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A Robust Design Criterion for Interpretable Fuzzy Models With Uncertain Data

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314 IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL. 14, NO. 2, APRIL 2006
A Robust Design Criterion for InterpretableFuzzy Models With Uncertain Data
Mohit Kumar
, Associate Member, IEEE
,
Regina Stoll
, and
Norbert Stoll
Abstract—
We believe that nonlinear fuzzy ﬁltering techniquesmay be turned out to give better robustness performance than theexisting linear methods of estimation (
and ﬁltering tech-niques), because of the fact that not only linear parameters (con-sequents), but also the nonlinear parameters (membership func-tions) attempt to identify the uncertain behavior of the unknownsystem. However, the fuzzy identiﬁcation methods must be robustto data uncertainties and modeling errors to ensure that the fuzzyapproximation of unknown system’s behavior is optimal in somesense. This study presents a deterministic approach to the robustdesign of fuzzy models in the presence of unknown but ﬁnite un-certainties in the identiﬁcation data. We consider online identiﬁ-cation of an interpretable fuzzy model, based on the robust so-lution of a regularized least-squares fuzzy parameters estimationproblem. The aim is to resolve the difﬁculties associated with therobustfuzzyidentiﬁcationmethodduetolackof
apriori
knowledgeabout upper bounds on the data uncertainties. The study derivesan optimal level of regularization that should be provided to en-sure the robustness of fuzzy identiﬁcation strategy by achieving anupper bound on the value of energy gain from data uncertaintiesand modeling errors to the estimation errors. A time-domain feed-backanalysisoftheproposedidentiﬁcationapproachiscarriedoutwith emphasis on stability, robustness, and steady-state issues. Thesimulation studies are provided to show the superiority of the pro-posed fuzzy estimation over the classical estimation methods.
Index Terms—
-optimality regularization, interpretability,least-squares,
-stability, min–max identiﬁcation, normalizedleast mean squares algorithm (NLMS) algorithm.
I. I
NTRODUCTION
M
ANY real-world physical processes are generally char-acterizedbythepresenceofnonlinearity,complexityanduncertainty. These processes cannot be represented by linearmodelsusedinconventionalsystemidentiﬁcation[1].Thecapa-bility of fuzzy systems paradigm for not only learning complexinput-output mappings butalso to interpret thesemappings withlinguistic terms stimulates the study of approximating ill-de-ﬁned and complex processes using a fuzzy inference system.Therefore, the fuzzy identiﬁcation of nonlinear systemsfrom input–output data has become an important topic of scientiﬁc research with a wide range of applications [2]–[9]. A
Manuscript received December 22, 2003; revised December 30, 2004,March 8, 2005, and May 2, 2005. This work was supported by European SpaceAgency under ESTEC Contract 14350/01/NL/SHMAP project AO-99-058.M. Kumar and R. Stoll are with the Institute of Occupational and SocialMedicine, Faculty of Medicine, University of Rostock, D-18055 Rostock, Ger-many (e-mail: mohit.kumar@uni-rostock.de; regina.stoll@uni-rostock.de).N.StolliswiththeInstituteofAutomation,College ofComputer ScienceandElectrical Engineering, University of Rostock, D-18119 Rostock-Warnemünde,Germany (e-mail: norbert.stoll@uni-rostock.de).Digital Object Identiﬁer 10.1109/TFUZZ.2005.861614
large number of techniques have been developed for the fuzzyidentiﬁcation of nonlinear systems from measured input-outputdata. These techniques can be grouped into three approachesand their combinations:
ad-hoc
data covering approaches[4], [10]), neural networks ([6], [11]), and genetic algorithms
[9], [12]–[14]. However, these methods do not consider the
situations when the training data is uncertain. Regularizationwas suggested as a method for improving the robustness of fuzzy identiﬁcation scheme in [7], [15], and [16]. However,
the choice of regularization parameter is usually not obviousand application dependent. An iterative method for the robust(min–max) identiﬁcation of fuzzy parameters with uncertaindata, was suggested in [3], by solving an equivalent optimallyregularized identiﬁcation problem. However, the method is of-ﬂineand requires
apriori
knowledgeaboutupperboundsonthedata uncertainties. At present, the literature lacks robustness,stability, and steady-state analysis of online fuzzy identiﬁcationmethods that don’t require the knowledge of upper bounds onuncertainties, in the deterministic or stochastic framework. Weconsider the fuzzy identiﬁcation problem for a special class of fuzzy systems (Sugeno type fuzzy systems), since they ideallycombine simplicity with good analytical properties [17]. Also,if we take into account the appropriate restrictions in termsof interpretability, the data-driven construction of Sugenofuzzy systems allows qualitative insight into the relationships[18]–[21].This paper starts with the mathematical formulation of Sugeno type fuzzy systems and considers the fuzzy parametersestimation with uncertain data, based on the robust solutionof a regularized least-squares problem. By a robust solutionwe mean one that attempts to alleviate the worst-case effect of data uncertainties on fuzzy parameters estimation performance.However, the knowledge of an upper bound on the values of data uncertainties is needed to compute the robust solution of fuzzy parameter estimation problem. Therefore, we consider inparticular, the derivation of a robust fuzzy identiﬁcation methodthat does not require
a priori
knowledge about upper bounds onthe data uncertainties, by providing a suitable choice of regular-ization parameters. The proposed method is shown to be robustand stable by providing a time-domain feedback analysis. Theanalysis highlights and exploits an intrinsic feedback structure,mapping the data uncertainties to the estimation errors. Theoptimal choice of regularization parameters is motivated by therobustness analysis of the mapping that can be associated withthe proposed method of fuzzy parameters estimation.The performance of a fuzzy parameter estimation schemeshould be measured with its transient behavior and itssteady-state behavior. The transient behavior is characterized
1063-6706/$20.00 © 2006 IEEE
KUMAR
et al.
: A ROBUST DESIGN CRITERION FOR INTERPRETABLE FUZZY MODELS WITH UNCERTAIN DATA 315
Fig. 1. Trapezoidal membership functions.
by the stability and convergence rate whereas the steady-statebehavior provides information about the mean-square-erroronce it reaches steady-state. In particular, the paper performstransient analysis in a purely deterministic framework withoutassuming
a priori
statistical information and steady-stateanalysis is carried out under the often realistic statisticalassumptions.In the
ﬁ
eld of estimation theory, the classical linear least-mean-squares (or ) estimation techniques and robust esti-mation methods (such as ), are considered as two extremesin terms of their goals. The estimation techniques, undercertain statistical assumptions on the signals, minimize the ex-pected estimation error energy. On the other hand, robust esti-mation methods, safeguard against the worst-case uncertaintiesand therefore make no assumptions on the statistical nature of signals. Therefore, the feasibility of proposed fuzzy estimationstrategy is shown through simulation studies by its comparisonwith the existing extreme methods of estimation ( and ).II. S
UGENO
F
UZZY
I
NFERENCE
S
YSTEM
Let us consider the problem of tuning a Sugeno fuzzy infer-ence system , mapping -dimensional inputspace to one dimensional real line,consisting of different rules. The rule is in the the form:
If is and is and is
,
then
; for all, where are nonempty fuzzysubsets of , respectively, such that membershipfunctions ful
ﬁ
llfor all . The values are real numbers.So, we have(1)We assume that belongs to a nonempty real intervals i.e.for all .We de
ﬁ
ne a real vector such that the membership functionscan be constructed from the elements of vector . To illustratethe construction of membership functions based on knot vector, consider the following examples.a)
Trapezoidal membership function:
Let,such that for input,holds . Now, trapezoidal membershipfunctions for input can bede
ﬁ
ned asif if otherwiseif if if otherwiseif if otherwiseFig. 1 shows an example with the choice of antecedentparameters as: , , , , ,and .b)
Gaussian membership functions with unit disper-sion:
Let, such that for input,holds for all . Now,Gaussian membership functions assuming unit dispersionfor input can be de
ﬁ
ned asFig. 2 shows an example with the choice of antecedentparameters as: , , and .Total number of possible rules depends on the number of membership functions for each input i.e. , whereis the number of membership functions de
ﬁ
ned overinput. Depending upon the choice of membership functions,(1) can be rewritten as function of , i.e.,(2)where(3)Let us introduce the notation
316 IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL. 14, NO. 2, APRIL 2006
Fig. 2. Gaussian membership functions.
, ,. Now, it can be written that(4)Hence, the output of above de
ﬁ
ned type of fuzzy inferencesystem comes out to be linear in consequent parameters butnonlinear in antecedent parameters. Expression (4) is a generalrepresentation that allows also to include multivariate member-ship functions. That is, if we consider a fuzzy modelIf is thensuch that antecedent fuzzy set is de
ﬁ
ned by a multivariatemembership function in the product-space domain of the vector, then we could de
ﬁ
neNow, for example, Gaussian-type multivariate membershipfunctions can be de
ﬁ
ned aswhere and denotes the Euclidean norm. In this case,we havewhereand, henceIII. R
OBUST
O
NLINE
F
UZZY
I
DENTIFICATION
Let us consider a nonlinear system, mapping -dimensionalinput vector to one-dimensional real line, described byfollowing unknown equation:Our aim is to identify the previous nonlinear systemusing an interpretable Sugeno type fuzzy model fromuncertain input
–
output identi
ﬁ
cation data sequence. Here, is themultiplicative uncertainty present in input vector , andis the uncertainty present in output measurement .A practical assumption about the uncertainty present inis that the inequality holds good. That is, theuncertainty present in the output measurement is less than100%.Assume that, is the true value of unknown fuzzyparameters, which approximate the nonlinear system, then wehaveIf we denote the additive uncertainties as and, then(5)Note that also includes modeling errors. If is the un-certainty vector in regression vector due to uncertainty in, then we havewhere , is some un-known bounded function of , , and . The vectoris bounded due to the fact that each element of vector has apositive value between zero and one and sum of all its elementsis equal to one [see (3)]. ThereforeAssume that we already have estimation of fuzzy parametersat time , say ( , ). Given the new measure-ments , one can seek to improve the estimate of , by solvingwhere are theregularization parameters. Also, we wantto preserve the interpretability of fuzzy system during learning.So the membership functions can be prevented from overlap-ping by imposing some constraints on the position of knots, for
KUMAR
et al.
: A ROBUST DESIGN CRITERION FOR INTERPRETABLE FUZZY MODELS WITH UNCERTAIN DATA 317
instance, in case of trapezoidal membership functions the con-straints can be formulated, i.e., for allfor allThese constraints can be formulated in term of a matrix in-equality (as in [3], [5], [7], [16], [22]). Let and
be the upper bounds on and respectively (i.e.,). Now, the robust online identi
ﬁ
ca-tion problem is reduced to(6)De
ﬁ
ne the quantitiesThe optimization problem (6) can be shown to be equivalent to(7)To show the equivalence of problems (6) and (7), we de
ﬁ
ne fora given value of ,andIf , then it follows from the triangle inequality of normsthat . Conversely, if , then de
ﬁ
ne for a given theperturbationsThen , , andso that . Hence,This shows that two variables in problem (6) can bereplaced by a single variable in (7). Problem (7) can be
rewritten as(8)Since each element of vector correspondsto the normalized
ﬁ
ring strength of a rule (3), having any non-negative value less than one and sum of all elements of vectoris equal to 1, therefore, . It followsfrom (5) thatTherefore, a reasonable condition on the estimate of is that it should also satisfyThis implies thatNoting , we haveSince by de
ﬁ
nition , thereforefor (9)This implies thatWe
ﬁ
nd that the minimum value of is equal to .Therefore, there is no harm, so far as robustness is concerned,in solving a more conservative optimization problem(10)The advantage for solving a more conservative problem is thatthemeasurementnoiseuncertainty canbeseenasanequiv-alent additional uncertainty of in the re-gression vector uncertainty , and that would allow the explicitsolution of the optimization problem. LetForany
ﬁ
xedvaluesofparameters ,weconsiderthemax-imization problemFor constant , the cost function is convexin , so that the maximization over is achieved at the
318 IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL. 14, NO. 2, APRIL 2006
boundary, . Therefore, the previous constrainedoptimizationproblemcanbereformulatedbyintroducinganon-negative Lagrange multiplier , as follows:Let and be the optimal solution of above optimizationproblem obtained by differentiating the cost function with re-spect to and which satisfy the equationsandAlso the solution should satisfy . This is because thehessianofthecostfunctionw.r.t. mustbenonpositive
–
def-inite. Now, the maximum cost becomeswhereif otherwiseNoting that , i.e.,, can be expressed asThe srcinal problem (10) is, therefore, equivalent toLet us introduce a three-variable cost function asThen, it can veri
ﬁ
ed thatThus, the srcinal min
–
max problem becomes equivalent toNow, to
ﬁ
nd the minimum of cost function over, the gradient of with respect to is set to zero,resulting in(11)Let be the optimal solution obtained by solving the previousequation,whichobviouslyisafunctionof and ,sowewriteNow, the optimization problem is reduced toLet us de
ﬁ
ne a measure of total data uncertainty asso that and .Note from (9) that is always positive and bounded for any
nonzeros output measurement . Now, solving for from(11), we getIt should be noted that . Applying matrix in-version lemma and then after some algebra, we get (assuming)Let us substitute the aforementioned value of intoand then simplify to getHence, the estimation equations can be written asandThus,weseethat
apriori
knowledgeofparameter isrequiredfor the computation of parameters using the previousestimation equations. However, in a practical situation, cannot be computed because of the lack of knowledge about noisesignal magnitude (i.e., ). Therefore, our concern in the

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