Digital Signal Processing Reference
In-Depth Information
It will even maintain the desired water level in the presence of a modest leak in the
tank. Furthermore, the closed-loop system does not require any action by the user.
It responds automatically to the water level.
However, feedback, if improperly applied, can introduce problems. It is well
known that the feedback interconnection of a perfectly stable plant with a perfectly
stable controller can produce an unstable closed-loop system. It is not so well
known, but true, that feedback can increase the sensitivity of the closed-loop system
to disturbances. In order to explain these potential problems it is first necessary to
introduce some fundamentals of feedback controller design.
The first step in designing a feedback control system is to obtain as much
information as possible about the plant, i.e., the system to be controlled. There
are three very different situations that can result from this study of the plant.
First, in many cases very little is known about the plant. This is not uncommon,
especially in the process industry where the plant is often nonlinear, time varying,
very complicated, and where models based on first principles are imprecise and
inaccurate. Good examples of this are the plants that convert crude oil into a variety
of products ranging from gasoline to plastics, those that make paper, those that
process metals from smelting to rolling, those that manufacture semiconductors,
sewage treatment plants, and electric power plants.
Second, in some applications a good non parametric model of the plant is known.
This was the situation in the telephone company in the 1920s and 1930s. Good
frequency responses were known for the linear amplifiers used in transmission lines.
Today, one would say that accurate Bode plots were known. These plots are a precise
and detailed mathematical model of the plant and are sufficient for a good feedback
controller to be designed. They are non parametric because the description consists
of experimental data alone.
Third, it is often possible to obtain a detailed and precise mathematical model of
the plant. For example, good mathematical descriptions of many space satellites can
be derived by means of Newtonian mechanics. Simple measurements then complete
the mathematical description. This is the easiest situation to describe and analyze so
it is discussed in much more detail here. Nonetheless, the constraints and limitations
described below apply to all feedback control systems.
In the general case, where the plant is nonlinear, this description takes the form
of a state-space ordinary differential equation.
x
(
t
)=
f
(
x
(
t
)
,
u
(
t
)
,
d
(
t
))
(1)
y
(
t
)=
g
(
x
(
t
)
,
u
(
t
)
,
d
(
t
))
(2)
z
(
t
)=
h
(
x
(
t
)
,
u
(
t
)
,
d
(
t
))
(3)
In this description,
x
(
t
)
is the
n
-dimensional state vector,
u
(
t
)
is the
m
-
dimensional input vector (the control input),
d
(
t
)
is the exogenous input,
y
(
t
)
is
the
p
-dimensional measured output (available for feedback), and
z
is the vector
of outputs that are not available for feedback. The dimensions of the
d
(
t
)
(
t
)
and
z
(
t
)
vectors will not be needed in this article.
