An
Introduction To Robust Control
Introduction
System
Models
Problems
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Why
Robust?
When you design a control system, your ultimate goal is to control a particular
system in a real environment. When you design the control system
you make numerous assumptions about the system and then you describe the
system with some sort of mathematical model. Using a mathematical
model permits you to make predictions about how the system will behave,
and you can use any number of simulation tools and analytical techniques
to make those predictions.
Whenever you make a prediction, your prediction is only as good as the
model you use to make your prediction. However there are always problems
when you model a system.
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There are inaccuracies
in any measurement, so if you measure an input and an output, neither of
those signals are measured with complete accuracy.
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If you fit a model to
input-output data, you know that you may not even know the order of the
system (order of differential equation relating input and output) very
well.
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If you attempt to model
a physical situation, you will probably use a model of a lumped system,
but many systems have distributed characteristics.
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If you are heating a system,
heat flow is described by partial differential equations that you might
simplify to get a reduced order model.
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If you are modelling an
airplane, the airflow is describe by partial differential equations, and
you surely end up using a model that doesn't take all of the dynamics into
consideration.
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And on and on. . .
You can see that your model may have many inaccuracies,
but there are other things that can cause errors and problems in a closed
loop control system.
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There may be disturbances
in the system. Actually, there will almost always be disturbances
in a system.
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One kind of disturbance
might be a change in the dynamics of a system.
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There may be changes in
a system. In an airplane, as a flight goes on, the fuel gets expended
and the weight of the aircraft changes, causing the dynamics of the aircraft
to change.
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Controlling the level
of liquid in a tank might be made more difficult if the density of the
liquid is not constant, or the pressure is not constant.
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There may be external
things which affect a system.
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An aircraft may have a
cross-wind, a tail-wind, etc.
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An electronic system may
have noise that affects the system.
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There may/will be noise
in the measurement of the output of the system - the feedback signal.
All of these possibilities could become reality,
even all within a single system. It's not that unusual. Yet,
your system has to work even when all of these things happen. In
this lesson we will examine some ways of alleviating some of these problems.
System
Models
Some of the problems noted above can be incorporated into the model you
use when you analyze and design your control system. The block diagram
below incorporates some of the items above.
This model incorporates two important problems
that are often encountered.
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A disturbance signal is
added to the control input to the plant. That can account for wind
gusts in airplanes, changes in ambient temperature in ovens, etc.
-
Noise is added to the
sensor output. That accounts for noise an inaccuracies in measurements
of the plant output - the controlled variable.
In both cases, you want to design a system that
minimizes the effect of the disturbance signal and the noise signal.
In order to minimize the effect of the disturbance signal and the noise
signal we will need to determine how the output of the system depends upon
the disturbance signal and the noise signal. We need to do a little
algebra. We know the following:
Y(s) = Gp(s)[D(s)
+ W(s)]
W(s) = Gc(s)E(s)
E(s) = U(s) - [X(s)
+ N(s)]
X(s) = Gs(s)Y(s)
Now, we can back-substitute, starting with
the last two equations. We have:
E(s) = U(s) - [Gs(s)Y(s)
+ N(s)]
Then, we can substitute the expression for
E(s) into the expression for W(s) to obtain:
W(s) = Gc(s)[U(s)
- (Gs(s)Y(s)+ N(s))]
Then, substitute the expression for W(s)
into the expression for Y(s), to get:
Y(s) = Gp(s){D(s)
+ Gc(s)[U(s) - (Gs(s)Y(s) + N(s))]}
Y(s) = Gp(s)D(s)
+ Gp(s)Gc(s)U(s) - Gp(s)Gc(s)Gs(s)Y(s)
- Gp(s)Gc(s)N(s)
Then, collect the Y(s) terms on the left
hand side of the equation to get:
Y(s)[1 + Gp(s)Gc(s)Gs(s)]
= Gp(s)D(s) + Gp(s)Gc(s)U(s)
- Gp(s)Gc(s)N(s)
Y(s) = [Gp(s)D(s)
+ Gp(s)Gc(s)U(s) - Gp(s)Gc(s)N(s)]/[1
+ Gp(s)Gc(s)Gs(s)]
The result says that the output is composed
of three terms.
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A term due to the input,
U(s),
-
Gp(s)Gc(s)U(s)/[1
+ Gp(s)Gc(s)Gs(s)]
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A term due to the disturbance,
D(s),
-
Gp(s)D(s)/[1
+ Gp(s)Gc(s)Gs(s)]
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A term due to the sensor
noise, N(s),
-
-Gp(s)Gc(s)N(s)/[1
+ Gp(s)Gc(s)Gs(s)]
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And, note that all three
of these terms have the same denominator.
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Further note that the
noise and the input are treated the same in this representation except
for the negative sign in the transfer function for the noise.
There
is one quick conclusion that we can draw from this analysis. Consider
the following.
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Since the input and the
noise are essentially treated the same, if the input and the noise "look
like each other" it may be difficult to make the system noise-insensitive.
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However, noise usually
will have more high frequency components than the typical input, while
the typical input may well have mostly low frequency components.
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That implies that you
may well want the closed loop transfer function [Gp(s)Gc(s)/[1
+ Gp(s)Gc(s)Gs(s)]to
be a low-pass filter.
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That might not be hard
to arrange since most plants are low pass filters.
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If the signal is predominantly
low frequency and we want the output to follow the input signal, we will
want the low frequency gain to be large.
These requirements lead
to a frequency response (for Gp(jw)Gc(jw)Gs(jw
)) that
looks generally like the one shown in the figure below.
Note the following:
-
The low frequency gain
has to be high so that the output follows the input signal at low frequencies.
-
There will probably be
a minimum bandwidth crossing because of speed requirements.
-
To filter out sensor noise,
keep the closed loop gain low at higher frequencies.
Adhering to the three rules above will help greatly when designing a system,
but they only address how the system responds to the input and how the
system responds to noise. There is also a disturbance input and,
generally, we want to minimize the effect of that input. Examining
the equation for the output, we see that the output consists of three terms.
Y(s) = [Gp(s)D(s)
+ Gp(s)Gc(s)U(s)
- Gp(s)Gc(s)N(s)]/[1
+ Gp(s)Gc(s)Gs(s)]
If we examine the response
to the input, U(s), we have:
Yinput(s)
= [Gp(s)Gc(s)U(s)]/[1
+ Gp(s)Gc(s)Gs(s)]
Normally, we want the
quantity multiplying U(s) - the closed loop transfer function - to be as
close to 1.0 as possible, at least with the frequency range where the input,
U(s), is large. In many situations, the sensor has a transfer function
that is a constant. For example, a tachometer will produce some fixed
number of volts for every thousand rpm.
Now, examine the response to the disturbance, D(s). Here we have:
Ydisturbance(s)
= [Gp(s)D(s)]/[1
+ Gp(s)Gc(s)Gs(s)]
The only difference
here is the factor Gc(s)
is missing in the numerator of this transfer function. If that is
the case, we should choose the controller to have a transfer function,
Gc(s),
to emphasize the input and de-emphasize the disturbance. In most
cases, that is