Proceedingsof the American Control Conference
Arlington, VA June 25-27,2001
Fuzzy Segregation based Identification and Control of Nonlinear Dynamic Systems
Aswin N. Venkat and Ravindra D. Gudi’
Department of Chemical Engineering, Indian Instituteof Technology, Bombay, Powai, Mumbai- 400076, India
Abstract : In this work, we propose a new method to model
and control complex nonlinear dynamic systems. The
suggested scheme employs an output curve methodology
to determine the initial set of dynamic clustering spaces.
The choice of the optimal dynamic clustering space is
made through an analysis of cross validation performance
and other indicative indices. A fuzzy clustering
methodology for dynamic model building is proposed. For
online control, a smooth internal model-switching strategy
based on fuzzy methods is proposed and shown to be
superior to other methods hitherto proposed in literature.
Two control structures based on the proposed
methodology are discussed. The efficacy of the proposed
fuzzy modeling and control schemes are demonstrated
through illustrative examples and by application to a high
purity distillationprocess.
1. INTRODUCTION
Chemical processesare invariablynonlinear. The
complex nature of such processes makes them difficult to
model and control. Non-linearities make it challenging to
develop a model or design a controller that performs
satisfactorily over the entire operating range. Linear
models/controllers are valid only around a small range of
operation. This often leads to complications especially
when operating in a different regime, as is the case during
startup or shutdown, grade changeovers in polymer
reactors and alteration in dynamics due to catalyst
deactivation.
Two approaches maybe employed to tackle these
problems. The first is the development of an overall
nonlinear model that performs sufficiently well over the
entire operating range. The other option would be to
devise a regimewise modeling strategy employing local
models. The former approach is often complicated in
terms of the difficulty in arriving at a suitable model. Such
a model is also invariably complex. The inherent
challenges in the latter approach, namely a multiple-model
based strategy, are in devising techniques for (i) Division
of the operating range into local regions (ii) Constructionof
local models, and (iii) Switching between models.
From the standpoint of model accuracy
/relevance in an operating region, ease of model switching
is of paramount importance in local modeling approaches.
Traditional methods in literature have looked to employ
scheduling algorithms to switch between local
models/controllers in such a manner that only the most
representativemodel/controller is active at any given time.
Such a hard switching strategy leads to poor transient
performance. Some more effective switchinglscheduling
approaches have been developed in recent times. A
multiple model adaptive control approach employing an
estimator based scheduling algorithm to weigh the local
controllers was proposedby Schott and Bequette ([18]) for
control of chemical reactors. Local model performance
indices to select local controllers have been employed by
Narendra et. a/. ([is]). A supervisor based technique for
local controller selection based on virtual control loop
feedback error has been developed by Kordon et. a/.
([IZ]). This approach assumes the availability of the
knowledge of the various operating regions -their centers
and their ranges. Wang, Tanaka and Griffin ([Z])
have
discussed model stability issues for Takagi-Sugeno fuzzy
models and have undertaken controller design using a
parallel-distributed compensation scheme.
In this work we have employed a fuzzy clustering
based strategy for decomposition of the operating space.
To account for the fact that the segregation of the data is
not crisp, we employ fuzzy clustering algorithms which by
assigning memberships to the data points or sets, allow
them to simultaneously belong to more than one cluster.
The advantage of the approach lies in the relatively
minimum apriori process knowledge essential for
implementation of the proposed technique. In the present
work, no assumption of local model homogeneity is made
and the proposed methodology is capable of handling
disparate local models as well. The structural identification
task here comprises of (i) selection of the appropriate
dynamic space structure for segregation into local
operating regions, and (ii) determination of an appropriate
local model structure for each of the local spaces. To
generate an initial estimate of the lag space in which the
classification is to be performed, we propose the use of an
output curve methodology as an initial guess.
Determinationof an optimal number of clusters, as well as
refining of the lag space generated by the output curve
method is then proposed. Identificationof local models for
each cluster is achieved based on concepts of overall
performanceand local model parsimony. It is shown that a
composite controller based on the fuzzy predictions
proposed above, facilitates the smooth switching of the
model used in the controllerand thus ensures good closed
loop performance over the entire range of operation. The
effectiveness of the proposed methodology is highlighted
through case studies. Two alternate control structures
based on the composite models have been proposed and
evaluated for the control of top product composition in the
high purity distillation column of Skogestad ([19]).
The paper is organized as follows. In section 2, a
brief overview of fuzzy clustering is provided and an
approximate methodology for
dynamic
space
determination is outlined. A detailed discussion on the
proposed local model based dynamic modeling
methodology is provided in section 3. In section 4, two
multiple model based control schemes- one a multiple
local internal model control scheme and the other a
composite model MPC strategy are proposed for handling
such systems. The effectiveness of the proposed
technique in modeling nonlinear dynamic systems is
demonstratedthrough three representativecase studies in
section 5. In section 6, control performance of both the
proposed schemes (outlined in section 4) is demonstrated
on the high punty distillation column of Skogestad ([19]),A
’ CorrespondingAuthor, E-mail: [email protected]
0-7803-6495-3/01/$10.00 0 2001 AACC
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discussion on the proposed technique and scope for future
work is developed in section 7.
2. Fuzzy Clustering
Fuzzy clustering has been extensively employed
for classification problems especially in pattern recognition
literature ([3],[ I I ] ) . Clustering enables division of the
complex nonlinear spaces into elementary subspaces,
which could then be modeled by simple elementary
models. Incorporation of fuzziness into the clustering
methodology allows for overlap of subspaces and thereby
smoother transitions between operating regions.
Let c be the number of clusters and m be the
fuzzy exponent (defined later), where m > 1. Partitioning
of the data, in the n-dimensional space, into each of these
c clusters would result in feature points that have similar
relationship (dynamic in our case) to lie closer to each
other in a cluster. Let bj therefore denote the membership
of the j* feature point in x to the i* cluster and let v k
denote the prototype point for the km cluster. A fuzzy
partition is characterized by the following equations:
N
0 c x p u < N
t / i = 1 , 2 ,......._.. . . , e
j=l
This partitioning involves the minimization of the fuzzy
objective function J.
I=1
J=1
The minimization of J yields a partition of the data with VI,
i=1,..., c as the cluster centers. It also produces a matrix
of membershipsof each feature vector to each cluster.
A number of fuzzy clustering algorithms are
available in literature ( [ I ] ,[2]).We have employed the
Gustafson-Kessel algorithm ([4]) ( G K ) for adaptive norm
based segregation. The alternate fixed norm based
clustering forces the clusters into characteristic norm
based topographies, even if such a structure is not present
in the data.
The first step in the f u n y clustering procedurefor
dynamic systems is the determination of an appropriate
dynamic lag space in which the clustering has to be
carried out. Most literature on fuzzy clustering pertains to
clustering of static systems or dynamic systems whose lag
space is assumed to be known. Cluster based techniques
utilizing static representations of systems have been
proposed in [4] and [8].In an actual modelingenvironment
such knowledge is rarely available. In the following
section, we first propose a methodology to ascertain the
appropriate lag space and in the succeeding sections
develop a cluster-based method for modeling dynamic
systems.
2.1 Clustering spaces for dynamic systems
Consider X" as the domain of the universe of discourse of
the input variable set PI, X2, .........., &}
X" = XI x x2 x ... ........x x,
For a static system of dimension (m x 1) (multiple input
single output), the attempt is to build a model A relating
the mdimensional input space X"
and the unit
dimensional output space Y such that
A: X" 3 Y
There must clearly be a relationship between data in Xmx
Y. Clustering in this space is therefore adequate to
develop the relation. For a dynamic system with 'm'
external inputs, the number of actual variables affecting
the process could be different from m. Clustering in the X"'
x Y space cannot satisfactorily capture all the inherent
relations. A dynamic model C2 therefore has to relate data
in the space Xpx Y such that
n: xp+ Y
where P denotes the number of past inputs and outputs
significantly affecting the process.
The major problem employing cluster-based
techniques to handle dynamic systems is the
determination of this space, which we term the 'Dynamic
Clustering Space' or DCS in short. Clustering in this
extended input-output space is necessary to represent the
association between the input and the output variables.
Two techniques to aid in the determination of the DCS are
proposed in this work The first methodology employs an
output curve approach to determine the possible DCS
structures (section 2.2). This technique is approximate and
hence a second methodology for accurate determination
of the DCS is developed (section 3.1).
2.2 Determination of DCS employing output curves
The output curve methodology as proposed in
this work, is an iterative scheme for structure
determination that is based along the lines of the fuzzy
curve approach of Lin and Cunningham 111 ([14]). It is a
visual/graphical technique that could be employed to
determine those variables with significant influence on the
process. The technique as proposed is approximate and
serves only as a guideline to reduce the combinatorial
space of the possible DCS. The method is based on the
evaluation of scatter plots of the measured output and the
linear model predictions. For nonlinear systems, this could
be a curve as well. This m&hod i m l v e s the determinatbn
of the set(s) of variables that result in minimum deviatim
of the (linear or nonlinear) scatter pM from the 45-degree
line. Such a set@) constitutes a candidate space for
clustering. Finetuning fm the optimal DCS is then cam&
aut mpicyring the "dproposed in Section 3.1. The
i m p b "
strategy dm the output cuwe mefhod is
discussed in detail in Table 1
The modeling step in this procedure is linear.
Hence for nonlinear systems, it is not possible to exactly
match the span of the measuredand predicted outputs. In
such a case, the output curve that spans the maximum
range for the minimum number of inputs is a good choice
for the DCS.
3 Cluster based dynamic modeling approach
3.1 Cluster based dynamic modeling
As mentioned in section 2.1, the first task for the
con. of kxaldynamicmoddsinvolves sekkmd
a suitabfe dustsing space.The quality ob the local model
obtained d e s an the nature o
f mapping within the
cluster and it is therefore important to cluster in the correct
dynamic clustering space. Funy clustering employing the
G-K algorithm provides a technique for adaptive nonuniform segregation of the operating space. It is evident
that successive decomposition of the DCS would yield
more accurate local models but at the cost of increased
overall complexity. We shall consider, in this work, the
concepts of model parsimony and stability in basing our
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choice of local models. The DCS based local modeling
strategy that we propose involves the following subtasks:
(A) Identification of the different operating regimes and
division of the dynamic space into localregions
The output curve methodology provides the
reduced set of possible candidate DCS. The division of the
operating regime requires the knowledgeof (i)the choice
of the DCS, and (ii) the number.of clusters necessary.
Determinationof the optimal numberof divisions is carried
out through evaluation of cluster validity measures for the
set of possible divisions. The determination of the optimal
number of clusters is important to maintain composite
model parsimony, In this work, two cluster validity
measures namely Fuzzy Hypervolume (Gath and Geva,
[SI) and Average within cluster distance (Krishnapuram
and Freg, [13]) are employed to identify the optimal
number of clusters.
(6)Selection of suitable dynamic model structures for the
localmodels and estimation of localmodel parameters
The procedure in (A) yields IC'clusters along with
the corresponding partition matrix for each of the clusters.
We then seek to build local dynamic modelswithin each of
the clusters. Local model structures are selected with a
view to maintain model parsimony. In addition, these local
models identified should be (a) stable, and (b) provide
good overall performance. Cluster wise regression
analysis (Yang and KO, [23]) is employed for local model
parameter estimation. For the purposes of performance
evaluation of the local models, we have employed
measures of final prediction error and analysis of the cross
validation errors obtained.
(C) Suitable combination of local model outputs to obtain
aggregated outputs.
The strategy proposed for handling transients
involves computation of the cluster wise memberships for
each new data point and weighting the local model output
by the funy cluster-wise membership. Direct use of this
technique for prediction is not possible as at any instant
'k', out of the 'n' dimensional mapping within each cluster,
only a maximum of 'n-I' dimensions are known. Two
strategies to overcome this problem are considered. The
first involves the projection of the 'n' dimensional cluster
onto the known ' P dimensional space. Gomez-Skarmeta
et. al. ([8]) have employed a similar technique to obtain
model predictions. In our work we have employed the
probabilistic estimate for the cluster membership. The
membership of a point X, to cluster 'i' is obtained as
An alternate technique proposed in this work, for
chemical systems, assumes that the instantaneous
change in the dynamics is small. Therefore, the variation
in memberships between two successive points is
negligible and hence, the previ6os witMdOw of data could
be utilized to compute the current weighPing factors. This
technique is henceforth referred to in this work as the
'Method of Quasi-Invariance'. A comparison of either
method is provided in case study 3. The overall prediction
is obtained as a membership weighted average of the
local model outputs
4. Control
ControlStrategy 7: Multiple controller based IMC
A widely employed strategy for control of
chemical systems is the IMC scheme (Morari and Zafiriou,
[15]). A strategy to handle nonlinear systems using the
IMC framework was outlined by Henson and Seborg ([IO]).
In the present work, a modified form of the IMC scheme
employing multiple local controllers, models and filters is
proposed. A schematicfor the contrd scheme is shown in
Figure 1 Each duster presents a l o d dynamic mapping
between the input and the miput variables. The focal
mapping implies an elementary nature for this relationship
within each cluster. This enables the existence an analytic
inverse for each local mapping. Filters are added to each
such local controllers to ensure causality. The filters are
tuned for optimal performance. The extent of influence of
each local controller is estimated based on the smooth
internal switching mechanism employed earlier.
Control Strategy 2: Multi-model MPC
MPC has been widely employed for control. An
introduction to linear DMC based control is provided by
Garcia et. al. ([5]). A number of applications involving the
use of nonlinear models within the MPC framework are
available in literature ([17], [20]). In this work we employ
cluster based local models within the MPC framework to
achieve control. The standard model is replaced by a
collection of suitable cluster based local dynamic models
aggregated using an appropriate smooth internal switching
strategy.
5. Illustrative Case Studies
Case Study 7: Output Curve Methodologyand Fuzzy
modeling
Consider a system of the form
y(t+l) = a [y(t)lo3+ b [u(t)lo2+ c y(t)u(t) + noise
where, a=0.42, b=0.65 and c=0.5
The output curve methodologyis employedto estimate the
candidate dynamic clustering spaces. The first variable
u(t) is considered and a scatter plot between the
measured and predicted output (ypr~(t+l)=Fu(t))IS
constructed. The plot (Figure 2(i)) shows considerable
deviation from the 45' line. A second variable u(t-I) is
added and a new scatter plot is constructed. The curve
though an improvement over the previous output curve, is
still unsatisfactoryand shows only a marginal shift towards
the 45" line. This variable is therefore discarded. Addition
of the next variable y(t) yields a scatter plot that is close to
'the 45" line (Figure 2(ii)). This space i.e. y(t+l) x y(t)x u(t)
is therefore a candidate DCS. Introduction of an extra lag
in each variable also yields an output curve that is very
close to the 45' line. The space y(t+l) x y(t) x u(t) x y(t-I) x
u(t-I) is another candidate DCS. Addition of a bilinear term
y(t)u(t) yields a scatter plot that is again very close to the
45'line. This space i.e. y(t+l)x y(t)x u(t)x y(t)u(t) is also a
possible DCS. It is noticeablethat the deviation introduced
by the addition of the extra lag or the bilinear term to the
variables {y(t+l): y(t), u(t)} is small. Therefore, the search
is terminated at this point and the candidate DCS are
considered for further analysis. From model simplicity and
parsimony considerations, the DCS of y(t) x u(t) x y(t+l)
appears to be a g w d starting point.
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An evaluation of the same system employing the
cluster based modeling methodology gives similar results.
The final prediction error and average cross validation
error for each case is shown in Table 2. It is observed that
the same three structures mentioned earlier give superior
performance as compared to other choices of the DCS
and the final decision of the choice of the DCS ((ii) in this
case) was based on minimum cross validation error
(minimum final prediction error). The cross validation
results obtained in this case are shown in Figure 2(iii)
Case Study 2: Fuzzy modeling of a Bilinear Plant
A bilinear plant of the form y(t+l) = a u(t) + b y(t) + c y(t)
u(t) + E is considered, where E is a noise term with a SNR
of 0.9.The values for the parameters a, b and c are 0.42,
0.65and 0.5 respectively. For the purposes of generality it
is assumed that no knowledge of the possible regimes is
available to adequate levels of confidence. The first task
involved here is in the determination of the DCS. The
output curve approach is used to determine most likely
candidate DCS structures. Based on this, preliminary
investigation, three candidate dynamic clustering spaces
are considered for further evaluation. The optimal number
of clusters for each choice of DCS is determined
employing cluster validity measures. Local models
satisfying conditions of simplicity and stability are
constructed for each cluster and aggregated employing
the methods discussed earlier. Cross validation
performance and final prediction errors are utilized as
measures for the final selection of the DCS. Table 3
provides the listing of parameter values obtained. Based
on the performance obtained, the y(t+l) x y(t) x u(t) space
is selected as the DCS. Local modeling in each of the
clusters leads to four linear models. The gain and time
constants of each cluster-based model are provided in
Table 4.
Case Study 3: High Purity Distillation Column (U)
In this case study, we consider the high purity
distillation column of [IQ].
The reflux rate is used to control
the top product composition. The system is highly
nonlinear and the construction of a global nonlinear model
is quite a complex procedure. Since the method proposed
is essentially data based, the first step involved here is the
generation of rich process data. Owing to nonlinear
characteristics, persistent excitation is difficult to ensure.
The signal design ( [ 7 ] ) is therefore based on the
characteristics of the region of nominal operation or
maximum operational probability. For our work, we have
employeda PRBS signal of suitable amplitude, switch time
and length. Some guidelines to this extent have been
specified in [21]. A noise to signal ratio of 10% is
employed.
The next step involves the selection of the
appropriate DCS. A procedure similar to that in earlier
case studies is followed. It is determined that the y(t+l) x
y(t) x u(t) space is the most suitable DCS among the
candidate spaces. The optimal number of clusters in this
case comes out to be 4. The parsimonious model
parameters for each cluster are shown in Table 5(a). It is
observed that the local linear model in cluster 3 is
.mstable. This results due to the assumption that a linear
-model is sufficient to capture the dynamics within each
cluster. A nonlinear structure is therefore utilized to
capture the dynamic relationships. In our work, we have
assumed a bilinear structure for the model in cluster 3.
This gave good mapping of the local dynamic relationship.
The local model parameters thus obtained are shown
Table 5(b).
For this case study, cross validation performance
comparison employing both the ‘Projection Method‘ and
the ‘Method of Quasi-Invariance’ is carried out. For
physical systems, it is observed that both methods work
equally well.
6. Control Case Study- High Purity Distillation Column
Control of the high purity distillation column of
case study 3 is attempted employing either of the
strategies discussed in section 5.
Control Strategy f : Multiple controller based IMC
The performance of the scheme is shown in
Figure 3. It can noticed from Figure 3 that the IMC scheme
proposed responds quickly even at very high purity
p0.995). At high purities, the process gain decreases
thereby imposing a demand for larger reflux rates to
increase purities. In the proposed IMC scheme, the
composite control action weighs that controller with
highest gain, the most. The control strategy also displays
good set point tracking and disturbance rejection.
Control Strategy 2: Multi-model MPC
Figure 4 shows the performance of the clusterbased multi-model MPC. It should be noted here that the
model employed .was the one trained in the range of the
output variable 0.88-0.995.
However, excellent closed loop
performance is observed in regions where model training
was not carried out (good control performance was
achieved for set points as low as 0.69). The extrapolation
ability of the proposed methodology is of great significance
as very limited data in real practice is available outside the
normal operating zone. It was noticed that at very high
purity (>=0.995),the response of the MPC scheme
proposed was slightly sluggish. To enhance performance,
a threetiered weight scheme is implemented based on
closed loop response characteristics. The response of the
weight-scheduled MPC is shown in Figure 5.
The proposed schemes are compared with the
performance of a standard PI controller, tuned for optimal
performance in the vicinity of 0.99 (top product
composition). It is noticeable that the PI controller
performance deteriorates as we move away from the
nominal value, failing at a set point of 0.95 as shown in
Figure 6.
7 Conclusion and Future Work
In this work, we propose a novel multiple model
methodology based on fuzzy segregation to handle
complex nonlinear dynamics. The effectiveness of this
strategy in terms of its ability to handle such systems is
investigated through case studies. The proposed scheme
shows good modeling and control performance. The
choice of the DCS still remains a combinatorial problem
though a method to reduce the combinatorial space has
been outlined in our work Extension of this methodology
to the MlMO framework may offer many challenges and
future work may be directed towards addressing this
problem. The choice of the number of optimal divisions of
the operating space based on cluster validity measures is
computationally expensive and efforts may be directed
towards developing more efficient procedures for the
same. Like most other data based methods, the efficacy of
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the method discussed in the paper depends on the quality
of data abstracted. Though extrapolation abilities are
observed to be good, the quality of data acquired is
primary to the success of this methodology. Attention
therefore has to also be focused on design of suitable
identification signals to generate rich data. Adaptation of
the procedure to online modeling and control may also be
an avenue worth Drobina.
Table i: Output Curve Methodology
Output Curve Methodology
Given a training data set 2 and a cross validation data set
Zvof inputatput data for a dynamic system 4. Also given,
the set 8 of possible inputs and past outputs (to various
lags) that are likely to affect the process.
1) Select the first variable from the set 8
2) Obtain a linear model using data in 2 and the selected
variables from 8
3) Perturb the linear model using Zv
4) Construct the scatter plot between the measured
output and model predicted output using data from Z"
5) Does the scatter plot show significant movement
towards the 45-degree line as compared to the base
case
If Yes, include the new input into the DCS structure
and reset it as the base case
If No,discard current variable
select new variable from 8
goto step 2
Select all possible parsimonious model structures for
further evaluation
DCS
I
1
y(t+l) x u(t)
y(t+l) x u(t) x u(t-
Final
Prediction
Error
65.055
16.365
~~
I
Average Prediction
Error
I
I
I
I
I
I
2.661
1.502
2.644
I
I
.J
I a U.
A
I
y(t)=a u(t)+b y(t)+c y(t) u(t)
3
I
for cluster 3
I
c -0.3424
I a 0.004, b 0.9986,
1
J
Figure 1: roposed IMC Structure. t-cc:kuny CJmposite
Controller, FCM: Funy Composite Model
I
62.504
15.1
11
Y(t+l) x Y(t) x u(t)
y(t+l)
_ . . x u(t) x u(tI)
x y(t) x Y(t-I)
Y(t+l) x Y(t) x u(t)
Table 5(a): Local Model parametersfor case study 3
Local Model
I ModelParameters
y(t)=a u(t)+b y(t)
1
I a n W30, b 0.9842
I
L
I a u.ud53. b 0.9502
I
I
I
2.4564
1.2781
2.3443
Table 3: DCS structures for Case Study 2
DCS
I Optimal I Average I Final
I nukberof I PredictTon I Prediction
clusters
Error
Error
3
37.345
38.482
y(t+l) x u(t)
8.96
4
8.272
Y(t+l) x Y(t) x
u(t)
4
9.933
11.67
Y(t+l) x Y(t) x
y(t-I) x u(t) x
u(t-I)
1
I
1
I
Figure 2: (a),(b)Scatter plots for case study 1 (c) cross
validation performance in the DCS of y(t+l) x u(t) x u(t-I)
089
00s
2
086
0%
so001
0 91
oe
Figure 3: Performanceof composite controller based IMC
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0.9
.
E
.E0.85 ",
0.8
D
.
c
-
0.75
lime
Figure 4: Performance of multiple model based MPC
0.565
- Set
Point
...__.
Plant output
1910
ana,
2500
3ooo
35[13
4Gm
4500
5om
55Do
T"0
Figure 5: Performance of weight-scheduled MPC at high
Punty
099
......
0.98
Plsmoutpm
SdPOlnt
1
091
os
0
5 m l a n l s m 2 m o Z a R a a o ~ 4 0 0 0 4 6 m
Ti"
Figure 6:Performance of PI controller for a set p i n t of
0.95 (top product composition)
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