A local recursive least squares approach for discrete-time adaptive fuzzy control
This paper proposes a local recursive least squares (RLS) estimation strategy for discrete-time adaptive fuzzy control of nonlinear systems represented in quasi-Linear Parameter Varying (qLPV)/Takagi--Sugeno (TS) form. Unknown nonlinear terms are approximated by a constant-consequent TS fuzzy model, and the consequent parameters are updated by a membership-function-weighted RLS law with a forgetting factor. The proposed estimator keeps a different covariance for each rule, considerably reducing the memory footprint of the least-squares updates. The adaptation is also simplified since each rule is only adapted when it is active. From these properties, we are capable of showing that the adaptation law ensures bounded local adaptation errors. Building upon this adaptation law, Linear Matrix Inequality (LMI) synthesis conditions are presented for matched, sector-bounded and norm-bounded unknown nonlinearities, guaranteeing ultimate uniform boundedness of the adaptive control system in closed loop. Three numerical examples are presented to illustrate the adaptive control conditions in the three cases: a planar manipulator with unknown gravity direction, a two-tank system with unknown coupling, and a Brushless DC (BLDC) motor acting as thrust for an efficiency vehicle.