Stabilization conditions of fuzzy systems under persistent perturbations and their application in nonlinear systems

The rejection of perturbations is a classical problem in control theory. Part of the interest in the rejection of persistent disturbances is focused on those perturbations affecting nonlinear systems [Chen, C.W., 2006. Stability conditions of fuzzy systems and its application to structural and mechanical systems. Advances in Engineering Software 37(9), 624–629; Isidori, A., Astolfi, A., 1992. Disturbance attenuation and H∞H∞-control via measurement feedback in nonlinear systems. IEEE Transactions on Automatic Control 37(9), 1283–1293]. This paper, firstly, analyzes some literature contributions which tackle the problems of using fuzzy models as representations of the nonlinear behaviour of the real system—and shows the drawbacks of these methods. Secondly, a method for designing parallel distributed compensation fuzzy state-feedback controllers is proposed. This new approach is based on minimizing the 1-norm between the disturbance input signal and output signal. Specifically, an upper bound on this norm, the ⋆⋆-norm, is minimized, which, unlike the 1-norm, can be formulated in terms of linear and bilinear matrix inequalities. This novel contribution presents two main advantages with respect to previous works—firstly, it considers a priori bounds on the applied persistent perturbation. Secondly, it provides a minimized upper bound on the output. The proposed method has been tested on a mechanical system with nonlinear behaviour. By means of this example, we show that the minimization of the ⋆⋆-norm can also limit the control actions, if they are very large.