Lyapunov functional-based stability analysis and control optimization in renewable energy management systems

Abstract

A minimax optimization formulation with respect to the Lyapunov functional for constrained control of nonlinear systems with bounded disturbances is introduced in this paper. This approach combines the Lyapunov-Krasovskii functional theory with minimax optimization, considering the stability constraint in the worst-case control problem and thus enabling both performance maximization and asymptotic convergence at the same time. An adaptive parameterization scheme for the Lyapunov matrix is devised to handle the case of parametric uncertainties in the system, while a control law is constructed analytically based on the introduced optimization model. The validity of the proposed approach is proved through three stability theorems that guarantee the positive definiteness of the introduced functional, asymptotic stability of the controlled system, and disturbance rejection. The proposed minimax optimization model is validated through the numerical study of the renewable energy management system, where its superiority over traditional PID, LQR, and robust control methods is demonstrated. In particular, the proposed approach achieves reduced settling time, minimal values of the Lyapunov functional residue, better tracking properties, and superior worst-case disturbance handling properties.