矩阵不等式在时滞系统中的应用

Application of matrix inequality in control theory

首先提出了矩阵不等式对解决控制论领域中时滞系统分析与综合问题的重要性和必要性。提出了贯穿文章以及时滞系统分析与综合问题的2个重要引理。再次,先利用一元二次函数的思想得到了詹森不等式,并在此基础上提出了用来解决时滞系统中不等式放缩问题的2个推论;接下来利用矩阵乘积巧妙地得到了著名的柯西不等式,并利用柯西不等式得到了进一步的矩阵不等式放缩方法——推论3,同时利用相似的证明方法得到了在解决带有不确定项的时滞系统时采用的方法——定理3;给出了解决时滞系统中问题的最常用的不等式放缩技术——凸组合技术的证明。最后,给出结论,指出文中定理和推论在控制论领域中时滞系统分析与综合问题中的有效作用。

This paper introduces the importance and necessity of a matrix inequality for solving time delay system analysis and synthesis problems in the field of control theory.It presents some related symbols and two important lemmas which are employed to prove the theorem and time delay system analysis and synthesis problems.Then,the paper reveals the Jensen inequality with quadratic function.Meantime,two corollaries which often used to solve inference delay systems inequality problem are proposed.Next,with matrix multiplication,it demonstrates the well-known Cauchy inequality,under which Corollary 3 dealing with the related problem is induced.Theorem 3,used in the time-delay system with uncertainties,has also been obtained with similar method.The paper also puts forward matrix convex combination technique to solve the commonest inequality zoom problems in time delay system.Finally,the conclusions illustrate the effectiveness and practicality of the theorem in this paper.