An Introduction To System Dynamics - First Order Systems
Introduction
Goals
The System
Impulse Response
Problems
You are at Basic Concepts - Time Response - 1st Order Step and Impulse Response

Why Worry About Time Response Behavior?

Time behavior of a system is important.  When you design a system, the time behavior may well be the most important aspect of its' behavior.  Points you might worry about include the following.

• How quickly a system responds is important.  If you have a control system that's controlling a temperature, how long it takes the temperature to reach a new steady state is important.
• Say you're trying to control a temperature, and you want the temperature to be 200o C.  If the temperature goes to 250o C before it settles out, you'll want to know that.  Control systems designers worry about overshoot and how close a system comes to instability.
• If you're trying to control speed of an automobile at 55mph and the speed keeps varying between 50mph and 60mph, your design isn't very good. Oscillations in a system are not usually desirable.
• If you are trying to control any variable, you want to control it accurately, so you will need to be able to predict the steady state in a system.
These examples are intended to show you that the ability to predict details of how a system responds is important when you design systems.  These are but a few of many different aspects of time behavior of a system that are important in control system design.  The examples above really are talking about aspects like:
• Speed of response
• Relative stability of the system
• Stability of the system
When you design systems or circuits you often need to worry about these aspects of the system's time behavior.  Before you build your system, you want to know how it will perform.  You need to make predictions.

In this lesson, we will begin to examine how it is possible to predict aspects of the time behavior of a system.  We'll do that by starting with a first order system and examining the parameters of that kind of system that control their time behavior.  We'll do that in ways that will let us generalize concepts to more complex systems - and there are lots of more complex systems you'll be worrying about.  With that under your belt, you will have the knowledge you need to predict how first order systems behave.  That will set the stage for learning about more complex systems.

Goals For This Lesson

There are a number of goals for you in this lesson.

First, if you have a first order system, you need to be able to predict and understand how it responds to an input, so you need to be able to do this.

Given a first order system,
Determine the impulse and step response of the system.
Secondly, you may go into a lab and measure a system, and if it is first order, you need to be able to do this.
Given the step response of a first order system,
Determine the parameters - DC gain and time constant of the system.
This second goal is considerably different from the first.  In the first goal, you are given information about the system and the input to the system, and have to determine how the system responde.  In the second goal, you are given information about the input and output of a system and have to determine what the system is.  That's a completely different kind of problem, but in both cases you will need to learn the material in the rest of this lesson.  There is a separate lesson on system identification where you look at input and output and work to determine a model for the system.  Click here to go to that lesson.

The System - And Some Examples

The simplest possible dynamic system is one which satisfies a first order, linear, differential equation.  Here's a generic form of the differential equation.  A block diagram representation of the system is given at the right.
tdx/dt + x(t) = Gdcu(t)

x(t)  =  Response of the System,
u(t)  =  Input to the System,
t  =  The System Time Constant,
Gdc  =  The DC Gain of the System.

The parameters you find in a first order system determine aspects of various kinds of responses.  Whether we are talking about impulse response, step response or response to other inputs, we will still find the following relations.
• t, the time constant, will determine how quickly the system moves toward steady state.
• Gdc, the DC gain of the system, will determine the size of steady state response when the input settles out to a constant value.
You also need to note that a system that satisfies the differential equation above has a transfer function of the form:
G1(s) = X(s)/U(s) = Gdc/(st + 1)

Some Examples of First Order Systems

The differential equation describes many different systems of many different types.  We can look at some of the systems whose behavior is described by a first order differential equation like the one above.

The parameters you find in a first order system determine aspects of various kinds of responses. Whether we are talking about impulse response, step response or response to other inputs, we will still have the following quantities and system parameters.

x(t)  =  Response of the System,
u(t)  =  Input to the System,
t  =  The System Time Constant,
Gdc  =  The DC Gain of the System.
Every system will have an input which we can call u(t), and a response we will denote by x(t).  Each system will also have a time constand and a DC gain.
• t, the time constant, will determine how quickly the system moves toward steady state.
• Gdc, the DC gain of the system, will determine the size of steady state response when the input settles out to a constant value.
Now, let's look at some example systems.  The first system is not entirely whimsical.

A Cartoon Biplane

.