Abstract
Starting from a nonlinear model for a pendulum-cart system, on which viscous friction is considered, a Takagi-Sugeno (T-S) fuzzy augmented model (TSFAM) as well as a TSFAM with uncertainty (TSFAMwU) is proposed. Since the design of a T-S fuzzy controller is based on the T-S fuzzy model of the nonlinear system, then, to address the trajectory tracking problem of the pendulum-cart system, three T-S fuzzy controllers are proposed via parallel distributed compensation: (1) a T-S fuzzy servo controller (TSFSC) designed from the TSFAM; (2) a robust TSFSC (RTSFSC) designed from the TSFAMwU; and (3) a robust T-S fuzzy dynamic regulator (RTSFDR) designed from the RTSFSC with the addition of a T-S fuzzy observer, which estimates cart and pendulum velocities. Both TSFAM and TSFAMwU are comprised of two fuzzy rules and designed via local approximation in fuzzy partition spaces technique. Feedback gains for the three fuzzy controllers are obtained via linear matrix inequalities approach. A swing-up controller is developed to swing the pendulum up from its pendant position to its upright position. Real-time experiments validate the effectiveness of the proposed schemes, keeping the pendulum in its upright position while the cart follows a reference signal, standing out the RTSFDR.
Highlights
A great number of nonlinear systems can be represented by Takagi-Sugeno (T-S) fuzzy models
Since the design of a T-S fuzzy controller is based on the T-S fuzzy model of the nonlinear system, to address the trajectory tracking problem of the pendulum-cart system, three T-S fuzzy controllers are proposed via parallel distributed compensation: (1) a T-S fuzzy servo controller (TSFSC) designed from the TakagiSugeno fuzzy augmented model (TSFAM); (2) a robust TSFSC (RTSFSC) designed from the TSFAM with uncertainty (TSFAMwU); and (3) a robust T-S fuzzy dynamic regulator (RTSFDR) designed from the RTSFSC with the addition of a T-S fuzzy observer, which estimates cart and pendulum velocities
New relaxed stability conditions and designs based on linear matrix inequalities (LMIs) for fuzzy control systems in continuous and discrete time have been presented in [5] and its utility is demonstrated with a fuzzy regulator and a fuzzy observer design
Summary
A great number of nonlinear systems can be represented by Takagi-Sugeno (T-S) fuzzy models. Since the design of a T-S fuzzy controller is based on the T-S fuzzy model of the nonlinear system, to address the trajectory tracking problem of the pendulum-cart system, three TS fuzzy controllers are proposed: (1) a T-S fuzzy servo controller (TSFSC) designed from the TSFAM; (2) a robust TSFSC (RTSFSC) designed from the TSFAMwU; and (3) a robust T-S fuzzy dynamic regulator (RTSFDR) designed from the RTSFSC with the addition of a T-S fuzzy observer, designed via PDC using the separation principle, which estimates cart and pendulum velocities Both TSFAM and TSFAMwU are comprised of only two fuzzy rules and designed via local approximation in fuzzy partition spaces technique.
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