In this paper we study the robust invariant sets generation problem for discrete-time switched polynomial systems subject to disturbance inputs within the optimal control framework. A robust invariant set of interest is a set of states such that every possible trajectory starting from it never leaves a specified safe set, regardless of actual disturbances. The maximal robust invariant set is shown to be the zero level set of the unique bounded solution to a Bellman type equation, which is a functional equation being widely used in discrete-time optimal control. This is the main contribution of this work. The uniqueness of bounded solutions enables us to solve the derived Bellman type equation using numerical methods such as the value iteration, which provides an approximation of the maximal robust invariant set. In order to increase the scalability of the Bellman equation based method, a semi-definite program, which is constructed based on the derived Bellman type equation, is also implemented to synthesize robust invariant sets. Finally, three examples demonstrate the performance of our methods.
翻译:在本文中,我们研究了在最佳控制框架内受扰动输入影响的离时交换多式系统产生的稳健的变数组生成问题。 稳健的变数组是一组状态, 使得从它开始的每一条可能的轨迹都永远不会留下特定的安全套件, 不论实际的扰动。 最大稳健的变数组显示是贝尔曼型方程式独有的捆绑解决方案的零水平组, 这是在离时最佳控制中广泛使用的一种功能方程式。 这是这项工作的主要贡献。 约束式解决方案的独特性使我们能够使用数值转换法等数值方法解决导出的贝尔曼型方程式。 为了增加基于贝尔曼方程式的方法的可缩放性, 以所生成的贝尔曼型方程式为基础构建的半定式程序也用于合成稳健的变数组。 最后, 三个例子展示了我们方法的性能。