Modeling of Response in Linear Systems

In a linear control system, the relationship between input y and output x can be described by a differential equation  

The order of the differential equation determines the order of the control system. Thus, a first order control system is described by  

After a long time (t °o), the following relationship is valid  

The solution to this differential equation, when the input variable is given a step change of 1, is the well-known exponential relationship of a first order control system. 

In reality, most systems do not behave in a truly linear fashion, but the assumption of linearity can usually be justified over a small region near the operating point (linearization). 

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