties of both phases, the wall heat flux and the capillary sizes on the flow stability is studied. A scenario of a possible process at small and moderate Peclet number is considered. The boundaries of stability separating the domains of stable and unstable flow are outlined and the values of the geometrical and operating parameters corresponding to the transition are estimated. 

The partial differential equations used to model the dynamic behavior of physicochemical processes often exhibit complicated, non-recurrent dynamic behavior. Simple simulation is often not capable of correlating and interpreting such results. We present two illustrative cases in which the computation of unstable, saddle-type solutions and their stable and unstable manifolds is critical to the understanding of the system dynamics. Implementation characteristics of algorithms that perform such computations are also discussed. 

The identification of plant models has traditionally been done in the open-loop mode. The desire to minimize the production of the off-spec product during an open-loop identification test and to avoid the unstable open-loop dynamics of certain systems has increased the need to develop methodologies suitable for the system identification. Open-loop identification techniques are not directly applicable to closed-loop data due to correlation between process input (i.e., controller output) and unmeasured disturbances. Based on Prediction Error Method (PEM), several closed-loop identification methods have been presented Direct, Indirect, Joint Input-Output, and Two-Step Methods. 

A PET scan requires a substance called a tracer. A suitable tracer must accumulate in the target organ, and it must be modified to contain unstable radioactive atoms that emit positrons. Glucose is used for brain imaging, because the brain processes glucose as the fuel for mental and neural activities. A common tracer for PET brain scans is glucose modified to contain radioactive fluorine atoms. Our molecular inset shows a simplified model of this modified glucose molecule. 

The model of the chain of hydrogen atoms with a completely delocalized (metallic) type of bonding is outlined in the preceding section. Intuitively, a chemist will find this model rather unreal, as he or she expects the atoms to combine in pairs to give H2 molecules. In other words, the chain of equidistant H atoms is expected to be unstable, so it undergoes a distortion in such a way that the atoms approach each other in pairs. This process is called Peierls distortion (or strong electron-phonon coupling) in solid-state physics ... 

The controller function will take on a positive pole if the process function has a positive zero. It is not desirable to have an inherently unstable element in our control loop. This is an issue which internal model control will address. 

Combining solids that have previously been equilibrated at different relative humidities results in a system that is thermodynamically unstable, since there will be a tendency for moisture to distribute in the system so that a single relative humidity is attained in the headspace. As shown in Fig. 7, moisture will desorb into the headspace from the component initially equilibrated at a higher relative humidity and sorb to the component initially equilibrated at a lower relative humidity. This process will continue until both solids have equilibrated at the final relative humidity. The final relative humidity can be predicted a priori by the sorption-desorption moisture transfer (SDMT) model [95] if one has moisture uptake isotherms for each of the solid components, their initial moisture contents and dry weights, headspace volume, and temperature. Final moisture contents for each solid can then easily be estimated from the isotherms for the respective solids. 

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