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.

中文翻译：

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多线性参数变化系统基于双线性矩阵不等式的非二次控制器设计

本文提出了宽松的双线性矩阵不等式（BMI）条件，以设计用于多目标线性参数变化（LPV）表示形式的非线性系统的增益调度控制器。基于非二次Lyapunov函数和并行分布式补偿器方案，得出获得的条件。控制器设计过程涉及一些新颖的空项，并导致BMI问题，这在以前的研究中几乎没有解决。此外，为了有效地解决BMI条件，提出了一种新颖的顺序方法，该方法将整个BMI问题转换为线性矩阵不等式（LMI）约束，并且将一些BMI条件简化为尺寸比原始条件小。最初，将LMI条件作为凸优化问题解决。第二，BMI项在LMI的可行解附近以及BMI约束的每个解候选附近进行迭代线性化。最后，将线性化条件作为特征值问题求解。为了显示所提出方法的优点，提供了一些数值比较和仿真。