Control theory

It has been suggested that Classical control theory be merged into this article. (Discuss) Proposed since March 2023.

Control theory

Control theory is a field of <a href="/en/Control_engineering" title="Control engineering" class="wl">control engineering</a> and <a href="/en/Applied_mathematics" title="Applied mathematics" class="wl">applied mathematics</a> that deals with the <a href="/en/Control_system" title="Control system" class="wl">control</a> of <a href="/en/Dynamical_system" title="Dynamical system" class="wl">dynamical systems</a> in engineered processes and machines. The objective is to develop a model or algorithm governing the application of system inputs to drive the system to a desired state, while minimizing any delay, overshoot, or steady-state error and ensuring a level of control <a href="/en/Stability_theory" title="Stability theory" class="wl">stability</a>; often with the aim to achieve a degree of <a href="/en/Optimal_control" title="Optimal control" class="wl">optimality</a>.

<div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none" about="#mwt1">Branch of engineering and mathematics</div>
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To do this, a controller with the requisite corrective behavior is required. This controller monitors the controlled <a href="/en/Process_variable" title="Process variable" class="wl">process variable</a> (PV), and compares it with the reference or <a href="/en/Setpoint_(control_system)" title="Setpoint (control system)" class="wl">set point</a> (SP). The difference between actual and desired value of the process variable, called the error signal, or SP-PV error, is applied as feedback to generate a control action to bring the controlled process variable to the same value as the set point. Other aspects which are also studied are <a href="/en/Controllability" title="Controllability" class="wl">controllability</a> and <a href="/en/Observability" title="Observability" class="wl">observability</a>. Control theory is used in <a href="/en/Control_system_engineering" title="Control system engineering" class="mw-redirect wl">control system engineering</a> to design automation that have revolutionized manufacturing, aircraft, communications and other industries, and created new fields such as <a href="/en/Robotics" title="Robotics" class="wl">robotics</a>. 

Extensive use is usually made of a diagrammatic style known as the <a href="/en/Block_diagram" title="Block diagram" class="wl">block diagram</a>. In it the <a href="/en/Transfer_function" title="Transfer function" class="wl">transfer function</a>, also known as the system function or network function, is a mathematical model of the relation between the input and output based on the <a href="/en/Differential_equation" title="Differential equation" class="wl">differential equations</a> describing the system.

Control theory dates from the 19th century, when the theoretical basis for the operation of governors was first described by <a href="/en/James_Clerk_Maxwell" title="James Clerk Maxwell" class="wl">James Clerk Maxwell</a>.<a href="#cite_note-1" style="counter-reset: mw-Ref 1;">[1]</a> Control theory was further advanced by <a href="/en/Edward_Routh" title="Edward Routh" class="wl">Edward Routh</a> in 1874, <a href="/en/Jacques_Charles_François_Sturm" title="Jacques Charles François Sturm" class="wl">Charles Sturm</a> and in 1895, <a href="/en/Adolf_Hurwitz" title="Adolf Hurwitz" class="wl">Adolf Hurwitz</a>, who all contributed to the establishment of control stability criteria; and from 1922 onwards, the development of <a href="/en/PID_control" title="PID control" class="mw-redirect wl">PID control</a> theory by <a href="/en/Nicolas_Minorsky" title="Nicolas Minorsky" class="wl">Nicolas Minorsky</a>.<a href="#cite_note-2" style="counter-reset: mw-Ref 2;">[2]</a>
Although a major application of <a href="/en/Mathematics" title="Mathematics" class="wl">mathematical</a> control theory is in <a href="/en/Control_Systems_Engineering" title="Control Systems Engineering" class="mw-redirect wl">control systems engineering</a>, which deals with the design of <a href="/en/Process_control" title="Process control" class="mw-redirect wl">process control</a> systems for industry, other applications range far beyond this. As the general theory of feedback systems, control theory is useful wherever feedback occurs - thus control theory also has applications in life sciences, computer engineering, sociology and <a href="/en/Operations_research" title="Operations research" class="wl">operations research</a>.<a href="#cite_note-3" style="counter-reset: mw-Ref 3;">[3]</a>

Control theory is a field of control engineering and applied mathematics that deals with the control of dynamical systems in engineered processes and machines. The objective is to develop a model or algorithm governing the application of system inputs to drive the system to a desired state, while minimizing any delay, overshoot, or steady-state error and ensuring a level of control stability; often with the aim to achieve a degree of optimality. To do this, a controller with the requisite corrective behavior is required. This controller monitors the controlled process variable , and compares it with the reference or set point . The difference between actual and desired value of the process variable, called the error signal, or SP-PV error, is applied as feedback to generate a control action to bring the controlled process variable to the same value as the set point. Other aspects which are also studied are  controllability and observability.  Control theory is used in control system engineering to design automation  that have revolutionized manufacturing, aircraft, communications and other industries, and created new fields such as robotics.   Extensive use is usually made of a diagrammatic style known as the block diagram. In it the transfer function, also known as the system function or network function, is a mathematical model of the relation between the input and output based on the differential equations describing the system. 