Control Theory

Control theory is a field of applied mathematics that did not become its own field until the late 1950s and early 1960s due to problems within economics and engineering. These problems were considered to be deviant from issues in differential equations and the calculus of variations. Originally, modifications of classical techniques such as calculus of variations and differential equations were used to solve problems, but eventually it was realized that these issues all had the same mathematical structure.

Control theory as a science corresponds to the control and regulation of machines, muscular coordination, metabolic processes within biological organisms, and design of prostheses. It also relates to broader aspects of coordinated activity within society such as optimization of business operations, control of economic activity by government policies, and regulation of political decisions by democratic processes. Control theory can be seen as the science of modifying its environment in the physical, biological, and social sense.

Control theory studies the behavior of dynamical systems that take inputs. Reference is the term used when referring to external inputs given to a system. A control is used to manipulate the inputs given to a system when one or more output variables are needed to follow a particular reference to obtain the effect on the system’s output.

The four functions of control theory include measure, compare, compute, and correct, which are completed by these five main elements  -  detector, transducer, transmitter, controller, and final control element. The detector, transducer, and transmitter are part of one unit as they comprise the measuring function. The controller completes the compare and compute functions which are used electronically by proportional control, PID controller, PI controller, or programmable logic controller. The final control function completes the correct function which is responsible for changing an input and output in a control system that affects the controlled variable.

Some examples of modern control systems include machines that cannot work without feedback control, control of machines, control of large systems, biocontrol, and robots. Many devices must be made in such a way that their behavior can be modified by way of an external control. For example, transistor amplifiers produce intolerable distortion in sound systems when used by themselves, but when modified by a feedback control can achieve a desired degree of fidelity (the degree to which an electronic device accurately reproduces its effect).  Jet aircraft cannot be operated without automatic control to aid the pilot.

Using machines to perform tasks can be directed manually but may be more convenient to the user when machines are directly connected to the measuring instrument, such as when a thermostat is used to turn on or off a refrigerator, oven, air conditioning unit, or heating system. Dimming headlights for cars, settings for the diaphragm of a camera, and finding the correct exposure for color prints can be automatically completed by connecting a photocell to the machine.

Places such as power plants, oil refineries, and chemical factories can be controlled and monitored manually but it is considered more feasible to use automatic controls to make systems work more efficiently. Biocontrol such as controlling artificial hearts or kidneys, nerve controlled prosthetic devices, and controlling brain functions by combining artificial biology and natural technology is still a work in progress because of the lack of knowledge about control principles being employed in the natural world. Pattern recognition and insight into thought processes has also slowed the development of robots that engage in purposeful behavior without direct input from humans. Industrial manufacturing robots exist but there need to be more breakthroughs and scientific advances to solve problems with pattern recognition and thought processes.

 

Sources

  1. https://www.britannica.com/science/control-theory-mathematics; accessed July 26 and 27
  2. http://www.mathcaptain.com/applied-mathematics/control-theory.html; accessed July 26 and 27

 

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