Digital computation computational complexity

The subject of adaptive control is one of current interest. New algorithms are presently under development, but these need to be field-tested before industrial acceptance can be expected. It is clear, however, that digital computers will be required for implementation of self-adaptive controllers due to their complexity. An adaptive controller is inherently nonlinear and therefore more complicated than the conventional PID controller. 

Renon s techniques valuable for the complexities of multicomponent systems and in particular the solution by digital computer. 

The ideal variable to measure is one that can be monitored easily, inexpensively, quickly, and accurately. The variables that usually meet these qualifications are pressure, temperature, level, voltage, speed, and weight. When possible the values of other variables are obtained from measurements of these variables. For example, the flow rate of a stream is often determined by measuring the pressure difference across a constriction in a pipeline. However, the correlation between pressure drop and flow is also affected by changes in fluid density, pressure, and composition. If a more accurate measurement is desired the temperature, pressure, and composition may also be measured and a correction applied to the value obtained solely from the pressure difference. To do this would require the addition of an analog or digital computer to control scheme, as well as additional sensing devices. This would mean a considerable increase in cost and complexity, which is unwarranted unless the increase in accuracy is demanded. 

To overcome these problems, we must learn another language Chinese. This is what we wilt call the frequency-domain methods. These methods are a little more removed from our mother tongue of English and a little more abstract. But they are extremely powerful and very useful in dealing with realistically complex processes. Basically this is because the manipulation of transfer functions becomes a problem of combining complex numbers numerically (addition, multiplication, etc.). This is easily done on a digital computer. 

However, in a more complex case with many equations where analytical solution would be diflicult, the equations can be solved numerically. All the and simultaneous equation solving packages available in most digital-computer libraries can be used. Keep in mind, however, that these are complex-variable simultaneous equations. A special-purpose method for solving the equations Tor a distillation column will be presented in the next example. 

Both the openloop and the closedloop frequency-response curves can be easily generated on a digital computer by using the complex variables and functions discussed in ( han. 12, Tlie freauencv-resnnnse curves for the closedloon... 

Recent decades have seen the emergence of a novel approach to scientific research, based on the exploitation of fast electronic digital computers. Computation provides a method of investigation which transcends the traditional division between theory and experiment. Computer-assisted simulation and design may afford a solution to complex problems which would otherwise be intractable to theoretical analysis, and may also provide a viable alternative to difficult or costly laboratory experiments. Though stemming from Theoretical Chemistry, Computational Chemistry is a field of research... 

Numerical analysis is important in digital-computer work from another viewpoint. Sometimes it is necessary to express complex functional relationships in a simpler form. Occasionally relationships may be given in a graphical or tabular form not directly suitable for processing on digital equipment. In these situations numerical methods for curve fitting and interpolation are techniques which will necessarily be employed. 

To extend the application of the Cochran- Zachariasen sign relation to more complex structures, in which the proportion of strong reflections is smaller, Cochran and Douglas (1955, 1957) have used an electronic digital computer to produce a number of alternative sets of signs, and to select the most probable ones on the basis of certain criteria. The most important criterion for correctness is that... 

The final possibility, a uniformly interesting movie, would have to depict a process with thousands or millions of critical steps occuring in a definite order, each step necessary to understand the next, as in an industrial process, the functioning of a digital computer, or the development of an embryo. Enzymes, having been optimized by natural selection, may be expected to have somewhat complex mechanisms of action, perhaps with several equally important critical steps, but not with thousands of them. There is reason to believe that processes with thousands of reproducible non-trivial steps usually occur only in systems that are held away from thermal equilibrium by an external driving force. They thus belong to the realm of complex behavior in continuously dissipative open systems, rather than to the realm of relaxation processes in closed systems. 

Discrete time controllers will not normally be stand-alone units but will be simulated within the software of a digital computer. The capacity of the computer can be used if necessary to produce more complex forms of feedback control than those provided by the standard algorithms of the classical fixed parameter controller. 

Finally, we might add a third independent variable, catalyst concentration, to type of catalyst and temperature of setting. We then would use a three-way analysis of variance to determine whether differences in means exist. Of course, as additional independent variables are added, the calculations become much more complex, so that they are better carried out on a digital computer. 

Batch Reactors. One of the classic works in this area is by Gee and Melville (21), based on the PSSA for chain reaction with termination. Realistic mechanisms of termination, disproportionation, and combination, are treated with a variety of initiation kinetics, and analytical solutions are obtained. Liu and Amundson (37) solved the simultaneous differential equations for batch and transient stirred tank reactors by using digital computer without the PSSA. The degree of polymerization was limited to 100 the kinetic constants used were not typical and led to radical lifetimes of hours and to the conclusion that the PSSA is not accurate in the early stages of polymerization. In 1962 Liu and Amundson used the generating function approach and obtained a complex iterated integral which was later termed inconvenient for computation (37). The example treated was monomer termination. 

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