Control theory is an interdisciplinary branch of engineering and mathematics, that deals with influencing the behavior of dynamical systems. Control theory is an interdisciplinary branch of engineering and mathematics that deals with the behavior of dynamical systems with inputs. The external input of a system is called the reference. When one or more output variables of a system need to follow a certain reference over time, a controller manipulates the inputs to a system to obtain the desired effect on the output of the system. The usual objective of a control theory is to calculate solutions for the proper corrective action from the controller that result in system stability, that is, the system will hold the set point and not oscillate around it. The inputs and outputs of a continuous control system are generally related by differential equations. If these are linear with constant coefficients, a transfer function relating the input and output can be obtained by taking their Laplace transform. If the differential equations are nonlinear and have a known solution, it may be possible to linearize the nonlinear differential equations at that solution.[1] If the resulting linear differential equations have constant coefficients one can take their Laplace transform to obtain a transfer function. The transfer function is also known as the system function or network function. The transfer function is a mathematical representation, in terms of spatial or temporal frequency, of the relation between the input and output of a linear time-invariant solution of the nonlinear differential equations describing the system. Extensive use is usually made of a diagrammatic style known as the block diagram. Overview Smooth nonlinear trajectory planning with linear quadratic Gaussian feedback (LQR) control on a dual pendula system. Control theory is a theory that deals with influencing the behavior of dynamical systems an interdisciplinary subfield of science, which originated in engineering and mathematics, and evolved into use by the social sciences, like psychology, sociology, criminology and in the financial system. Control systems can be thought of as having four functions; Measure, Compare, Compute, and Correct. These four functions are completed by five elements; Detector, Transducer, Transmitter, Controller, and Final Control Element. The measuring function is completed

y the detector, transducer and transmitter. In practical applications these three elements are typically contained in one unit. A standard example of a measuring unit is a Resistance thermometer. The compare and compute functions are completed within the controller which may be completed electronically through a Proportional control, PI Controller, PID Controller, Bistable, Hysteretic control or Programmable logic controller. Older controller units have been mechanical, as in a Centrifugal governor or a Carburetor. The correct function is completed with a final control element. The final control element changes an input or output in the control system which affect the manipulated or controlled variable. [edit]An example Consider a car's cruise control, which is a device designed to maintain vehicle speed at a constant desired or reference speed provided by the driver. The controller is the cruise control, the plant is the car, and the system is the car and the cruise control. The system output is the car's speed, and the control itself is the engine's throttle position which determines how much power the engine generates. A primitive way to implement cruise control is simply to lock the throttle position when the driver engages cruise control. However, if the cruise control is engaged on a stretch of flat road, then the car will travel slower going uphill and faster when going downhill. This type of controller is called an open-loop controller because no measurement of the system output (the car's speed) is used to alter the control (the throttle position.) As a result, the controller cannot compensate for changes acting on the car, like a change in the slope of the road. In a closed-loop control system, a sensor monitors the system output (the car's speed) and feeds the data to a controller which adjusts the control (the throttle position) as necessary to maintain the desired system output (match the car's speed to the reference speed.) Now when the car goes uphill the decrease in speed is measured, and the throttle position changed to increase engine power, speeding the vehicle. Feedback from measuring the car's speed has allowed the controller to dynamically compensate for changes to the car's speed. It is from this feedback that the paradigm of the control loop arises: the control affects the system output, which in turn is measured and looped back to alter the control.

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