Estabilização de sistemas descritores dependentes de parâmetros: aplicação ao controle de um pêndulo invertido

Abstract

This work proposes a methodology for the design of controllers for descriptor systems dependent on time-varying parameters, using linear matrix inequalities (LMIs) as the main tool. In this context, the descriptor matrix can be singular, implying the existence of states described by algebraic equations without free erivatives, or it can be non-singular, in which case the state derivatives are free. The adopted modeling, based on linear parameter-varying (LPV) systems in descriptor form, allows for flexible representation of nonlinear dynamic systems, capturing their variations throughout operation. A distinctive feature of this approach lies in the parametric dependence of the descriptor matrix, which enhances the modeling and control capabilities for these systems. The developed control technique is grounded in Lyapunov stability theory and semidefinite programming to guarantee the system’s asymptotic stability via state feedback. Beyond mere feasibility, performance optimization is incorporated through weighting matrices that balance response speed and control effort. As a case study, the methodology is applied to the control of the rotational inverted pendulum, a classic and challenging control system. Through simulations, three theorems encompassing different LMI formulations are analyzed, highlighting how parameter choices influence feasibility, performance, and dynamic behavior. The results demonstrate the effectiveness and flexibility of the proposal, emphasizing the strategic role of parameter tuning in controller design. Finally, limitations and future work prospects are discussed, including automatic parameter optimization, extension of feasibility regions, and robustness evaluation against disturbances and uncertainties, positioning the approach as a promising foundation for LPV descriptor system control.