Generally, in real-world engineering disciplines a dynamical system is nonlinear, having multi-input and multi-output (MIMO) variables, and high-level parameter uncertainties. Although there are many approaches proposed in the literature for system modeling and optimization, it remains a challenging topic to derive the precise mathematical models to characterize complex, dynamic and globally described systems. If training data in a real- world system are available, artificial neural network theories can be applied for system parameter recognition and optimization. The objective of this work is to develop a new fuzzy formulation based on the semi-tensor product (STP) method to construct fuzzy logic models for MIMO systems in a matrix representation. It involves the following processing operations:​ fuzzy modeling, structure and parameters identification, system optimization, and adaptive control of closed-loop fuzzy systems based on the fuzzy relation matrix (FRM) models and STP algorithms. The related contributions are summarized below:​

• The STP operations of logic matrices are proposed for fuzzy logic reasoning by the extension of STP for conventional matrices. And then some properties are developed for the fuzzy STP operations.
• A vector expression of fuzzy variables and matrix representation of fuzzy relations, are proposed for advanced research. Matrix expression of multi-dimensional data is applied to multi-variable fuzzy relations and fuzzy rules.
• Two modeling methods, direct modeling and indirect identification are proposed to identify the FRM models for MIMO fuzzy systems.
• A universal approximation with the approximation accuracy is proposed for MIMO fuzzy models based on FRMs and the fuzzy logic STP algorithm.
• A neural-fuzzy STP network is developed to train the parameters in MIMO fuzzy systems based on the FRM and fuzzy logic STP algorithm, and a hybrid optimization method is adopted to train parameters for FRM models.
• Based on the FRM model and fuzzy logic STP algorithm, an indirect adaptive FRM control law is suggested to improve the performance of FRM control systems.

The effectiveness of the proposed modeling, optimization, and adaptive control design techniques in the multi-variable FRMS and STP algorithms platform is validated by simulation tests in the Matlab environment.