Fuzzy logic inherently enables incorporating of a prior knowledge about the system into the identification algorithms of nonlinear dynamic systems using measured input-output data, a process that is called grey-box modelling. A prior knowledge is often available from physical grounds, e.g. exact knowledge of steady-state input-output characteristics, its monotonicity, monotonicity of step response, approximate knowledge of partial derivatives of the outputs along the particular inputs or other qualitative properties that are either global or valid only in some regions or for particular inputs. Nevertheless, it is not easy to incorporate such a rough knowledge usually described linguistically into the analytical formulas of black-box identification.
State of the art:
In the past there were few attempts to incorporate the monotonicity condition for multi-input mapping corresponding to Mamdani fuzzy logic. Unfortunately, usability of all the algorithms is restricted to a special choice of membership functions and the derived conditions are very conservative that results in poor approximation capability of fuzzy mapping. A simple and intuitive result for membership functions with finite support telling that monotonicity with respect to all inputs is enforced if both input and output membership functions are ordered in the same manner was presented in . In  monotonicity conditions were derived for Gaussian input membership functions with the same variance and in  it was proven that such a system is universal approximator of monotonic functions. The conditions were used for ensuring monotonicity of steady-state input-output characteristics for some applications in . In  conditions for convexity of single-input single-output fuzzy mapping with triangular input membership functions were derived and their universal approximation ability of convex functions was proven at the same time.
- Incorporation of exact knowledge of steady-state input-output characteristics into the input-output data identification algorithms of Mamdani and Takagi-Sugeno fuzzy systems.
- Incorporation of monotonicity condition of steady-state input-output characteristics into the input-output data identification algorithms of Mamdani and Takagi-Sugeno fuzzy systems.
Paper in IEEE Transactions on Fuzzy Systems and/or Fuzzy Sets and Systems
Hušek, P.: Modelling ellipsoidal uncertainty by multidimensional fuzzy sets, Expert Systems with Applications, vol. 39, no. 8, 2012, pp. 6967–6971