In this paper, a nonlinear dynamic system is first approximated by N fuzzy-based linear state-space subsystems. To track a trajectory dominant by a specific frequency, the reference models with desired amplitude and phase features are established by the same fuzzy sets of the system rule. Similarly, the same fuzzy sets of the system rule are employed to design robust fuzzy sliding-mode control (RFSMC) and adaptive fuzzy sliding-mode control (AFSMC). The difference between RFSMC and AFSMC is that AFSMC contains an updating law to learn system uncertainties and then an extra compensation is designed. It is different from the most previous papers to learn the whole nonlinear functions. As the norm of the sliding surface is inside of a defined set, the updating law starts; simultaneously, as it is outside of the other set, the updating law stops. For the purpose of smoothing the possibility of discontinuous control input, a transition between RFSMC and AFSMC is also assigned. Under the circumstances, the proposed control [robust adaptive fuzzy sliding-mode control (RAFSMC)] can automatically tune as a RFSMC or an AFSMC; then the advantages coming from the RFSMC and AFSMC are obtained. Finally, the stabilities of the overall system of RFSMC, AFSMC, and RAFSMC are verified by Lyapunov stability theory. The compared simulations among RFSMC, AFSMC, and RAFSMC are also carried out to confirm the usefulness of the proposed control scheme.
IEEE Transactions on Fuzzy Systems 12(5), pp.676-687