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#### Mathematical Models

Mathematical models may assume different forms. Depending on the particular system and the particular circumstances, one mathematical model may be better suited than other other models, Once a mathematical model of a system is obtained, various analytical and computer tools can be used for analysis and synthesis purpose.

#### Simplicity vs Accuracy

In obtaining a mathematical model, one must make compromise between the simplicity of the system and the accuracy of the result of the analysis. In deriving a reasonably simplified model, it is necessary to ignore certain inherent physical properties of the system.

While solving a new problem, it is desirable to build a simplified model so that a general feeling for the solution should be felt as we feel when we have found solution to a very complex mathematical problem. A more complete model then can be built and used for a more accurate analysis. A linear lumped parameter model which may be valid in low frequency operations may not be valid at sufficiently high frequency. Since the neglected property of distributed parameters may become an important factor in the dynamic behaviour of the system.

#### Linear System

A system is called linear if the principle of superposition applies. The principle of superposition states that the response produced by the simultaneous application of two different forcing functions is the sum of the two individual response. For the linear system, the response of several inputs can be calculated by treating one input at a time and adding the results. It is this principle that allows one to build up complicated solutions to the linear differential equation from simple solutions. In an experimental investigation of a dynamic system, if cause and effect are proportional thus implying that the principle of superposition holds then the system can be considered linear.

#### Linear Time Invariant System

A differential equation is linear if the coefficients are constants or functions only of the independent variables. Dynamic systems that are composed of linear time invariant lumped parameter components may be described linear time invariant (constant coefficient) differential equations. Such systems are called linear time invariant system.

Systems that are represented by differential equation and coefficients are functions of time then it is called as linear time varying system.

#### Controlled variable and manipulated variable

The controlled variable is the quantity or condition that is measured and controlled

The manipulated variable is the quantity or condition that is varied by the controlled variable so as to affect the value of the controlled variable.

Normally controlled output is the output of system.

Control means measuring the value of the controlled variable of the system and applying the manipulated variable to the system to correct or limit the deviation of the measured value from a desired value.

#### Plant

A plant may be a piece of equipment perhaps just a set of machine parts functioning together to perform a particular operation/task.

#### Processes

The process is defined in various ways depending on the context for which it is used. The Merriam Webster dictionary defines a process to be

a natural, progressively continuing operation or development marked by a series of gradual changes that succeed one another in relatively fixed way and lead toward a particular end or result. It may also be defined as an artificial or voluntary, progressively continuing operation that consists of a series of controlled actions or movements systematically directed toward a particular result or end.

Example :- chemical process, biological process.

#### System

A system is a combination of components that act together and perform a certain objective. A system is not limited to physical ones. The concept of the system can be applied to abstract, dynamic phenomena such as those encountered in economics.

#### Disturbances

A disturbance is a signal that tends to adversely affect the value of the output of a system. If disturbances is generate within the system, it is called

*internal*while an*external disturbance*is generated outside the system and is an input.
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