基于LQR冗余控制输入问题中非线性矩阵方程的研究

Redundant control inputs systems plays an important part in engineering, and we give the applications of new solution bounds of nonlinear matrix equation in relevant redundant control problems, which is practical and meaningful from both the engineering and the mathematical point of view. In this project, for the relationship between the solution of nonlinear matrix equation and controller gain, we study the more precise estimation of the solution bounds of the continuous algebraic Riccati equation, and their specific application in LQR controller gain of quadratic optimal control with redundancy of control inputs.. We will focus on the following questions: 1) the solutions and numerical solutions of nonlinear matrix equation, the properties of the solutions, the judgement and the properties of the relevant special matrix, the conditions about the existence and uniqueness of the solutions and the iterative algorithms; 2) based on the LQR redundant control inputs systems the more precise estimation of the solution bounds of the nonlinear matrix equation; 3)using the bounds in the LQR redundant control inputs systems, study the bounds of the controller gain.. These problems are not only the deep problems in matrix theory and matrix computation, but also the impractical problems that the theoretical research is applied to solve the actual LQR redundant control input problems. We use special matrix, matrix equation, matrix inequality and other related theoretical knowledge, and apply the contraction mapping principle and the fixed point theorem to study and solve these problems.

冗余控制输入系统在工程中占有重要地位，非线性矩阵方程解的界估计应用于LQR冗余控制输入系统中，具有重要的实用价值和理论意义。本项目针对非线性矩阵方程的解与冗余控制输入系统之间的关系，探讨非线性矩阵方程的解和数值解的更精确的界估计，及其在LQR冗余控制输入问题中的具体应用。. 我们将重点研究以下问题：1）非线性矩阵方程的解、数值解、解的性质、相关的特殊矩阵的性质和判定、解的存在唯一性条件和迭代算法；2）基于LQR冗余控制输入系统的非线性矩阵方程解的更精细的界估计；3）将得到的非线性矩阵方程解的新界估计应用于LQR冗余控制输入中，研究控制器增益的界估计。. 上述问题不仅是矩阵理论和计算的深刻问题，而且是把理论研究应用于解决实际的LQR冗余控制输入问题。我们结合特殊矩阵、矩阵方程、矩阵不等式等相关理论知识，运用压缩映像原理、不动点定理等原理，研究和解决这些问题。

本项目研究非线性矩阵方程解的新界估计和解的迹的界估计，将这些估计运用在LQR冗余控制输入中，分别根据迹的性质和奇异值的性质来分析控制器增益的变化。当系统中控制输入增加的时候，研究了控制器增益的范围。同时，讨论了离散代数Riccati方程和离散耦合代数Riccati方程的数值解，并给出了有效的迭代算法；讨论了连续耦合代数Riccati方程解的上界估计，并给出了更紧上界估计的有效算法。这些研究对于LQR的冗余控制输入系统具有非常重要的实用价值和理论意义。