Grid- and transformation-free model order reduction for linear parameter varying systems

本文研究线性变参数系统的模型阶数约简问题。LPV系统是动态系统，其状态空间矩阵是一个时变参数向量的函数，该参数向量在每个时刻都是未知的，但可以测量的。LPV框架的主要优点是它允许将线性控制器设计技术扩展到非线性和时变系统，从而通过求解线性矩阵不等式（lmi）来解决分析和综合问题。然而，随着状态维和参数维的增加，求解lmi的计算复杂度迅速增长，并且在处理大型系统时需要模型逼近。本课题解决了LPV模型的平衡截断问题，解决了现有方法中存在的三个开放性问题：—通过开发基于全块s过程的无网格约简方法，避免了在参数空间中求解网格上LMI条件的需要。-通过将约简问题表示为等效的固定结构控制器综合问题，从而实现无变换约简过程，可以避免对模型进行参数依赖的变换。-将上述固定结构方法与已有模态匹配结果相结合，解决LPV模型模态分解这一开放性且具有重要现实意义的问题。

This project considers model order reduction for linear parameter varying (LPV) systems. LPV systems are dynamic systems where the state space matrices are functions of a time-varying parameter vector which is not known in advance but measurable at each time instant. The main advantage of the LPV framework is that it allows to extend linear controller design techniques to nonlinear and time-varying systems, such that analysis and synthesis problems are being solved by solving linear matrix inequalities (LMIs). The computational complexity solving LMIs grows however rapidly with increasing state and parameter dimensions, and model approximation is required when dealing with large scale systems.In this project balanced truncation for LPV models is addressed, and three open problems arising in existing approaches will be solved:- The need of solving LMI conditions on a grid in parameter space will be avoided by developing a grid-free reduction method based on the full-block S-procedure.- The need of parameter-dependent transformations of the model will be avoided by representing the reduction problem as an equivalent fixed-structure controller synthesis problem, leading to a transformation free-reduction procedure. - The open and practically important problem of modal decomposition of LPV models will be addressed by combining the fixed-structure approach above with existing modal matching results.