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

本计画研究线性变参数系统之模型降阶问题。LPV系统是动态系统，其中状态空间矩阵是时变参数向量的函数，该时变参数向量是事先未知的，但在每个时刻是可测量的。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，使得分析和合成问题通过求解线性矩阵不等式（LMI）来解决。求解线性矩阵不等式的计算复杂性随着状态和参数维数的增加而迅速增加，当处理大规模系统时需要模型近似。本项目讨论了LPV模型的平衡截断，并解决了现有方法中存在的三个问题：-通过开发基于全块S-过程的无网格简化方法，避免了在参数空间网格上求解线性矩阵不等式条件的需要。通过将降阶问题表示为等价的固定结构控制器综合问题，避免了模型的参数相关变换的需要，从而导致无变换降阶过程。- 将上述固定结构方法与现有的模态匹配结果相结合，解决了LPV模型的模态分解这一开放的、实际上重要的问题。

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.