## Response of 1st Order Systems

A first order system is described by

In this model, x represents the measured and controlled output variable and f(t) the input function. (The forcing function of the ODE.) The equation is often rearranged to the form

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

This type of transfer function is known as a first order lag with a steady state gain of 1.0.

### Step Response

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

First order kinetics are used for the rate of disappearance of A.

Rearrange to a convenient form:

and it becomes clear that the time constant for the system is:
or in terms of the residence time
The rearranged ODE model of the system is then:
This can be shifted to deviation variables, Laplace transformed, and converted to transfer function 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 (CA0=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, CA0.

### Initial Response

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

### Time Constant

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

Thus, when one time constant has elapsed, the process output will have achieved 63.2% of its final value (in the plot, 1.26).

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.

### Gain

The response of the system was given by

It can be observed that when t=0, CA=0, and as t approaches infinity,

We will define the steady state gain, KP, 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:

so the response equation can be rewritten as
If we observe that,
then the original differential equation can be shown to be
a useful, general form.

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

### Ramp and Impulse Response

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, CA0=Rt. Then

Next, look at the response to an impulse function.

Notice that the impulse response is the derivative of the step response. In some cases, it is easier to find the impulse response function by taking the derivative of the step response than by integrating the impulse forms.

References:

1. Luyben, W.L. Process Modeling, Simulation and Control for Chemical Engineers (2nd Edition), McGraw-Hill, 1990, pp. 167-171, 177-182.
2. Marlin, T.E., Process Control: Designing Processes and Control Systems for Dynamic Performance, McGraw-Hill, 1995, pp. 61-65.
3. Riggs, J.B., Chemical Process Control (2nd Edition), Ferret, 2001, pp. 182-85.
4. 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