In this paper, we consider a particular class of optimal switching problem for the linear-quadratic switched system in discrete time, where an optimal switching sequence is designed to minimize the quadratic performance index of the system with a switching cost. This is a challenging issue and studied only by few papers. First, we introduce a total variation function with respect to the switching sequence to measure the volatile switching action. In order to restrain the switching magnitude, it is added to the cost functional as a penalty. Then, the particular optimal switching problem is formulated. With the positive semi-definiteness of matrices, we construct a series of exact lower bounds of the cost functional at each time and the branch and bound method is applied to search all global optimal solutions. For the comparison between different global optimization methods, some numerical examples are given to show the efficiency of our proposed method.

中文翻译：

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最佳的开关信号设计，降低了开关动作

在本文中，我们考虑了离散时间线性二次切换系统的一类最佳切换问题，其中设计了一个最佳切换序列，以最小化具有切换成本的系统二次性能指标。这是一个具有挑战性的问题，只有很少的论文对此进行了研究。首先，我们针对开关序列引入总变化函数，以测量易失性开关动作。为了限制切换幅度，将其作为代价添加到成本函数中。然后，提出特定的最佳切换问题。利用矩阵的正半定性，我们每次构造了一系列成本函数的精确下界，并且采用分支定界法搜索所有全局最优解。