Optimal Open-Loop Control

After the identification of a model, the open-loop optimal control policy for the nominal plant can be determined by solving the following optimization problem. 

This approach enables the incorporation of input, output, and final-time constraints and is flexible with respect to the crystallizer configuration, the objective function definition, and the choice of manipulated variables. Examples of the use of nonlinear programming to solve this problem are given subsequently. 

The population and mass balances for batch crystallization without fines destruction and with size-independent growth are as follows  

It should be emphasized that the expressions for growth rate and nucleation rate given above are valid for operation in the metastable zone only (i.e., operation in which the concentration is maintained between the saturation concentration and the metastable limit, the concentration above which homogeneous nucleation occurs). 

The determination of the temperature profile that is optimal in some sense can be formulated as a nonlinear programming problem as follows  

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The operability measure is based on the idea that the time spent away from the desired set point is linked to potential losses due to off-specification products and economic penalties for non-optimal performance. Different types of feedback controllers can be utilized to evaluate this operability measure. However, a performance measure independent of the feedback controller to be used and capable of assessing the inherent limitations of the process is desirable. The minimum-time optimal controller suits these demands very well. The approach is based on an implied assumption that a feedback controller exists that will deliver a closed-loop dynamic operability close to the one calculated here by use of the optimal open-loop controller. For similar reasons, Carvallo [63] employed the minimum time optimal controller to calculate the time it would take the process to respond to the worst disturbance and/or set-point change. 

The PI controller, even when optimally tuned, is also unable to prevent surge. Furthermore, it is unable to stop surge once it occurs. In the above situation, the operator would correctly identify the problem as instability of the closed-loop PI controller. The only viable action would be to open the closed control loop by placing the controller in manual, thereby freezing the valve open. In this scenario, open-loop control will stop surge. 

Several methods have been investigated to find correlations between physical properties of fuel gas mixtures and the excess air ratio to optimize the combustion procedure. In spite of the varying composition of natural gas it is said to be possible to control a heater system by measurements of the dynamic viscosity of the gas [7]. One explanation could be the correlation between Wobbe number and viscosity With increasing Wobbe numbers the viscosity decreases, and if the Wobbe number of a gas is known, the excess air ratio can be adjusted, resulting in an open loop control. 

As presented in the earlier chapters, the operating policy for a batch distillation column can be determined in terms of reflux ratio, product recoveries and vapour boilup rate as a function of time (open-loop control). Under nominal conditions, the optimal operating policy may be specified equivalently in terms of a set-point trajectory for controllers manipulating these inputs. In the presence of uncertainty, these specifications for the optimal operating policy are no longer equivalent and it is important to evaluate and compare their performance. 

The controls that are not given by optimal control laws are often called open-loop controls. They simply are functions of independent variables and specific to the initial system state. The application of open-loop controls is termed open loop control, which is the subject matter of this book. 

Maximizing the objective function Z under these constrants with proper limits of the operating parameters will change the related parameters to the values under which economical value Z will be maximal. In this case, the operating parameters achieve optimal values, which can be used either as set points for the closed-loop or open-loop control. 

Vicente M, SayerC, Leiza JR, Arzamendi G, Lima EL, Pinto JC, Asua JM. Dynamic optimization of non-linear emulsion copolymerization systems - open-loop control of composition and molecular weight distribution. Chem Eng J 2002 85 339-349. 

The weak maximum principle can certainly be used to maximize the performance index of Eq. 13.33, for instance, for the independent deactivation problem given by Eqs. 13.40 through 13.45 with the addition of a heat balance at the pellet-bulk fluid interface for the pellet surface temperature 7 [( )s]. Nevertheless, the optimal inlet temperature will be a function of time. Thus, the inlet temperature is to be manipulated in a prescribed manner in time without any regard to what happens to the reactor. This open-loop control is rarely practiced in actual applications because of the uncertainties regarding the model, measurements, and disturbances. Rather, closed-loop control using feedback from the process is usually practiced. This fact should not discourage one from using the maximum principle for optimization, for it will at least indicate what the best possible performance is in a relative sense. At the same time it can yield in some simple cases a very powerful optimal policy such as the constant conversion policy, which can be implemented by a feedback control scheme. 

As stated in Chapter 1, control refers to a closed-loop system where the desired operating point is compared to an actual operating point and a knowledge of the error is fed back to the system to drive the actual operating point towards the desired one. However, the optimal control problems we consider here do not fall under this definition of control. Because the decision variables that will result in optimal performance are time-dependent, the control problems described here are referred to as optimal control problems. Thus, use of the control function here provides an open-loop control. The dynamic nature of these decision variables makes these problems much more difficult to solve as compared to normal optimization where the decision variables are scalar. 

Optimal control strategy based on a mechanistic model Open-loop control... 

In an open-loop control problem, we compute the optimal control input trajectory and then fully implement it over the entire period to < t < in- For this, the direct approach... 

Bahri, P Bandoni, A., and Romagnoli, J. A. (1996). Effect of disturbances in optimizing control Steady-state open-loop backoff problem. AIChE J. 42,983-995. 

Optimized fan station controls include automatic fan cycling and a damper position to discharge pressure cascade loop to keep the most open user supply damper at 90% opening. 

Positioning systems can use either an open-loop or a closed-loop control system. In closed-loop motion control, such as the optimized positioning of solar collectors based on measuring their shadows, the positions of both the collector and the shadow are continuously detected. Based on this feedback, the position and velocity of the collector can both be controlled. The reported position is continuously compared to the desired one, and the collector is moved to reduce the error between the two. This is called servo control (Figure 3.154). 

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