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:

x(t) = response

C(t) = Control Effort
as a function of time (CE in the block diagram above)
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:

C(t) = K_{p}
* E(t)

C(t) = K_{p}
* (D(t)  x(t))
Then, use this expression
in the original differential equation:

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

t(dx(t)/dt)
+ x(t) = K * K_{p} * (D(t)  x(t))
Then, gather terms in
x(t) on the left hand side of the differential equation.

t(dx(t)/dt)
+ x(t) = K * K_{p} * (D(t)  x(t))

t(dx(t)/dt)
+ x(t)[ 1 + K * K_{p} ]= K * K_{p} * D(t)

[t/[
1 + K * K_{p} ]](dx(t)/dt)
+ x(t)= D(t) * K * K_{p}/[
1 + K * K_{p} ]
This is exactly the same
form
as the original differential equation describing the plant. That's
really a general
differential equation describing a timeconstant system. Since
the differential equation for the whole system (i.e. the closed loop system)
is of the same form, it is also a timeconstant 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.

Closed loop time constant
= t/[
1 + K * K_{p} ]

Closed loop constant (DC
gain? corresponding to K in the plant's equation) = K * K_{p}/[
1 + K * K_{p} ]
Now, you should observe
that "closing the loop" (i.e. implementing proportional control) for a
first order plant causes two effects.

The time constant of the
closed loop system goes down by a factor of [ 1 + K x K_{p}
]. In other words, the closed loop system is faster by the
same factor  taking the time constant as a measure of how fast the system
is.

The constant of proportionality
for steady state behavior (the "closed loop DC gain") is transmogrified
into a form that can be very close to 1.0 if the factor [ K * K_{p}]
is large.