Optimization steady-state

In practice, one will seek to obtain an estimate of the elimination constant kp and the plasma volume of distribution Vp by means of a single intravenous injection. These pharmacokinetic parameters are then used in the determination of the required dose D in the reservoir and the input rate constant k (i.e. the drip rate or the pump flow) in order to obtain an optimal steady state plasma concentration... 

Therefore we conclude not only that feedback control is useful to stabilize an optimal unstable steady state such as depicted in Figures 4.34 to 4.37 for the original set of parameter data, but feedback control can also help ensure the robustness of an otherwise stable optimal steady state over a larger region of parameters and system perturbations. Proper feedback control is also helpful in damping temperature explosions. 

There are many other interesting and complex dynamic phenomena besides oscillation and chaos which have been observed but not followed in depth both theoretically and experimentally. One example is the wrong directional behavior of catalytic fixed-bed reactors, for which the dynamic response to input disturbances is opposite of that suggested by the steady-state response [99, 100], This behavior is most probably connected to the instability problems in these catalytic reactors as shown crudely by Elnashaie and Cresswell [99]. Recently Elnashaie and co-workers [102-105] have also shown rich bifurcation and chaotic behavior of an anaerobic fermentor for producing ethanol. They have shown that the periodic and chaotic attractors may give higher ethanol yield and productivity than the optimal steady states. These results have been confirmed experimentally [105],... 

Arkun, Stephanopoulos and Morari (1978) have added a new twist to control system synthesis. They developed the theory and then demonstrated on two example problems how to move from one control point to another for a chemical process. They note that the desirable control point is likely at the intersection of a number of inequality constraints, the particular set being those that give optimal steady-state performance for the plant. Due to process upsets or slow changes with time, the point may move at which one wishes to operate. Also, the inequality constraints themselves may shift relative to each other. Arkun, Stephanopoulos and Morari provide the theory to decide when to move, and then develop alternative paths along which to move to the new... 

However, the reactor performance obtained under optimal steady-state conditions does not determine a potential limit for a heterogeneous catalytic system. This performance can be improved further using forced unsteady-state operation. Such an operation is capable of substantial extending a gamut of the process features and allows for better use of nonlinear properties inherent in a catalytic reaction system. The positive effect can be generated by two major factors [1] ... 

A set of the possible unsteady or cyclic states of the dynamic system comprises a narrower set of steady states. Therefore, the maximum value of an object or performance function /max obtained under optimal unsteady-state conditions may not be lower than that for optimal steady-state JJ ", i.e. 7max > "ax. However, efficient or proper periodic operation includes only the regimes which provide higher performance in comparison with the optimal steady-state, Jmax >... 

The latter inequality is not always satisfied. The goal of general mathematical theory is in establishing whether the optimal steady-state operation can be improved using forced unsteady-state conditions. Then, if the answer is positive, the optimal operation should be found. The last stage obviously is the cost-analysis of... 

After satisfying all of the basic regulatory requirements, we usually have additional degrees of freedom involving control valves that have not been used and setpoints in some controllers that can be adjusted. These can be utilized either to optimize steady-state economic process performance (e.g., minimize energy, maximize selectivity) or to improve dynamic response. 

If concentration of the intermediate [BZ] responds slower than the concentration of [AZ], periodic forcing of concentration of the component A increases selectivity compared to the optimal steady-state. An intermediate frequency periodic control is preferred. Even larger selectivity results from simultaneous variation of both inlet concentrations A and B (Fig. 1). 

It has been recognised for long time that the performance of a plant depends much more on its design characteristics than on the sophistication of the control algorithms. The practice showed also that the optimal steady state design is not always the best in operation. Only slightly more expensive alternative could posses much better dynamic properties, and become more profitable when the cost of time is taken into account. Hence, there is always a trade-off between steady state economics and controllability. 

The main objective of this section is to provide a quantitative example in which the besf process is not the optimal steady-state economic design. In this example reactor system, temperature control is the measure of product quality. For this type of system, dynamic controllability is improved by increasing the heat transfer area in the reactor. 

The basic idea behind constraint-based approaches is to take the optimal Steady-state design and dclerniinc how far away from this optimal point the plant must... 

Optimal periodic control involves a periodic process, which is characterized by a repetition of its state over a fixed time period. Examples from nature include the circadian rhythm of the core body temperature of mammals and the cycle of seasons. Man-made processes are run periodically by enforcing periodic control inputs such as periodic feed rate to a chemical reactor or cyclical injection of steam to heavy oil reservoirs inside the earth s crust. The motivation is to obtain performance that would be better than that imder optimal steady state conditions. 

In this chapter, we first describe how to solve an optimal periodic control problem. Next, we derive the pi criterion to determine whether better periodic operation is possible in the vicinity of an optimal steady state operation. 

The pi criterion is a sufficient condition for the existence of a periodic solution that is better than the neighboring optimal steady state solution of an optimal periodic control problem. Using the criterion, we would hke to know, for example, whether the time-averaged product concentration in a periodic process can be more than what the optimal steady state operation can provide. In other words, we would hke to check if oscillating the optimal steady state control with some frequency and time period improves on the steady state solution. 

Let the pair (y, u) denote the optimal steady state solution with the corresponding multipliers A, fi, and P. The necessary conditions for this solution are provided in Appendix 8.A (p. 260). Since this solution minimizes J, satisfying the state equation and the constraints,... 

In deriving the above equation, we have considered the fact that all equality constraints are satisfied at the optimal steady state, and the multipliers are zero for inactive inequality constraints (see Section 4.5, p. 109). 

We ultimately need to determine whether in the vicnity of the optimal steady state pair (y, u), the objective functional I could reduce further for some admissible pair of state and corresponding control, (y, u), where... 

The first assumption implies that the constraint qualification or normality condition (see Section 4.3.2, p. 96) is satisfied. It ensures that an infinite rnunber of solution pairs exist near the optimal steady state solution pair (y, ti). This pair, as well as ti, is called normal when the normality condition is satisfied. 

Admissible ( y, u) at the Optimal Steady State We propose the following control variation... 

We now utilize Equation (8.14), which provides the change in I from its optimal steady state value. For the present problem, SF is zero since (y, u) is optimal. Expanding in Equation (8.14), we obtain... 

The optimal steady state pair (x, u) is proper if there exists a neighboring pair (x, u) that provides a lower objective functional value I than /, which is given by (x, u). Assuming that the pair (x, u) is normal, the sufficient condition for its properness is that the right-hand side of Equation (8.25) should be less than zero, i.e., the pi criterion... 

As long as the optimal steady state controls are normal and lie within but not at the boundaries of the set of admissible control values, the negative-... 

The sufficient conditions for properness of the optimal steady state pair (x, u) stay the same as long as u is less than Umax- However, if a uj = Uj max,... 

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