Steady State Decoupling

One possibility is that one of the controllers could be de-tuned so that it reacted very slowly to disturbances. For example, it might be acceptable for the level to deviate from SP for long periods and changes in its SP are likely to be rare. Priority would then be given to the temperature controller. While the interaction still exists, this would avoid instability. 

We can apply relative gain analysis to determine the level of interaction. If Fi is the flow and F2 the other then, if the cross-sectional area of the drum is A, then the rate of change of level is given by 

If Ti is the temperature of one stream and T2 the other then, if the specific heat is the same for both streams, the combined temperature is given by 

We can see that as Fi approaches F2, i.e. because the average of the two stream temperatures is close to the target temperature, the relative gains all approach 0.5 and the process becomes uncontrollable using the chosen MVs. 

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This paper extends previous studies on the control of a polystyrene reactor by including (1) a dynamic lag on the manipulated flow rate to improve dynamic decoupling, and (2) pole placement via state variable feedback to improve overall response time. Included from the previous work are optimal allocation of resources and steady state decoupling. Simulations on the non-linear reactor model show that response times can be reduced by a factor of 6 and that for step changes in desired values the dynamic decoupling is very satisfactory. 

Timm, Gilbert, Ko, and Simmons O) presented a dynamic model for an isothermal, continuous, well-mixed polystyrene reactor. This model was in turn based upon the kinetic model developed by Timm and co-workers (2-4) based on steady state data. The process was simulated using the model and a simple steady state optimization and decoupling algorithm was tested. The results showed that steady state decoupling was adequate for molecular weight control, but not for the control of production rate. In the latter case the transient fluctuations were excessive. 

In this work only steady state decoupling was used. This is accomplished with the matrix G(3x3), defined by... 

McAvoy, T. J., "Steady State Decoupling Sensitivity with Application to Distillation Control," AIChE National Meeting Houston, Texas, April, 1979. 

A highly non-linear polymerization reactor can be controlled satisfactorily using controller designs based on the linearized case. Steady-state decoupling of outputs appears to be adequate under the conditions studies. It has also been shown that when the decoupling matrix is not square, some form of optimization technique can be used to provide "best" values of the input parameters. 

The last equation describes the necessary steady-state decoupler which cancels any effects that the flow control loop might have on the composition control loop. 

Find the steady-state decouplers for the two control loops selected in the process of Example 24.4. 

The advantage of static decoupling is that less process information is required namely, only steady-state gains. Nonlinear decouplers can be used when the process behavior is nonlinear. 

It is desirable to "decouple" the system so that each manipulated variable appears to affect only one of the output variables. This requires a matrix upstream of the process which, in response to a change in only one of its input signals, will manipulate all the actual process inputs simultaneously so that only the desired output signal changes. If decoupling is to be accomplished only at steady state conditions this matrix is a set of constants. However, if decoupling is required during transient operation the matrix must contain dynamic transfer functions, some of which may not be physically realizable. 

A control algorithm has been derived that has improved the dynamic decoupling of the two outputs MW and S while maintaining a minimum "cost of operation" at the steady state. This algorithm combines precompensation on the flow rate to the reactor with state variable feedback to improve the overall speed of response. Although based on the linearized model, the algorithm has been demonstrated to work well for the nonlinear reactor model. 

Assuming further that decoupling has been turned on a long time ago before the carbon-13 measurement so that a steady state is reached, one ends up with a steady state carbon polarization, I, obtained by setting dh /dt to zero ... 

Based on the linearized models around the equilibrium point, different local controllers can be implemented. In the discussion above a simple proportional controller was assumed (unity feedback and variable gain). To deal with multivariable systems two basic control strategies are considered centralized and decentralized control. In the second case, each manipulated variable is computed based on one controlled variable or a subset of them. The rest of manipulated variables are considered as disturbances and can be used in a feedforward strategy to compensate, at least in steady-state, their effects. For that purpose, it is t3q)ical to use PID controllers. The multi-loop decoupling is not always the best strategy as an extra control effort is required to decouple the loops. 

It is interesting to note that in chaotic regime, the flow rate outlet stream, which is manipulated by the control valve CVl (see Figure 12), and the reactor volume, are driven by the PI controller to the equilibrium point without chaotic oscillations. However, the other variables have a chaotic behavior as shown in Figure 18. So it is possible to obtain a reactor behavior, in which some variables are in steady state and the others are in regime of chaotic oscillations, due to the decoupling or serial connection phenomena. In this case the control system and the volumetric flow limitation of coolant flow rate through the control valve VC2, are the responsible of this behavior. Similar results can be obtained from model. 

In the steady stagnation-flow formulation the thermodymanic pressure may be assumed to be constant and treated as a specified parameter. The small pressure variations in the axial direction, which may be determined from the axial momentum equaiton, become decoupled from the system of governing equations (Section 6.2). The small radial pressure variations associated with the pressure-curvature eigenvalue A are also presumed to be negligible. While this formulation works very well for the steady-state problem, it can lead to significant numerical difficulties in the transient case. A compressible formulation that retains the pressure as a dependent variable (not a fixed parameter) relieves the problem [323],... 

It remains now to solve Eq. (2.3). Here, there are various approaches, depending on the conditions. When a non-steady-state solution is required, one can introduce the decoupling approximation of Sumi and Marcus, if there is the difference in time scales mentioned earlier. Or one can integrate Eq. (2.3) numerically. For the steady-state approximation either Eq. (2.3) can again be solved numerically or some additional analytical approximation can be introduced. For example, one introduced elsewhere [44] is to consider the case that most of the reacting systems cross the transition state in some narrow window (X, X i jA), narrow compared with the X region of the reactant [e.g., the interval (O,Xc) in Fig. 2]. In that case the k(X) can be replaced by a delta function, fc(Xi)A5(X-Xi). Equation (2.3) is then readily integrated and the point X is obtained as the X that maximizes the rate expression. The A is obtained from the width of the distribution of rates in that system [44]. 

In the case of solar powered systems, decoupling of heat source and chemical plant facilitates the compensation of fluctuating and intermittent available power input. This is particularly important if units of the chemical system require steady-state conditions over long periods. Beyond that decoupled systems allow for an easier integration of thermal and chemical storage units to compensate daily or seasonal variation of solar supply. The same applies for hybrid operation, i.e. the combination with burner firing or with a nuclear heat source. 

Figure 4. Half-cycles in dissipative (d) maintenance metabolism with steady-state ATP turnover, decoupled by futile cycling with, in this example, fructose 6-phos-phate/fructose 1,6-bisphosphate. The net enthalpy change is calculated exclusively due to the catabolic half-cycle because both the ATP and the futile cycles contain equal but opposite exothermic and endothermic components (after Gnaiger, 1990).

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