区间时变时滞离散系统的稳定性分析及控制器设计

含有区间状态时滞的系统在实际中有重要的应用,如它可以方便地描述一类网络控制系统.本文对含有区间状态时滞的线性离散系统考虑了稳定性分析与状态反馈控制器设计问题.为研究该类系统的稳定性,通过引入二重求和和三重求和构造了新型的Lyapunov-Krasovskii泛函.在对所构造的泛函处理其差分的过程中,利用了基于Abel引理的有限和不等式技术以及时滞分割方法,进而提出了该类系统稳定的线性矩阵不等式充分性条件.相对于以往的Lyapunov泛函,本文所提出的Lyapunov-Krasovskii泛函包含更多的时滞信息,而且所采用的差分处理方法不涉及原系统的模型变化,因而所提出的渐近稳定充分条件具有较低保守性.另外,相对于以前存在的自由权矩阵方法,本文所提出的方法具有较少的可行性变量矩阵,因而能有效地降低计算量.基于所提出的稳定性条件,本文进一步提出了无记忆状态反馈镇定控制器的设计方法.所提出镇定控制器设计方法也通过线性矩阵不等式给出,具有很好的数值稳定性.最后,通过数值算例验证了所提方法的正确性和有效性.

Stability analysis and controller design for discrete-time systems with interval time-varying delays

The systems with interval time-delay appearing in state variable have wide application, and can be used to describe a class of networked control systems. The stability analysis and the state feedback controller design are investigated for linear discrete-time systems with interval time delays. In order to investigate the stability, a new Lyapunov-Krasovskii functional is proposed by introducing the double summation and triple summation. In order to estimate the difference of the Lyapunov-Krasovskii functional, the Abel lemma based finite sum inequality technique and the time delay segmentation method are used as tools, and thus a sufficient condition is presented for the asymptotic stability of the considered systems in terms of linear matrix inequalities. Compared to the traditional Lyapunov functional, since the proposed Lyapunov-Krasovskii functional contains more information on delays, and the approach for estimating the difference does not involve model transformation, thus the presented sufficient condition for the asymptotic stability of the system is less conservative. In addition, compared with the previous free-weighting matrix method there exist less feasible matrix variables in the current method. Thus, the computational load can be effectively reduced. According to the derived stability criterion, a design approach for non-memory state feedback controllers is presented in terms of linear matrix inequalities, and thus has good numerical stability. Finally a numerical example is employed to illustrate the effectiveness of the methods proposed in this paper.