To deal with the real-world problems characterized by complexities and uncertainties, the design of the fuzzy filters is an important issue since the real-world applications require the filtering of uncertainties from the experimental data. Therefore, we developed a mathematical framework for the design and analysis of fuzzy based intelligent systems taking into account the underlying uncertainties in a sensible way. This framework facilitate the design of algorithms optimized for performance and complexity making them suitable for a real-time operation. Fuzzy filtering algorithms, i.e. the strategies for estimating the parameters of a nonlinear fuzzy filter, were developed based on different mathematical criteria. The studied mathematical criteria include
1. Robust Regularized Least-Squares Estimation
3. Least-Squares Estimation
4. Generalized Least-mean-squares like p−norm Algorithms
5. Risk-sensitive Estimation
A mathematical theory was developed for the stability, robustness, and steady-state analyses of fuzzy filtering algorithms.
- M. Kumar, R. Stoll, and N. Stoll, “A robust design criterion for interpretable fuzzy models with uncertain data,” IEEE Transactions on Fuzzy Systems, vol. 14, pp. 314-328, Apr. 2006.
- M. Kumar, R. Stoll, and N. Stoll, “A min-max approach to fuzzy clustering, estimation, and identification,” IEEE Transactions on Fuzzy Systems, vol. 14, pp. 248-262, Apr. 2006.
- M. Kumar, N. Stoll, and R. Stoll, “An energy-gain bounding approach to robust fuzzy identification,” Automatica, vol. 42, pp. 711-721, May 2006.
- M. Kumar, R. Stoll, and N. Stoll, “Deterministic approach to robust adaptive learning of fuzzy models,” IEEE Transactions System, Man, and Cybernetics, Part B: Cybernetics, vol. 36, pp. 767-780, Aug. 2006.
- M. Kumar, N. Stoll, and R. Stoll, “Adaptive fuzzy filtering in a deterministic setting,” IEEE Transactions on Fuzzy Systems, vol. 17, no. 4, pp. 763-776, Aug. 2009.
- M. Kumar, N. Stoll, and R. Stoll, “On the estimation of parameters of takagi-sugeno fuzzy filters,” IEEE Transactions on Fuzzy Systems, vol. 17, no. 1, pp. 150-166, Feb. 2009.