Minimum-time control

FIG 5.13. Minimum-time control of a dead-time plus integrating process. 

To explore what is required for minimum-time control, consider the application to a dead-time plus integrating process. In Fig. 5.13 the tracks of both the intermediate variable, i.e., the output of the integrator, and the controlled variable are plotted. Minimum time requires that m be switched from 100 percent to equal the load. 5 before the set point is reached. 

The program consists of switching from 100 percent output at ei to 0 percent output for a specified time tj, after which the steady-state value is selected. Values of ei and h necessary for minimum-time control can be found by solving the following pair of equations ... 

To improve the efficiency. Shin and McKay [2] developed a minimum-time control criterion to achieve the maximum efficiency by applying full acceleration capability along the entire path. In their study, the minimization of traveling time is the only concern. Several important factors, such as the nonlinear stiffness of the transmission mechanism and the structural dynamics of the manipulator arm, are neglected. Without considering these factors, the suddenly changed... 

Kang G. Shin Neil. D. McKay," Minimum-Time Control of Robotic Manipulators with Geometric Path Constraints," IEEE Transaction Automatic Control, June 1985. 

Tests using a constant stress (constant load) normally by direct tension have been described in ISO 6252 (262). This test takes the specimen to failure, or a minimum time without failure, and frequently has a flaw (drilled hole or notch) to act as a stress concentrator to target the area of failure. This type of testing, as well as the constant strain techniques, requires careful control of specimen preparation and test conditions to achieve consistent results (263,264). 

P,n and the roll compaction time control compact density. Generally speaking, as compaction time decreases (e.g., by increasing roll speed), the minimum necessary pressure for quahty compacts increases. There may be an upper limit of pressure as well for friable materials or elastic materials prone to delamination. 

Although it is often possible to predict the effect of the solvent on retention, due to the unique interactive character of both the solvents and the enantiomers, it is virtually impossible to predict the subtle differences that control the separation ratio from present knowledge. Nevertheless, some accurate retention data, taken at different solvent compositions, can allow the retention and separation ratios to be calculated over a wide range of concentrations using the procedure outlined above. From such data the phase system and the column can be optimized to provide the separation in the minimum time, a subject that will be discussed later in the treatment of chromatography theory. 

Microbiocides may be toxic to humans therefore, care must be taken when used. When selecting the microbiocide, the field engineer can obtain pertinent information on chemicals from the service company providing the chemicals. The microbiocide selected must be compatible with the system in which it is being used. Some chemicals such as quaternary amines have dual functions one as microbiocides and the other as film-forming corrosion inhibitors. Insufficient concentrations of this type of chemical may not be enough to coat the whole surface of metal and can cause pitting corrosion. The selection must also depend on chemicals that can produce the desired control in minimum time limits and... 

A closer look at the nonisothermal and isothermal policy results reveals some additional interesting features with regard to optimization. As mentioned earlier, isothermal policies were determined by two factors. One was the M, value and the other was the dead end polymerization caused by depletion of initiator. It was also observed that the minimum time from a nonisothermal policy was considerably less than the minimum time due to the isothermal policy whenever H>, was the controlling factor in the isothermal policy when the isothermal policy was controlled by initiator depletion, a nonisothermal policy did not show significant improvement in minimum time relative to the isothermal one. 

This time is considerably shorter than the galvanostatic minimum time (Eq. 15). Since a theoretical analysis of this type of unsteady-state free convection offers severe difficulties, there is no way to confirm this difference except by controlled experiments of the kind undertaken, on a limited scale, by Hickman (H3) and by Fenech (F2). 

Morrall2 used a HPLC system with two columns. The first column was loaded with the controlled pore glass (CPG) to be modified. The second column was used for separation of the reaction effluents. This column was coupled to a refractive index detector, allowing for quantitative detection of the effluents. The reaction was initiated by injecting an APTS/toluene mixture and stopped by injection of pure toluene. With this so-called stop-flow mechanism reaction times down to 18 seconds could be used. From these analyses it became evident that upon mixing of the aminosilane with the silica, a very rapid physisorption occurs. The initial adsorption of the APTS (from toluene solution on dried CPG) occurred before the 18 second minimum time delay of the stop-flow apparatus. For non-aminated silanes the adsorption proved to be much slower. This study also revealed the pivotal role of surface water in the modification of siliceous surfaces with alkoxysilanes, as discussed in the previous chapter. 

The integration or exposure time of the arrays is controlled by the clock and a series of counters. The minimum integration time at room temperature of an array running with a 20 kHz clock is 15 milliseconds the maximum integration time is 205 milliseconds. The minimum time is limited by the scan time of the array the maximum time by thermal charge. 

The minimum time problem is also known as the time optimal control problem. Coward (1967), Hansen and Jorgensen (1986), Robinson (1970), Mayur et al. (1970), Mayur and Jackson (1971), Mujtaba (1989) and Mujtaba and Macchietto (1992, 1993, 1996, 1998) all minimised the batch time to yield a given amount and composition of distillate using conventional batch distillation columns. The time optimal operation is often desirable when the amount of product and its purity are specified a priori and a reduction in batch time can produce either savings in the operating costs of the column itself or permit improved scheduling of other batch operations elsewhere in a process. Mathematically the problem can be written as ... 

Recently there has been great interest in discrete-time optimal control based on a one-step ahead optimization criterion, also known as minimum variance control. A number of different approaches for minimum variance control has been developed in the last decade. MacGregor (51) and Palmor and Shinnar (52) have provided overviews of these minimum variance controller design techniques. 

As shown in the above works, an optimal feedback/feedforward controller can be derived as an analytical function of the numerator and denominator polynomials of Gp(B) and Gn(B). No iteration or integration is required to generate the feedback law, as a consequence of the one step ahead criterion. Shinnar and Palmor (52) have also clearly demonstrated how dead time compensation (discrete time Smith predictor) arises naturally out of the minimum variance controller. These minimum variance techniques can also be extended to multi-variable systems, as shown by MacGregor (51). 

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