This article addresses the problem of distributed controller design for linear discrete‐time systems. The problem is posed using the classical framework of state feedback gain optimization over an infinite‐horizon quadratic cost, with an additional sparsity constraint on the gain matrix to model the distributed nature of the controller. An equivalent formulation is derived that consists in the optimization of the steady‐state solution of a matrix difference equation, and two algorithms for distributed gain computation are proposed based on it. The first method consists in a step‐by‐step optimization of said difference matrix equation, and allows for fast computation of stabilizing state feedback gains. The second algorithm optimizes the same matrix equation over a finite time window to approximate asymptotic behavior and thus minimize the infinite‐horizon quadratic cost. To assess the performance of the proposed solutions, simulation results are presented for the problem of distributed control of a quadruple‐tank process, as well as a version of that problem scaled up to 40 interconnected tanks.

中文翻译：

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离散线性系统的分布式控制器设计和性能优化

本文解决了线性离散时间系统的分布式控制器设计问题。问题是使用经典的状态反馈增益优化框架解决了无限水平的二次成本，并在增益矩阵上附加了稀疏性约束来建模控制器的分布式特性。推导了包括优化矩阵差分方程稳态解的等效公式，并在此基础上提出了两种分布式增益计算算法。第一种方法是对所述差分矩阵方程进行逐步优化，并允许快速计算稳定状态反馈增益。第二种算法在有限的时间窗口内优化了相同的矩阵方程，以逼近渐近行为，从而最小化了无限水平二次成本。为了评估所提出解决方案的性能，针对四罐工艺的分布式控制问题以及该问题的一个版本（最多可扩展到40个互连罐），给出了仿真结果。