A practical test for assessing the reachability of discrete-time Takagi–Sugeno fuzzy systems

Abstract

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.

Fault‐tolerant fusion‐based MPC with sensor recovery for constrained LPV systems

“…Remark We will provide conditions, via attractive invariant set construction (see Section 3), which will ensure the closed‐loop stability (in terms of boundedness) of the system. Hence, stronger conditions, such as those in the work of Witczak et al for systems in convex polytopic description, are not required.…”

“…Denoting A = A( ), B = B( ), E = E( ), and K = K( ) for space reasons, the dynamics of the plant tracking error (15) can be written, using (1), (15)- (18) and (22), as…”

“…The constraint set (x, x s , u s ) in (41) for the nominal system (6) is defined using (3), (11), (15), (18)…”

“…We can now establish the stability properties of the nominal controller in (47). Note thatū is forced to satisfy the tighter constraint in (45), which from the bounding property of  z and * and from (18) ensures u − u s ∈  . Similarly, the tighter constraint forx in (45) ensures that x − x s ∈  is satisfied for system (1).…”

“…9. Compute the estimate-based reference-tracking feedback control u(k) from (18) and send it to the plant. 10.…”