An Introduction to Control Systems - Proportional Control

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.

C(t) = K_p * E(t)

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 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.

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.