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A robust design criterion for interpretable fuzzy models with uncertain data

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 robustnessof fuzzy identification strategy byachieving anupper bound on the value of energy gain from data uncertaintiesand modelingerrors to the estimation errors. Atime-domain feed-backanalysisoftheproposedidentificationapproachiscarriedoutwithemphasisonstability,robustness,andsteady-stateissues.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:;,College ofComputer ScienceandElectrical Engineering, University of Rostock, D-18119 Rostock-Warnemünde,Germany (e-mail: 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 thesituations 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
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