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This article proposes relaxed sufficient bilinear matrix inequality (BMI) conditions to design a gain-scheduling controller for nonlinear systems described by polytopic-linear parameter varying (LPV) representations. The obtained conditions are derived based on a nonquadratic Lyapunov function and a parallel distributed compensator scheme. The controller design procedure involves some novel null terms and leads to a BMI problem, which hardly has been solved in previous researches. Furthermore, to effectively solve the BMI conditions, a novel sequential approach is proposed which convert the overall BMI problem into linear matrix inequality (LMI) constraints and some simpler BMI conditions with fewer dimensions than the original one. Initially, the LMI conditions are solved as a convex optimization problem. Second, the BMI terms are iteratively linearized near the feasible solutions of the LMIs and each solution candidates for the BMI constraints. Finally, the linearized condition is solved as an eigenvalue problem. To show the merits of the proposed approach, several numerical comparisons and simulations are provided.

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This paper presents a methodology for frequency regulation in a microgrid involving renewable energy sources (RES) using a dynamic controller, which is an output feedback controller (OFC). The parameters of OFC are tuned by searching the design space of the controller. Since the RES model is not exactly known, the uncertain model is derived and the OFC is considered for it. The goal of controller tuning is to find appropriate parameters of the controller such that the norm of frequency deviations, even in presence of uncertainties in the RES parameters is minimized. An algorithm based on searching the controller design space is suggested to find the suitable controller gains. The algorithm assumes the controller parameters lie in a convex space and searches the space systematically such that an appropriate solution is found. The method is proved mathematically and two theorems are mentioned, accordingly. Finally, a simulated model of a RES is utilized for algorithm evaluation and the results demonstrate the algorithm capability in optimal frequency regulation.