Control theory is a branch of mathematics and engineering in an interdisciplinary way. The general objective of this theory is to evaluate solutions from the controller for the proper corrective measure that will help in stabilizing the system. This means that the system will not oscillate around the set point rather will hold it.

Definition

Control theory deals with the study or behavior of dynamical systems that takes inputs. Reference is the term used for external inputs given to a system. A controller is used to manipulate the inputs that are given to a system in the case when one or more output variables are required to follow a particular reference for certain time so as to obtain the required effect on the system's output.

In a continuous control system, the inputs and the outputs are usually related by differential equations in some or other manner. In the case when these differential equations are linear along with constant coefficients as well, then we can obtain a transfer function that will relate the input and output simply by finding the Laplace transform.

In the case when these differential equations are not linear and also if their solution is known, then there is a possibility that we can linearize the non linear differential equation at that known solution. And then if the resulting linear equation has constant coefficients, we can find its Laplace transform in order to obtain the transfer function.

The extensive use is usually made of a kind of diagram commonly known as the block diagram of a feedback control system. Basically the definition of control theory has two parts:

1)   It deals with influencing the behavior of dynamical systems.

2)    It is a subfield of science, originated in engineering and mathematics in interdisciplinary manner and is also evolved in use by social sciences as in psychology, criminology, sociology and financial system as well.
It is being considered having four functions that are measure, compare compute and correct which are completed by five main elements which are detector, transducer, transmitter, controller and final control element. The detector, transducer and transmitter are contained in one unit in practicality as they together complete the measuring function.

The controller completes the compare and compute functions which are implemented electronically by proportional control, PID controller, PI controller or programmable logic controller. The final control element completes the correct function. This element is responsible for changing an input and output in a control system that further affects the controlled variable.

Basic Control Theory

Feedback is introduced by control theory system so as to overcome the limitations of an open loop controller. Feedback is used by a closed loop controller so as to control the outputs generated of a dynamical system. The name feedback comes from the path of the information in the system which is: the inputs of the process have a sure effect on the outputs of the process, which is then measured with the help of sensors and is also worked on by the controller, and the final result is “fed back” as a new input to the process, in a way closing the loop.Advantages of closed loop controllers over open loop controllers:
1)    Rejection of disturbance.

2)    The performance is guaranteed even with existence of uncertainties.

3)    The unstable processes can also be stabilized.

4)    Sensitivity to variations in the parameters is reduced.

5)   The performance of tracking the reference is improved.
In many systems, both the open and the closed loop controllers are used together simultaneously. In such cases, the open loop control is also known as “feed-forward”. It helps in improving the performance of tracking the reference in this case.
Classification of Systems
1)  Linear System Control:
In MIMO systems, pole placements are performed mathematically with the use of state space representation in open loop system with calculation of the feedback matrix by assignment of poles in required locations. In more complicated systems this might need computer aided assistance.
2)  Non Linear Systems Control:
In control theory sometimes it is easier to line arise such functions. But sometimes they might require scratching theories needing permission of control of non linear systems.
3)  Decentralized Systems:
When there exist multiple controllers for a system, decentralization of controls is required. This decentralization helps in many ways, likewise, it helps in controlling the operations of systems over a bigger area geographically. The different agents of a decentralized system can interact with each other via various communication channels in order to coordinate their actions and functions properly.

Applied Control Theory

It utilizes the time domain state space representation rather than a frequency domain analysis as in classical theory.

This representation is a model in mathematics of a physical system that has a set of input, output and state variables which are related by first order differential equations. The variables here are expressed as vector quantities so as to abstract from the number of outputs, inputs and states. Also, both the differential and algebraic equations are represented in matrix form.

The state space or time domain representation provides an easy, convenient, brief and compact way for modeling and analyzing the systems that have multiple inputs as well as outputs. Otherwise we would have to write the Laplace transform for inputs and outputs in order to encode all the information about any system.

Main Topics in Control Theory:

1) Stability:
For a BIBO (bounded input bounded output) stability in a linear system, we say it is stable if for any bounded input given to it the output is also bounded. In a non linear system, stability is ISS (input to state stability), which is combination of a part similar to BIBO stability along with Lyapunov stability.

2) Controllability & Observability:

They are the main issues faced while analyzing the system before deciding on the best control strategy that can be applied, or if the system can be stabilized or not.

3) Control Specification:

Stability, rejection of a step disturbance, time response are some of the specifications.

4) Model Identification & Robustness:

The properties of a robust controller does not change so much as it is applied to a system even slightly different from the mathematical one that was used for its synthesis.