On the interpretation and identification of dynamic Takagi-Sugeno fuzzy models

Dynamic Takagi-Sugeno fuzzy models are not always easy to interpret, in particular when they are identified from experimental data. Ideally, it is desirable that a dynamic Takagi-Sugeno fuzzy model should give accurate global nonlinear prediction and at the same time that its local models are close approximations to the local linearizations of the nonlinear dynamic system. The latter is important in many applications where the constituent local models are used individually and aids validation and interpretation of the model considerably. This defines a multi-objective identification problem, namely, the construction of a dynamic model that is a good approximation of both local and global dynamics of the underlying system. While these objectives are often conflicting, it is shown that there exists a close relationship between dynamic Takagi-Sugeno fuzzy models and dynamic linearization when using affine local model structures, which suggests that a solution to the multi-objective identification problem exists. However, it is also shown that the affine local model structure is a highly sensitive parameterization when applied in transient operating regimes, i.e., far away from equilibrium. The reason is essentially that the constant term in the affine local model tends to dominate over the linear term during transients. In addition, it is inherently more difficult to design informative experiments in transient regions compared to near-equilibrium regions. Due to the multi-objective nature of the identification problem studied here, special considerations must be made during model structure selection, experiment design, and identification in order to meet both objectives. Some guidelines for experiment design are suggested and some robust nonlinear identification algorithms are studied. These include constrained and regularized identification and locally weighted identification. Their usefulness in the present context is illustrated by examples.