Repetitive Set-based Learning Robust Predictive Control for Lur’e Systems

Abstract

Robust control of uncertain nonlinear systems subject to constrants often leads to conservatism. Such behaviors can be improved by updating the model of the uncertainty with the data collected during the operation time or by bounding the parameters. This paper proposes an approach to robustly control the discrete-time Lur’e system subject to states and input constraints, where the unknown memoryless nonlinearity is sector-bounded and its Lipschitz constant is assumed to be given. In the first phase of operation, when no data has been collected, a robust MPC controller obtained from solving linear matrix inequalities is used. This formulation is also used to compute the safe region, in which the system can operate safely. After sufficient data has been collected, an upper and lower bound of of the nonlinearity can be constructed by using the Lipschitz constant. A controller based on tube-based MPC is used, which results in less conservatism and provides more flexibility. Data of the nonlinearity can be further updated to reduce uncertainty, and hence, to decrease the size of the tube. Under additional conditions, the controller can safely explore the region outside the safe regions to collect more data of the unknown nonlinearity to improve performance and region of attraction.