A first order system is described by

Tau is designated the *time constant* of the process.

This model is linear as long as *f(t)* is not a function of x,
thus it can be transformed into a transfer function

To illustrate the step response of a first order system, let's use the example of an isothermal, constant volume, CSTR. The component balance for this process is

Rearrange to a convenient form:

The forcing function in this system is the feed composition. Lets
examine the dynamic response (change in outlet composition with time)
when we make a step change in the feed composition. At all times
prior to the start, the feed composition is zero (C_{A0}=0 for
t<0).

At time zero, the feed composition shifts to some new, constant value. This will make the output composition change. To determine how it changes, we must solve the differential equation component balance. Since this is a straightforward 1st order ODE, it can be solved by the integrating factor method:

Alternately, the solution can be found using the transfer function model:

The plot of the output response has a shape that will become very familiar. It is an example of the "step response" of a 1st order system. All first order systems forced by a step function will have a response of this same shape. The unit step response of a system with time constant 2.0 is shown in the figure. "Unit step response" means that the forcing function (the step) has magnitude 1.0.

As the system approaches steady state, the response approaches a constant value.
In the plot, this value is 2.0; in the equations for the reactor it is
There are two parts to this value -- the natural behavior
of the system is given by the denominator, and the magnitude of the
forcing function by the numerator, C_{A0}.

The initial response (time close to zero) has a slope of 1.0. This is true of all first order systems.

Next, consider what happens to the function when the elapsed time is equal to one time constant.

We can use these known behaviors -- the steady state value, the value
at one time constant -- to *identify* the model of an unknown
first order process from raw process data. The procedure is called
*step testing*.

The response of the system was given by

We will define the *steady state gain*, K_{P}, to be the magnitude of the
change in output at steady state divided by the magnitude of the
change in the input. This gives us:

From this, it should be apparent that a first order system can be completely described by two parameters: the gain and a time constant.

Briefly, let's take a look at the response of the first order process to two additional types of inputs.

First, consider a *ramp function*, C_{A0}=Rt. Then

Next, look at the response to an *impulse function*.

**References:**

- Luyben, W.L. Process Modeling, Simulation and Control for Chemical Engineers (2nd Edition), McGraw-Hill, 1990, pp. 167-171, 177-182.
- Marlin, T.E., Process Control: Designing Processes and Control Systems for Dynamic Performance, McGraw-Hill, 1995, pp. 61-65.
- Riggs, J.B., Chemical Process Control (2nd Edition), Ferret, 2001, pp. 182-85.
- Seborg, D.E., T.F. Edgar, and D.A. Mellichamp, Process Dynamics and Control, John Wiley, 1989, pp. 105-110.

R.M. Price

Original: 10/12/93

Modified: 1/17/97, 5/26/2003

Copyright 1997, 2003 by R.M. Price -- All Rights Reserved