Research on analysis and synthesis method of nonlinear control systems based on gain scheduling theory

为了建立非线性控制问题的求解方法和设计算法，我们研究了稳定性分析和综合问题、具有L^2增益性能的控制问题和鲁棒控制问题。首先，利用反映模型特殊结构的扩展二次Lyapunov函数，提出了状态反馈控制和输出反馈控制的设计方法。通过对实际系统模型的仿真研究，验证了该方法的有效性。对于该方法中出现的参数依赖线性矩阵不等式的求解方法，借鉴了增益调度算法中线性矩阵不等式求解方法的思想，提出了一种新的基于有限维线性矩阵不等式的求解方法。其次，针对一类具有k次多项式型非线性的特殊非线性系统，基于可达集和一维线性矩阵不等式，提出了一种新的非线性系统L^2增益分析方法。我们研究非线性控制问题的主要观点是将非线性系统的全部或部分状态作为调度参数，并将线性系统的分析和综合方法应用于具有调度参数的线性系统。在这种方法中，出现的一个问题是如何先验估计调度参数的范围和/或调度参数的变化范围。我们还成功地开发了解决这个问题的方法。

To establish the solution method and the design algorithm for nonlinear control problems, we investigated stability analysis and synthesis problems, control problems with L^2 gain performance and robust control problems. First, using an extended quadratic Lyapunov function which reflects a special structure of model, we proposed design methods of state feedback control and output feedback control. Through simulation studies in real system models, we showed efficiencies of the proposed method. As for the solution method of parameter-dependent linear matrix inequalities which appear in the proposed method, we also proposed a new solution method via finite-dimensional linear matrix inequalities, which borrows an idea of solution method of linear matrix inequalities in gain scheduling algorithms. Next, focusing on a special type of nonlinear systems which have kth-degree polynomial type of nonlinearlity, we developed a new analysis method of L^2 gain of nonlinear systems based on reachable sets and fin ite-dimensional linear matrix inequalities. The main point of our approach to nonlinear control problems is to consider full state or a part of state of nonlinear systems as a scheduling parameter and to apply analysis and synthesis methods for linear systems to the linear systems with scheduling parameter. In this approach, one problem which arises is how to estimate a priori the range of scheduling parameter and/or the range of the variation of scheduling parameter. We succeeded also in developing an solution method for this problem.