- Key People:
- Lev Semyonovich Pontryagin
control theory
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What is control theory in mathematics?
Control theory, a field of applied mathematics, focuses on the control of specific physical processes and systems. It addresses challenges in engineering and economics through modifications of classical techniques.
Control theory uses mathematical models to represent systems, and it depends on the agreement between available models and the actual behavior of the system to be controlled. It involves understanding the behavior of a system through mathematical description and defining the control's purpose and environment mathematically. The task of control theory is to study the mathematical quantification of these two basic problems and then to deduce applied mathematical methods whereby a concrete answer to optimization can be obtained.
The principles of control are expressed mathematically and applicable across various fields like engineering, physics, biology, and economics.
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What is the main goal of control theory?
Control theory seeks to modify the environment, be it physical, biological, or social, through mathematical principles applicable to any concrete situation. It addresses how to quantify system behavior and the purpose of control to deduce mathematical methods for optimization.
The scientific formulation of a control problem relies on a mathematically precise description of the system's behavior and a specification of the control's purpose and environment. This involves understanding the system's dynamics, defining a control law based on the system's state, and implementing control through a controller, often a special-purpose computer. The goal is to achieve desired outcomes by applying appropriate control actions, with the limitations of the theory depending on the accuracy of the mathematical models used.
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What are some examples of systems that use control theory?
Control theory is a field of applied mathematics used to control physical processes and systems. It has broad applications across various domains:
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Engineering: Control theory is critical in designing and operating jet aircraft, helicopters, and robots. It is also used to enhance the fidelity of transistor amplifiers in sound systems through feedback control systems.
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Economics: Control theory can be applied to optimize business operations and to study how government policies control economic activity. It is also relevant to the problem of economic stabilization.
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Social Sciences: Control theory extends to coordinated activity in the social sphere, including the control of political decisions through democratic processes.
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Biological Systems: Control theory is applicable to muscular coordination and metabolism in biological organisms, as well as the design of prosthetic devices.
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What is feedback, and how is it used in control theory?
In control theory, feedback is a system response that affects the system's ongoing activity or productivity. Part of the output is used as input to change and improve the system's subsequent output. Control theory is a branch of applied mathematics relevant to controlling physical processes and systems.
Feedback control uses process information to correct a machine's operation. A closed-loop control system, or servomechanism, relies on a motor to adjust the controlled quantity based on the difference between actual and desired values. Stability is crucial, and damping is often added to prevent hunting, where the system overcorrects itself.
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What is the difference between open-loop and closed-loop control systems?
In control systems, the key difference between open-loop and closed-loop systems lies in feedback mechanisms.
An open-loop system lacks self-correcting capabilities, meaning it doesn't adjust based on the output. Conversely, a closed-loop system uses feedback to regulate an operation, process, or mechanism automatically. It relies on the difference between the desired and actual values to control the system.
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How do mathematicians use equations or models to predict and control systems?
Control theory uses mathematical models to analyze and design systems that achieve desired behaviors. These models precisely describe system behavior and control objectives.
To control a system, control theory uses a list of numbers called a state vector to express the quantitative effect of all external influences on the plant. Control action at a given time can be specified as some function of the state at that time, called a control law.
The controller, a special-purpose computer, builds an optimal filter to determine the best estimate of the present state vector and determines an optimal control law by substituting the estimate of the state vector. These principles are applicable to any situation, including engineering, physics, and economics.
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control theory, field of applied mathematics that is relevant to the control of certain physical processes and systems. Although control theory has deep connections with classical areas of mathematics, such as the calculus of variations and the theory of differential equations, it did not become a field in its own right until the late 1950s and early 1960s. At that time, problems arising in engineering and economics were recognized as variants of problems in differential equations and in the calculus of variations, though they were not covered by existing theories. At first, special modifications of classical techniques and theories were devised to solve individual problems. It was then recognized that these seemingly diverse problems all had the same mathematical structure, and control theory emerged.
As long as human culture has existed, control has meant some kind of power over the environment. For example, cuneiform fragments suggest that the control of irrigation systems in Mesopotamia was a well-developed art at least by the 20th century bc. There were some ingenious control devices in the Greco-Roman culture, the details of which have been preserved. Methods for the automatic operation of windmills go back at least to the European Middle Ages. Large-scale implementation of the idea of control, however, was impossible without a high level of technological sophistication, and the principles of modern control started evolving only in the 19th century, concurrently with the Industrial Revolution. A serious scientific study of this field began only after World War II.
Although control is sometimes equated with the notion of feedback control (which involves the transmission and return of information)—an isolated engineering invention, not a scientific discipline—modern usage favours a wider meaning for the term. For instance, control theory would include the control and regulation of machines, muscular coordination and metabolism in biological organisms, and design of prosthetic devices, as well as broad aspects of coordinated activity in the social sphere such as optimization of business operations, control of economic activity by government policies, and even control of political decisions by democratic processes. If physics is the science of understanding the physical environment, then control theory may be viewed as the science of modifying that environment, in the physical, biological, or even social sense.
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Much more than even physics, control is a mathematically oriented science. Control principles are always expressed in mathematical form and are potentially applicable to any concrete situation. At the same time, it must be emphasized that success in the use of the abstract principles of control depends in roughly equal measure on basic scientific knowledge in the specific field of application, be it engineering, physics, astronomy, biology, medicine, econometrics, or any of the social sciences.
Examples of modern control systems
To clarify the critical distinction between control principles and their embodiment in a real machine or system, the following common examples of control may be helpful.
Machines that cannot function without (feedback) control
Many basic devices must be manufactured in such a way that their behaviour can be modified by means of some external control. Generally, the same effect cannot be brought about (in practice and sometimes even in theory) by any intrinsic modification of the characteristics of the device. For example, transistor amplifiers introduce intolerable distortion in sound systems when used alone, but properly modified by a feedback control system they can achieve any desired degree of fidelity. Another example involves powered flight. Early pioneers failed, not because of their ignorance of the laws of aerodynamics but because they did not realize the need for control and were unaware of the basic principles of stabilizing an inherently unstable device by means of control. Jet aircraft cannot be operated without automatic control to aid the pilot, and control is equally critical for helicopters. The accuracy of inertial navigation equipment cannot be improved indefinitely because of basic mechanical limitations, but these limitations can be reduced by several orders of magnitude by computer-directed statistical filtering, which is a variant of feedback control.
Control of machines
In many cases, the operation of a machine to perform a task can be directed by a human (manual control), but it may be much more convenient to connect the machine directly to the measuring instrument (automatic control); e.g., a thermostat may be used to turn on or off a refrigerator, oven, air-conditioning unit, or heating system. The dimming of automobile headlights, the setting of the diaphragm of a camera, and the correct exposure for colour prints may be accomplished automatically by connecting a photocell directly to the machine in question. Related examples are the remote control of position (servomechanisms) and speed control of motors (governors). It is emphasized that in such cases a machine could function by itself, but a more useful system is obtained by letting the measuring device communicate with the machine in either a feedforward or feedback fashion.
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