You are at Basic Concepts - Time
Response - 1st Order Step and Impulse Response
Return
to Table of Contents 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 200^{o}
C. If the temperature goes to 250^{o}
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) = G_{dc}u(t)
x(t) =
Response of the System,
u(t)
= Input to the System,
t
= The System Time Constant,
G_{dc}
= 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.
G_{dc},
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)
= G_{dc}/(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,
G_{dc}
= 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.
G_{dc},
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.