Joint Block Structure Sparse Representation for Multi-Input–Multi-Output (MIMO) T–S Fuzzy System Identification

Takagi-Sugeno (T-S) fuzzy system identification provides a reasonable framework for modeling and approximation by decomposition of a nonlinear system into a collection of local linear models. In this paper, we extend the multidimensional output fuzzy rule from the corresponding single-output fuzzy rule with the important difference in the variable definition of output, namely the output of each fuzzy rule is represented by a multidimensional vector instead of a scalar value. In this case, the consequent of the fuzzy rule with the multidimensional output variable share a common antecedent part. Different from the traditional methods that separate the multi-input-multi-output fuzzy system into a group of multi-input-single-output fuzzy models, in this paper, we take into account the block structure information in the T-S fuzzy system and cast the problem of multidimensional output fuzzy model identification as a joint structure sparse optimization problem, where the consequent parameters are estimated with a common block structured sparsity pattern over all dimensions of the output variable. Furthermore, we exploit a joint block sparse orthogonal-match-pursuit algorithm to reduce the number of fuzzy rules in terms of all dimensions of the output variable and prove the sufficient conditions in consideration of the multidimensional output together with the block structure in the T-S fuzzy model. This method is efficient and shows good performance in well-known benchmark datasets and real-world problems.