Proportional control action
In a proportional control action, there is a continuous linear relation between the output m (t) and the actual error signal e (t). Evidently the control valve or the regulating unit can take up any position right from the fully open to fully closed. Depending upon the shift of the measured variable from the desired value, an adjustment is made of the amount of energy supplied (i.e., the manipulated variable) to the process or plant. The change in the valve position is proportional to the amount of deviation and is not affected by the rate or duration of the deviation.

The proportional control follows the law:
                                                            M (t) =k1 e (t) +m

Where m is manual reset constant whose value determine the normal value of controller output. The constant kp, called the proportional sensitivity or gain,

                                                Kp = Change in controller
                                                                               Change in the deviation                                   

 In Laplace transform we can write as;                                                         
 K= m(s) /  e(s)
For a step or sudden change in deviation we have:
                                                                                    E=0 for t<0
                                                                                    E=e for t>0

Where e is constant
Substitution of this value in equation (i) gives:                              
                                                                          M = kp e+m

Or                                                                       m-m = kp e

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