Observer-based output feedback control using invariant polyhedral sets for fuzzy T-S models under constraints

Abstract

In this work, is proposed a numerical method for the computation of state observer-based output feedback controllers for fuzzy Takagi-Sugeno (T-S) systems subject to con-straints based on invariant set theory. In this regard, Positively Invariant (PI) polyhedral sets are used to ensure that state and input control constraints are satisfied at all times. Sufficient conditions are established for a polyhedron defined in the augmented state- space (state + estimation error) to be PI. The problem of tracking a constant reference signal is also considered, for which the concept of robust positive invariance is used along with a stabilizing Integral-Proportional (I-P) controller. Sufficient conditions are estab- lished for a polyhedron defined in the augmented state-space (state + estimation error + tracking error integral state) to be PI in the presence of a constant reference signal, which can be interpreted as a bounded disturbance. From the invariance conditions, a bilinear optimization problem is formulated to simultaneously compute the controller’s gains and the positively invariant polyhedron, guaranteeing the satisfaction of the constraints. The two types of observer found in the literature of fuzzy T-S systems are considered: the first considers the membership functions dependent only on the system output; the sec- ond, in turn, refers to the general case, where these functions can be associated with any state variables. In the simplest case, although the membership functions depend only on the output, the estimated state feedback results, in general, in controllers with better per- formance and with larger sets of admissible states associated with them than the output static feedback control. For the general case, as membership functions depend on non-accessible states, an estimation mechanism is needed to calculate these variables. In both cases, this role is played by the fuzzy T-S observer