Single-input Single-output Convex Fuzzy Systems as Universal Approximators for Single-input Single-output Convex Functions

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On approximation properties of smooth fuzzy models

This Paper Addresses The Approximation Properties Of The Smooth Fuzzy Models. It Is Widely Recognized That The Fuzzy Models Can Approximate A Nonlinear Function To Any Degree Of Accuracy In A Convex Compact Region. However, In Many Applications, It Is Desirable To Go Beyond That And Acquire A Model To Approximate The Nonlinear Function On A Smooth Surface To Gain Better Performance And Stability Properties. Especially In The Region Around The Steady States, When Both Error And Change In Error Are Approaching Zero, It Is Much Desired To Avoid Abrupt Changes And Discontinuity In The Approximation Of The Input-Output Mapping. This Problem Has Been Remedied In Our Approach By Application Of The Smooth Compositions In The Fuzzy Modeling Scheme. In The Fuzzy Decomposition Stage Of Fuzzy Modeling, We Have Discretized The Parameters And Then Calculated The Result Through Partitioning Them Into A Dense Grid. This Could Enable Us To Present The Formulations By Convolution And Fourier Transformation Of The Parameters And Then Obtain The Approximation Properties By Studying The Structural Properties Of The Fourier Transformation And Convolution Of The Parameters. We Could Show That, Irrespective To The Shape Of The Membership Function, One Can Approximate The Dynamics And Derivative Of The Continuous Systems Together, Using The Smooth Fuzzy Structure. The Results Of The Paper Have Been Tested And Evaluated On A Discrete Event System In The Hybrid And Switched Systems Framework

Parameterized Convex Minorant for Objective Function Approximation in Amortized Optimization

Parameterized convex minorant (PCM) method is proposed for the approximation of the objective function in amortized optimization. In the proposed method, the objective function approximator is expressed by the sum of a PCM and a nonnegative gap function, where the objective function approximator is bounded from below by the PCM convex in the optimization variable. The proposed objective function approximator is a universal approximator for continuous functions, and the global minimizer of the PCM attains the global minimum of the objective function approximator. Therefore, the global minimizer of the objective function approximator can be obtained by a single convex optimization. As a realization of the proposed method, extended parameterized log-sum-exp network is proposed by utilizing a parameterized log-sum-exp network as the PCM. Numerical simulation is performed for parameterized non-convex objective function approximation and for learning-based nonlinear model predictive control to demonstrate the performance and characteristics of the proposed method. The simulation results support that the proposed method can be used to learn objective functions and to find a global minimizer reliably and quickly by using convex optimization algorithms.Comment: 12 pages, 4 figure

Theoretical Interpretations and Applications of Radial Basis Function Networks

Medical applications usually used Radial Basis Function Networks just as Artificial Neural Networks. However, RBFNs are Knowledge-Based Networks that can be interpreted in several way: Artificial Neural Networks, Regularization Networks, Support Vector Machines, Wavelet Networks, Fuzzy Controllers, Kernel Estimators, Instanced-Based Learners. A survey of their interpretations and of their corresponding learning algorithms is provided as well as a brief survey on dynamic learning algorithms. RBFNs' interpretations can suggest applications that are particularly interesting in medical domains

Parameter identification for piecewise-affine fuzzy models in noisy environment

AbstractIn this paper the problem of identifying a fuzzy model from noisy data is addressed. The piecewise-affine fuzzy model structure is used as non-linear prototype for a multi–input, single–output unknown system. The consequents of the fuzzy model are identified from noisy data which are collected from experiments on the real system. The identification procedure is formulated within the Frisch scheme, well established for linear systems, which is extended so that it applies to piecewise-affine, constrained models