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290 IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS—PART C: APPLICATIONS AND REVIEWS, VOL. 29, NO. 1, FEBRUARY 1999
In order to verify the performance of the proposed method,
experiments with the NIST numeral database have been carried out
and the performance of the proposed method has been compared
with that of the previous ISR methods. The experimental results
revealed that the proposed method had much better discrimination
and generalization power than the previous ISR methods.
R
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648–652, June 1996.
Analytical Analysis and Feedback
Linearization Tracking Control of the General
Takagi-Sugeno Fuzzy Dynamic Systems
Hao Ying
Abstract—Takagi–Sugeno (TS) fuzzy modeling technique, a black-box
discrete-time approach for system identification, has widely been used to
model behaviors of complex dynamic systems. Analytical structure of TS
fuzzy models, however, is presently unknown, nor is its possible connec-
tion with the traditional models, causing at least two major problems.
First, the fuzzy models can hardly be utilized to design controllers for
control of the physical systems modeled. Second, there lacks a systematic
technique for designing a controller capable of controlling any given TS
fuzzy model to achieve desired tracking or setpoint control performance.
In this paper, we provide solutions to these problems. First of all, we
have proved that a general class of TS fuzzy models is nonlinear time-
varying Auto-Regressive with the eXtra input (ARX) model. The fuzzy
models in this study are general because they use arbitrary continuous
input fuzzy sets, any types of fuzzy logic AND operators, TS fuzzy rules
with linear consequent and the generalized defuzzifier which contains
the popular centroid defuzzifier as a special case. Furthermore, we have
established a simple necessary and sufficient condition for analytically
determining local stability of the general TS fuzzy dynamic models.
The condition can also be used to analytically check quality of a TS
fuzzy model and invalidate the model if the condition warrants. More
importantly, we have developed a feedback linearization technique for
systematically designing an output tracking controller so that output
of a controlled TS fuzzy system, which may or may not be stable, of
the general class achieves perfect tracking of any bounded time-varying
trajectory. We have investigated stability of the tracking controller and
established a necessary and sufficient condition, in relation to stability of
nonminimum phase systems, for analytically deciding whether a stable
tracking controller can be designed using our method for any given TS
fuzzy system. Three numerical examples are provided to illustrate the
effectiveness and utility of our results and techniques.
Index Terms—AR models, feedback linearization, fuzzy control, fuzzy
modeling, stability.
I. INTRODUCTION
Numerous successful industrial applications have shown the power
of Takagi–Sugeno (TS) fuzzy modeling approach [10], which is a
black-box discrete-time modeling approach developed for modeling
complex dynamic systems [1], [5], [16], [17], [19]. Compared with
the conventional black-box modeling techniques [7] that can only
utilize numerical data, TS modeling approach allows one to take
advantage of both qualitative and quantitative information [8]. This
advantage is practically important and even crucial in many circum-
stances. Qualitative information, such as expert/operator knowledge
and experience about a physical system to be modeled, can readily
be incorporated into TS fuzzy models in the form of fuzzy sets,
fuzzy logic, or fuzzy rules. Virtually all the TS fuzzy models in
the literature use linear functions of input variables as consequent
of the fuzzy rules. Many learning schemes have been developed
to automatically configure one or more components of TS fuzzy
models so that a TS fuzzy model can quickly be established when
qualitative/quantitative information is available [6], [15]. Despite of
Manuscript received November 10, 1996; revised September 17, 1998. This
work was supported in part by Texas Higher Education Coordinating Board
Grant 004952-054.
The author is with the Department of Physiology and Biophysics, Biomed-
ical Engineering Center, The University of Texas Medical Branch, Galveston,
TX 77555-0456 USA.
Publisher Item Identifier S 1094-6977(99)02771-6.
1094–6977/99$10.00 1999 IEEE
IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS—PART C: APPLICATIONS AND REVIEWS, VOL. 29, NO. 2, FEBRUARY 1999 291
success of TS fuzzy systems in practice, analytical study has been
scarce [24], [25], compared with the existing analytical results on
Mamdani fuzzy systems (e.g., [18], [20]–[23]).
TS fuzzy modeling scheme will be more powerful if several major
problems associated with it are overcome. First, almost all the TS
fuzzy models developed so far were used merely for empirically
mimicking measured input-output data sets of the physical systems
modeled. However, a fuzzy model that can mimic a number of
measured input-output data sets does not necessary mean the model
is a valid one. More rigorous methods are needed to ensure model
quality. Yet, there exists no analytical means for theoretically check-
ing quality of a TS fuzzy model and possibly invalidate it. Second,
TS fuzzy models were seldom utilized as control models (the only
exceptions appear to be the studies [11], [12], [14] in which some TS
fuzzy controllers were developed to control one type of TS fuzzy
models. Stability and robustness of the closed-loop fuzzy control
system were the main subjects in these research). This is in sharp
contrast to the conventional black-box models, like Auto-Regressive
(AR), Moving-Average (MA), and Auto-Regressive with the eXtra
input (ARX) models [7], which were developed to facilitate design
of a controller for controlling the physical system modeled. Third,
analytical structure of TS fuzzy models is currently unknown, letting
alone possible connection between the TS fuzzy models and the
conventional models. The fuzzy models have always been treated
and used as black boxes. Finally, there exist no systematic method
that can be used to design a controller to control a given TS fuzzy
model and achieve not only system stability but user-desired tracking
or setpoint control performance.
The objectives of this research were to solve these problems for
a general class of TS fuzzy dynamic systems that use arbitrary
continuous input fuzzy sets, any types of fuzzy logic AND operators,
fuzzy rules with linear consequent and the generalized defuzzifier
which contains the popular centroid defuzzifier as a special case.
II. A G
ENERAL CLASS OF TS FUZZY DYNAMIC SYSTEM MODELS
A TS fuzzy model is composed of input fuzzy sets, fuzzy logic
AND operators, fuzzy rules with linear functions of input variables,
and a defuzzifier. For the general TS fuzzy models studied in this
paper, we denote the
th TS fuzzy rule as being
the total number of the fuzzy rules)
IF is AND is AND
AND is
THEN
(1)
where
and are, respectively, model output and input at
time
a positive integer, represents sampling time where
is sampling period). Here, and are constant parameters.
is a fuzzy set fuzzifying and we denote its membership
function as
which may be any shape but required to
be continuous. We suppose that there are
different fuzzy sets for
fuzzification of
in all the fuzzy rules. Subsequently, there
exist
different combinations of the fuzzy sets
and that many fuzzy rules are needed to cover all the combinations.
To combine the membership values of the fuzzy sets in the rule
antecedent, any types of fuzzy logic AND operators may be used
and different types of AND operators may be used in different rules.
Using
as a symbol to represent an arbitrary type of fuzzy logic
AND operator, the combined membership for
in the rule
consequent in the
th rule is
where
We represent
0by 0
and
in such a case is expressed as
To produce crisp model output, the generalized defuzzifier [3] is
used and the model output is
(2)
where
is a design parameter. Different defuzzifi-
cation strategies can be realized by using different values for
The
popular centroid defuzzifier and mean of maximum defuzzifier are
just two special cases when
1 and respectively [3].
III. A
NALYTICAL STRUCTURE AND LOCAL STABILITY OF
THE
GENERAL TS FUZZY DYNAMIC SYSTEM MODELS
In this section, we first reveal the analytical structure of the above-
defined general TS fuzzy dynamic system models and relate the
resulting structure to ARX model. We then establish a necessary and
sufficient condition for analytically judging local stability of the fuzzy
dynamic system models at the equilibrium point (i.e., origin). Finally,
we show how to use the condition to check and possibly invalidate
a TS fuzzy model.
Theorem 1: The general TS fuzzy dynamic system models are
nonlinear time-varying ARX dynamic models.
Proof: We rewrite (2) as
(3)
where
and
(4)
292 IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS—PART C: APPLICATIONS AND REVIEWS, VOL. 29, NO. 1, FEBRUARY 1999
The result of (3) can be expressed as
(5)
Recall that the linear time-invariant ARX dynamic model [7] is
(6)
where
and are constant parameters and
represents
random error. Comparing (5) with (6), one sees that the general
TS fuzzy dynamic system models are nonlinear time-varying ARX
models without the
term. The nonlinearity and time-variation
of the fuzzy models are due to
and
whose values
are determined by
’s that change with and hence with
time.
This theorem shows that, in the context of traditional dynamic
system modeling, TS fuzzy modeling is indeed a rational and vi-
able way to construct nonlinear time-varying dynamic models. A
significant and unique advantage of the fuzzy modeling approach is
that both qualitative information (e.g., knowledge and experience of
system expert/operator) and quantitative information (e.g., measured
numerical data) can be utilized during modeling. Further, traditional
modeling mainly focuses on linear time-invariant dynamic systems
whereas, as we show here, TS fuzzy modeling scheme can handle
nonlinear time-varying dynamic systems. Therefore, TS fuzzy mod-
eling approach may be more desirable and effective when dealing
with complex systems.
Disclosure of the analytical structure of the general TS fuzzy
dynamic system models makes it possible to theoretically and pre-
cisely investigate various aspects of the fuzzy system models. In
present work, we focused on stability of the fuzzy models, which
characterizes one of the most important aspects of physical systems.
There exist two types of system stability: global stability and local
stability, and one type cannot replace the other as each has its
distinctive advantages and disadvantages. Generally speaking, global
stability conditions for nonlinear systems are, in most cases, sufficient
conditions, and necessary ones are uncommon. Except for linear
systems, it is rare that a global stability condition is a necessary and
sufficient condition. The most widely used methodology for global
stability determination is the one developed by Lyapunov, which
requires a Lyapunov function to be found for the system involved.
Regardless of methodologies, their foremost assumption/requirement
is that the complete and analytical expression of the system is
explicitly available. This is impractical to the general TS fuzzy
systems as their complete structures are usually not analytically
derivable. A TS fuzzy system model is made up of several inter-
related nonlinear components: input fuzzy sets, fuzzy rules, fuzzy
logic AND operators and a defuzzifier. As such, the structures of
most of the fuzzy systems are inherently complex and can be any
nonlinear and time-varying forms, making analytical derivation of the
complete structures for the whole input space virtually impossible.
Aside from the structure availability, even when the assump-
tion/requirement is met, properly determining global stability is still
very difficult or likely impossible. Constructing Lyapunov functions
is more an art than science and heavily involves trial and error. Due to
the structural and parametric complexity of the general fuzzy systems,
finding a proper Lyapunov function for all the systems is practically
impossible.
In view of these difficulties as well as nonlinear and time-varying
nature of the TS fuzzy system models and generality of their
components, we decided to concentrate on local stability. Determining
local stability requires much less information. For the general TS
fuzzy systems, we only need to know:
1) the structure of the fuzzy system around the equilibrium point
(i.e.,
;
2) the linearizability of the fuzzy system at the equilibrium point.
Furthermore, the stability condition that we developed is a necessary
and sufficient one, making it practically useful.
We now establish the local stability condition. Since stability is an
inherent property of a system, it is unrelated to system input. In other
words, stability of fuzzy dynamic system models (3) is determined
by the following nonlinear time-varying difference equation:
(7)
If (7) is linearizable at
0, then Lyapunov’s linearization
method [9] can be utilized to judge local stability of the resulting
linear difference equation, which will provide stability information
about nonlinear systems (7) around the equilibrium point. Local
stability can practically be determined by the following simple
necessary and sufficient condition.
Theorem 2: If nonlinear difference equation (7) for a TS fuzzy
dynamic system model of the general class (3) is linearizable at the
equilibrium point, the fuzzy model is locally stable at the equilibrium
point if and only if its corresponding linearized system
(8)
is stable, where according to (4)
(9)
Proof: If nonlinear difference equation (7) of a TS fuzzy system
model is linearizable at the equilibrium point, that is, if
unique constant for all (10)
then the linearized system model is
Using Lyapunov’s linearization method, the establishment of Theo-
rem 2 immediately follows.
The easiest way to determine stability of (8) is to use the -
transform. That is, (8) is stable if and only if all the roots of the
corresponding
-transform equation
are inside the unit circle. Later in this paper, we will use an unstable
TS fuzzy system model as an example (see Example 1) to show how
easily Theorem 2 can be employed for the determination of local
stability.
In addition to the local stability determination, another use of
Theorem 2 is to qualitatively check quality of a TS fuzzy system
model. If the physical system modeled is known to be stable at the
equilibrium point, the result of applying Theorem 2 to the fuzzy
system model should confirm it. If the confirmation occurs, the model
IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS—PART C: APPLICATIONS AND REVIEWS, VOL. 29, NO. 2, FEBRUARY 1999 293
builder can be more confident about the quality of the fuzzy system
model, at least about its behaviors around the equilibrium point.
Otherwise, the fuzzy system model is incorrect and a new fuzzy
system model needs to be established. This simple qualitative model
verification can practically be important and useful for the TS fuzzy
system modeling technique as there previously did not exist any
analytical means for checking and invalidating a TS fuzzy dynamic
system model. At present, the common practice on validation of
a fuzzy system model is using computer simulation, which is not
only time-consuming but, most importantly, can lead to erroneous
validation because fuzzy systems are nonlinear and time-varying and
no simulation can be comprehensive enough to cover all possible
situations.
System identification and controller design are two closely related
issues in theory and practice of conventional control and modeling
[13]. Knowing the analytical structure of the general TS fuzzy
dynamic system models enabled us to develop a design technique
to systematically design an output tracking controller for them in
achieving perfect tracking of any desired trajectory that is bounded
and time-varying. Without the derivation of the above analytical
structure of the fuzzy dynamic models, it is impossible to develop
the design technique presented below.
IV. S
YSTEMATIC DESIGN OF OUTPUT TRACKING
CONTROLLERS BASED ON FEEDBACK LINEARIZATION
In this section, we develop a systematic controller design tech-
nique for output tracking control of the general TS fuzzy dynamic
systems (3). We assume that (1) the fuzzy system model is a true
representation of the physical system to be controlled; and (2) the
fuzzy system can either be stable or unstable. Our control objective
is to make output of the general TS fuzzy dynamic systems achieve
perfect tracking of any bounded time-varying trajectories. We denote
such a trajectory as r(n). Our another requirement is that the output
of the controller that we design must always be bounded, including
when
The objective of our controller design is to produce
such controller output that
all the time (i.e., perfect
tracking). The principle underlying our design method is feedback
linearization, a well-established nonlinear controller design technique
[4], [9]. The essence of this technique is using feedback to cancel
internal nonlinearities of the system to be controlled and making
the closed-loop control system linear so that linear controller design
techniques can be used.
Note that at time
we know the values of
and we can calculate
the values of
To be general, supposedly we do not know explicit expressions
of
(indeed, one
will not be able to obtain them in many cases). We assume
that
for any meaning output of the fuzzy
systems always depends upon input of the systems. Using feedback
linearization, we choose tracking controller for the general TS fuzzy
dynamic systems (3) as follows:
(11)
Substituting (11) into (5), we obtain the output of the closed-loop
fuzzy control systems
for any
which means that we have achieved perfect tracking. The perfect
tracking always starts from the beginning of the control, i.e., from
time
0.
Practically, whether a controller so designed can achieve perfect
tracking for the actual physical system represented by the fuzzy
model used in the design depends on how accurate the model is. The
perfect tracking can be achieved if the model accurately describes the
physical system. Otherwise, the perfect tracking control performance
will not be guaranteed, owing to incomplete cancellation of the
system nonlinearities. The extent of the performance degradation
relates to the degree of mismatch between the fuzzy model and the
real system. The issue here is about the robustness of the resulting
physical control system. Obviously, this issue is not peculiar to the
control of the fuzzy systems; rather it is a general, difficult and still
open issue to nonlinear system control as whole. It has hardly been
addressed [2].
A controller designed by our feedback linearization technique can
always achieve perfect tracking, starting at time
0, for any given
desired trajectory. However, the controller output may or may not
be bounded, i.e., the controller is not guaranteed to be stable. A
controller is practically meaningless if its output is not bounded,
because such a controller cannot physically be realized. We now study
what determines stability of the controller and under what conditions
the controller is stable or unstable. For better presentation, we divide
general fuzzy systems (3) into two groups: the general TS fuzzy
systems with
0 and the general TS fuzzy systems with
1,
and study the controller stability accordingly.
A. Controller Stability for the General TS Fuzzy
Dynamic Systems with
According to (3), the general TS fuzzy dynamic systems with
0 are described by
(12)
This class of fuzzy dynamic systems is widely used in theory and
practice of fuzzy control and modeling. According to (11), controllers
designed using our method for these fuzzy systems are
(13)
Because a desired trajectory
is always bounded and
for
are bounded, too. Thus,
is always bounded and the controllers are always stable.
B. Controller Stability for the General TS Fuzzy
Dynamic Systems with
1
We now study the controller stability for the general TS fuzzy
dynamic systems with
1 in (3). For this group of fuzzy systems,
if desired trajectory constantly varies, the controller output will,
too. Because of the time-varying and nonlinear nature of the fuzzy
systems, it is difficult to analyze the controller stability if desired
trajectory endlessly changes. A related important question is: if a
desired trajectory does not change forever, say it is only a step
function, will the controller designed be guaranteed always stable?
The answer, as we will show now, is no.
Assume a desired trajectory has a final and fixed position. Our
tracking control task is to make output of the general TS fuzzy
dynamic systems with
1 follow a desired trajectory to reach
a final and fixed position within a finite period of time. One example
of such tracking control is to park a car while another one is to
reach a still object by a robot arm. Without loss of generality, we
294 IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS—PART C: APPLICATIONS AND REVIEWS, VOL. 29, NO. 1, FEBRUARY 1999
assume that the desired trajectory varies with time before time and
becomes unchanged thereafter. Mathematically, a desired trajectory is
described by a time series:
where
is the final fixed position and
when
Since a controller designed using our method always achieves
perfect tracking, output of the fuzzy systems is always
after
time
This means that
when
Additionally, when and
in (11) become constants because
becomes constant (i.e.,
We denote and
as respective values of and
when
Using all these facts, nonlinear time-varying controller (11) becomes
a linear time-invariant controller when
(14)
In order for the controller to be stable (i.e.,
is bounded), all the
roots of the
-transform equation of (14)
(15)
must be inside the unit circle. One sees that whether a controller
is stable depends on
which are the parameter values of
the fuzzy dynamic system to be controlled when
The
controller stability depends not only on the parameters of the fuzzy
system but also on the final fixed position of the desired trajectory,
For the same fuzzy system, it is possible that the controller is
stable for one final position but unstable for another one. We will
show this in Examples 2 and 3 in next section.
If a controller is stable, the controller output corresponding to
designated as can be computed by letting
in (14), which yields
(16)
It can easily be proved that if the denominator of (16) is replaced by
computed will be for the general TS fuzzy systems with
0, meaning (16) contains the steady-state controller output for
those fuzzy systems as a special case. Controller output will reach
and stay at
after time for the fuzzy systems with 0. For
the fuzzy systems with
1, controller output will reach and stay
at
after time
where According to (14), how large
is depends on how stable the controller is, which is determined by
The more stable the controller, the smaller the
Requiring all the roots of (15) be inside the unit circle is equivalent
to requiring the general TS fuzzy dynamic systems be minimum phase
systems when
(note that the fuzzy systems become linear
time-invariant systems when
A discrete-time system
that has open-loop zeros outside the unit circle is a nonminimum
phase system [9]. All the fuzzy systems with
0 are always
minimum phase systems, regardless of the desired trajectory. As
such, controllers designed using our feedback linearization method
are always stable. For any fuzzy system with
1, it belongs to
one of the following three situations:
1) it is a minimum phase system for any value of
;
2) it is a nonminimum phase system for any value of
;
3) it is a minimum phase system for some values of
and is a
nonminimum phase system for the remaining values.
We summarize these controller stability results in the form of theorem
as follows.
Fig. 1. Illustrative membership function definition of the six fuzzy sets used
in Examples 1, 2, and 3. The mathematical definitions are given in (17) and
(18) and the values of the parameters are listed in Table I.
TABLE I
V
ALUES OF THE PARAMETERS IN THE SIX MEMBERSHIP
FUNCTIONS USED IN EXAMPLES 1, 2, AND 3. THE
MATHEMATICAL DEFINITIONS ARE GIVEN IN (17) AND
(18)
Theorem 3: Controller (11) designed for general TS fuzzy dy-
namic systems (3) with
0 is always stable for any bounded
time-varying trajectory. Controller (11) designed for a fuzzy system
with
1 is stable at a given value of if and only if the fuzzy
system is a minimum phase system at that value.
Since the designed fuzzy control system always achieves perfect
tracking, the controller can be regarded stable between time 0 and
If the system satisfies Theorem 3, it is stable for the rest of
the time. Therefore, the fuzzy control system is stable in a global
sense, not in a local sense (i.e., around the origin only).
For any given fuzzy system with
1, before utilizing our design
method, one should use (15) to check stability of the controller to
be designed. If the controller is determined to be stable, then design
it. Otherwise, the desired perfect tracking is not achievable for the
given fuzzy system model because the stability condition stated by
Theorem 3 is a necessary and sufficient one.
V. N
UMERICAL EXAMPLES
We now demonstrate three examples that are related to each
other to show how to use our new results and methods. The first
example displays how to use Theorem 2 to analytically determine
local stability of a TS fuzzy dynamic system model. We purposely
use an unstable fuzzy system.
Example 1: Suppose that we have identified a physical system
using the TS fuzzy modeling technique. Assume the resulting TS
IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS—PART C: APPLICATIONS AND REVIEWS, VOL. 29, NO. 2, FEBRUARY 1999 295
fuzzy system model has the following eight TS fuzzy rules:
IF is AND is AND
is
THEN
IF is AND
is AND
is
THEN
IF is AND
is AND
is
THEN
IF is AND is AND
is
THEN
IF is AND is AND
is
THEN
IF is AND is AND
is
THEN
IF is AND is AND
is
THEN
IF is AND is AND
is
THEN
Here, and 0, 1, 2 and stands for “Positive” whereas
stands for “Negative”) are six fuzzy sets (Fig. 1 illustrates the
definitions). The membership functions of
are described by
(17)
whereas the membership functions of
are defined by
(18)
where
is or
The other parameters define
the shape of the membership functions and their values are listed in
Table I. Product AND fuzzy logic is used for all the AND’s in the
rules. Also, the popular centroid defuzzifier is used (i.e.,
1).
The question is: is this TS fuzzy dynamic system model stable
around
0?
Solution: According to Theorem 1, this TS fuzzy dynamic system
model is a nonlinear time-varying ARX system
In order to obtain the corresponding linearized system
we first need to determine whether the nonlinear system is linearizable
at
0. For this specific system, we can find it out without the
explicit expressions of
and The nonlinear system
is linearizable because the conditions expressed in (10) hold. This is
due to:
1) all the membership functions for
and
are differentiable at 0;
2)
yielded by product AND fuzzy logic are differen-
tiable at
0.
As a result,
and are differentiable at 0.
The values of
and can easily be computed using (4), and
the resulting linearized system at
0is
(19)
The corresponding
-transform equation is
The three roots are 0.9342 and The
last two roots are outside the unit circle. Hence, the given TS fuzzy
dynamic system is unstable at
0. Fig. 2 shows the system
output when a very small initial value is given (
0.0001). The
system output diverges with time, clearly demonstrating instability of
the system and confirming our analytical result.
In the second example below, we exhibit how to use our feedback
linearization design method to design a stable controller for the
unstable TS fuzzy system given in Example 1 and achieve perfect
output tracking performance.
Example 2: Using the feedback linearization method presented in
this paper, design a tracking controller for the TS fuzzy dynamic
system in Example 1 so that the output of the fuzzy system perfectly
follows the following trajectory (Fig. 3):
Is the controller so designed stable?
296 IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS—PART C: APPLICATIONS AND REVIEWS, VOL. 29, NO. 1, FEBRUARY 1999
Fig. 2. Simulated output of the TS fuzzy dynamic system given in Example
1, confirming local instability of the fuzzy system, which is determined
analytically by our necessary and sufficient stability condition. The initial
system output is set 0.0001.
Fig. 3. Output of the unstable TS fuzzy dynamic system controlled by an
output tracking controller in Example 2, which is designed using our feedback
linearization technique. Sign
represents the desired output trajectory whereas
sign
represents the fuzzy system output. The figure shows that perfect
tracking is achieved. Note that the final fixed position of the desired trajectory,
is 0.4.
Solution: Before designing the controller, we should use Theorem
3 to determine whether the controller to be designed will be stable.
According to the given desired trajectory, the final fixed position
is:
0.4. According to (15), the -transform equation for the
controller stability determination is
whose root is
It can be calculated easily from the given TS fuzzy system that
and and hence the root is
0.8622, indicating that the fuzzy system is a minimum phase
Fig. 4. Output of the output tracking controller designed using our feedback
linearization technique in Example 2. The controller is stable, confirming the
result of the analytical determination. The steady-state output of the controller
is 1.8461, the same as the value computed using (16).
system when 0.4, for Thus, the tracking
controller to be designed will be stable. The tracking controller is
According to (16), the steady-state output of the designed controller
at
is
From the given fuzzy system, we compute the values of
and as:
and 1.2623. Consequently,
1.8461. Fig. 3 displays the system output along with the
desired trajectory. The trajectory is always perfectly tracked. The
corresponding controller output is exhibited in Fig. 4. The controller
is stable and indeed the steady-state output is 1.8461, as expected.
In the last example, we show that the controller designed in
Example 2 becomes unstable for the same fuzzy system at a different
value of
Example 3: In Example 2, if the final fixed position of the desired
trajectory is 0.7 instead of 0.4, for 51
100, will the designed
controller still be stable?
Solution: Now
0.7. One can calculate that
1.2081 and 1.4867, and hence the root is 1.2307
(outside the unit circle). This means that the fuzzy system becomes a
nonminimum phase system when
0.7 and consequently
the designed controller becomes unstable for the new final position
of the trajectory. Although the perfect tracking is still achieved, as
shown in Fig. 5, the controller output grows without bound and the
controller is unusable (Fig. 6), as predicted.
IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS—PART C: APPLICATIONS AND REVIEWS, VOL. 29, NO. 2, FEBRUARY 1999 297
Fig. 5. Output of the unstable TS fuzzy dynamic system controlled by an
output tracking controller in Example 3, which is designed in Example 2 using
our feedback linearization technique. Sign
represents the desired output
trajectory whereas sign
represents the system output. The figure shows that
perfect tracking is achieved. Note that the final fixed position of the desired
trajectory,
is 0.7, instead of 0.4 shown in Fig. 3 for Example 2.
Fig. 6. Output of the output tracking controller in Example 3. Because of the
change of the final position of the desired trajectory from 0.4 in Example 2 to
0.7 in Example 3, the controller becomes unstable, as predicted by using (15).
VI. CONCLUSION
We have proved that a general class of TS fuzzy dynamic systems
is nonlinear time-varying ARX systems and have established a simple
necessary and sufficient condition for analytically determining local
stability of the fuzzy systems. The condition can also be used to check
quality of a TS fuzzy model against the physical system modeled and
invalidate the model if the conditions warrant. Based on the revealed
structure of the fuzzy systems, we have developed a feedback
linearization method for systematically designing a controller to
control any given TS fuzzy system of the general class, stable or not,
so that perfect output tracking is obtained. Our design method always
produces stable controllers for a large portion of the general TS fuzzy
systems that are commonly encountered [those with
0 in (3)],
regardless of desired trajectories as long as they are bounded. For the
remaining TS fuzzy systems [those with
1 in (3)], we have proved
that whether the designed controller is stable depends on the fuzzy
system to be controlled as well as the desired trajectory. We have
derived a simple and practical necessary and sufficient condition, in
relation to stability of nonminimum phase systems, for analytically
determining the controller stability. We have given three concrete
numerical examples to demonstrate practicality and utility of our
new results.
The results obtained in this paper cover a very general class of
TS fuzzy dynamic systems, which are actually, as we have shown,
nonlinear and time-varying ARX systems. Our results are not only
unique and hence valuable to fuzzy systems but also useful to the
conventional studies of output tracking control of nonlinear time-
varying systems.
R
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A Color Texture Based Visual Monitoring
System For Automated Surveillance
George Paschos and Kimon P. Valavanis
Abstract—This paper describes a visual monitoring system that per-
forms scene segmentation based on color and texture information. Color
information is combined with texture and corresponding segmentation
algorithms are developed to detect and measure changes (loss/gain) in
a given scene or environment over a period of time. The
color
space is used to represent the color information. The two chromaticity
coordinates
are combined into one, thus, providing the chrominance
(spectral) part of the image, while
describes the luminance (intensity)
information. The proposed color texture segmentation system processes
luminance and chrominance separately. Luminance is processed in three
stages: filtering, smoothing, and boundary detection. Chrominance is pro-
cessed in two stages: histogram multi-thresholding, and region growing.
Two or more images may be combined at the end in order to detect scene
changes, using logical pixel operators. As a case study, the methodology
is used to determine wetlands loss/gain. For comparison purposes, results
in both the
and
color spaces are presented.
I. INTRODUCTION
Texture has been widely accepted as a feature of primary impor-
tance in image processing and computer vision since it provides
unique information about the physical characteristics of surfaces,
objects, and scenes [1], [2]. An image may represent a specific
textural pattern, while in other cases, an image may be composed
of two or more textural patterns. In the first case, the problem
encountered is that of classification, since a single texture has to
be recognized. In the second case, one has to separate the different
textures from each other within a single image, thus, performing an
image segmentation task.
There has been considerable research in the area of texture analysis
(i.e., description, segmentation, classification) [3]–[5]. However, most
of the work has focused on methods using gray-level images, where
only the luminance (intensity) component of the image signal is
utilized. Only limited work has been reported in the literature related
to the use of color in texture analysis [6], [7]. In order to incorporate
the chromatic information into texture analysis, assuming that the
RGB color space is used, the following choices exist.
1) Each color band (i.e.,
is processed separately.
2) Information across different bands (e.g., cross-correlations
,
, ) is extracted.
3) Both individual color band and cross-band information is used.
Manuscript received September 9, 1996; revised May 10, 1998.
G. Paschos is with the Computer Science Division, Florida Memorial
College, Miami, FL 33054 USA.
K. P. Valavanis is with the Robotics and Automation Laboratory, A-CIM
Center, University of Southwestern Louisiana, Lafayette, LA 70504 USA.
Publisher Item Identifier S 1094-6977(99)02770-4.
4) A composite measure to describe the chromatic information is
used.
Methods based on one of the first three choices have been recently
reported [6], [7]. The fourth alternative is explored in this research
using the
color space [8]. The proposed Color Texture Analysis
System is shown in Fig. 1. The main goal of the system is to separate
a given image into two parts, namely, a Region of Interest (ROI),
and the rest of the image (i.e., the background). A ROI is typically
an area of the image that represents something meaningful in the
corresponding real-world scene. For example, an aerial image may
capture a piece of land surrounded by water. The land, in this case,
is the ROI, and the surrounding water is the background. The system
performs analysis on luminance and chrominance in parallel, and, at
the final stage, results are combined to detect changes (i.e., loss/gain)
in a specific area of the image (ROI).
Processing starts by transforming a given image from
to
(Fig. 1). This produces the luminance component
directly,
whereas the two chromaticity values
are combined to provide
for a single-valued chrominance. Textural information, such as sizes
and orientations of basic image features (e.g., edges, blobs), is
contained in the luminance component. Thus, a set of filters tuned to
different sizes and orientations is applied on luminance and produces
a corresponding set of filtered images. Smoothing of the filtered
images follows, thus, eliminating spurious/negligible regions. The
smoothed images are combined into a single image, based on a
neighborhood pixel similarity measure, and boundaries of potential
ROI’s are extracted using a perceptron-type processing mechanism.
The result of luminance processing is, thus, a Boundary Image.
Crominance processing proceeds in two stages. First, the chromi-
nance histogram is computed and multiple thresholds are identified.
Secondly, these thresholds are used to segment the chrominance
image into a corresponding number of regions (i.e., potential ROI’s).
Thus, the result of chrominance processing is a Region Image. Using
a region expansion algorithm, the Boundary and Region Images are
combined to locate the desired Region of Interest (e.g., wetland area).
The result is a ROI Image showing the identified ROI.
The final stage involves the comparison of two or more ROI images
to locate possible scene changes. Typically, two or more images of
the same real-world scene are taken at different times. Each of these
images will result in a corresponding ROI Image, after going through
the various segmentation stages (i.e., luminance and chrominance
processing). Change detection and measurement is performed by
comparing two such ROI Images using logical pixel operators.
The end result of this research is threefold:
1) incorporation of texture and color attributes for scene analysis;
2) development of computationally efficient and easily imple-
mentable algorithms for the analysis of color textures;
3) development of appropriate neural network architectures for
image segmentation and classification.
One of the main applications of the proposed system is in the
monitoring of wetlands. Such environments experience changes over
time (i.e., partial loss/gain of wetland area). The development of
autonomous surveillance systems capable of collecting data over a
period of time and analyzing them using a variety of visual properties
in order to identify such changes is, thus, important. The methodology
presented in this paper provides the analysis component of such an
autonomous system. It incorporates color and texture visual attributes
into a unified framework and utilizes them to detect and measure
loss/gain.
1094–6977/99$10.00 1999 IEEE
