This paper provides a necessary and sufficient condition for the reachability of discrete-time Takagi–Sugeno fuzzy systems that is easy to apply, such that it constitutes a practical test. The proposed procedure is based on checking if all the principal minors associated to an appropriate matrix are positive. If this condition holds, then the rank of the reachability matrix associated to the Takagi–Sugeno fuzzy system is full for any possible sequence of premise variables, and thus the system is completely state reachable. On the other hand, if the principal minors are not positive, the property of the matrix being a block P one with respect to a particular partition of a set of integers is studied in order to conclude about the reachability of the Takagi–Sugeno system. Examples obtained using an inverted pendulum are used to show that it is easy to check this condition, such that the reachability analysis can be performed efficiently using the proposed approach.

This work has been funded by the National Science Centre of Poland under Grant: UMO-2013/11/B/ST7/01110, by the Spanish Ministry of Science and Technology through the projects CICYT ECOCIS (ref. DPI2013-48243-C2-1-R) and CICYT HARCRICS (ref. DPI2014-58104-R), by AGAUR through the contracts FI-DGR 2014 (ref. 2014FI_B1 00172) and FI-DGR 2015 (ref. 2015FI_B2 00171), and by the DGR of Generalitat de Catalunya (ref. 2014/SGR/374).

Peer Reviewed