On Monotonicity of Takagi–Sugeno Fuzzy Systems With Ellipsoidal Regions

This paper presents sufficient conditions on monotonicity of a Takagi–Sugeno (T–S) fuzzy system with linear submodels in the consequents of the rules where the input space is considered to be partitioned to ellipsoidal regions. Such regions commonly arise in practice if clustering algorithms are used to identify a fuzzy model from measured data. By the monotonicity, it is meant that the partial derivatives of the output of the mapping represented by the fuzzy model with respect to all inputs are nonnegative on a given universe of discourse. The conditions are given in the form of linear matrix inequalities with respect to the parameters of the submodels that may be useful in solving associated optimization problems via efficient semidefinite programming techniques. The proposed conditions reduce conservatism of the existing ones from two reasons. First, the conservatism is introduced only once, since the conditions are not separated between antecedent and consequent parts of fuzzy rules. Second, the domain of interest where monotonicity is enforced may be restricted. The proposed algorithm is illustrated on least-squares approximation of a multivariate function by a monotonic T–S fuzzy system.