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.

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) = Kp * 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) = Kp * E(t)
• C(t) = Kp * (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 * Kp * (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 * Kp * (D(t) - x(t))
• t(dx(t)/dt) + x(t)[ 1 + K * Kp ]= K * Kp * D(t)
• [t/[ 1 + K * Kp ]](dx(t)/dt) + x(t)= D(t) * K * Kp/[ 1 + K * Kp ]
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 * Kp ]
• Closed loop constant (DC gain? corresponding to K in the plant's equation) = K * Kp/[ 1 + K * Kp ]
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 Kp ].  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 * Kp] is large.