Control Systems Intro - Time Constants in First Order Systems.

        If you have a first order system in a closed loop control system, you can find the time constant of the closed loop system knowing the time constant of the system being controlled - the plant - and the constants in the system.  Here is a block diagram of the system.

We start with the equation describing the plant+ actuator.

t(dx(t)/dt) + x(t) = K * C(t)

       In this situation, the variables are:

Now, we also know that the control effort is proportional to the error. Using the expression for the error (E(t) = D(t) - x(t), where D(t) is the desired response, and x(t) is the response), we have: Then, use this expression in the original differential equation: Then, gather terms in x(t) on the left hand side of the differential equation. This is exactly the same form as the original differential equation describing the plant.  That's really a general differential equation describing a time-constant system.  Since the differential equation for the whole system (i.e. the closed loop system) is of the same form, it is also a time-constant system.  However, the closed loop system does not have the same parameters as the plant.  The new parameters (the closed loop parameters) are given by the expressions below. Now, you should observe that "closing the loop" (i.e. implementing proportional control) for a first order plant causes two effects.