When discrete-time Markov jump linear systems are prone to the damaging effects of polytopic uncertainties, it is necessary to address all the vertices of each Markov mode in order to properly design robust controllers. To this end, we propose a robust recursive linear–quadratic regulator for this class of systems. We define a quadratic min–max optimization problem by combining least-squares and penalty functions in a unified framework. We design a one-step cost function to encompass the entire set of vertices of each mode altogether, while maintaining its quadratic structure and the convexity of the problem. The solution is then obtained recursively and does not require numerical optimization packages. We establish conditions for convergence and stability by extending the matrix structure of the recursive solution. In addition, we provide numerical and real-world application examples to validate our method and to emphasize recursiveness and diminished computational effort.

中文翻译：

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多面体不确定性马尔可夫跳跃线性系统的调节

当离散时间马尔可夫跳跃线性系统容易受到多面体不确定性的破坏性影响时，有必要解决每个马尔可夫模式的所有顶点，以便正确设计鲁棒控制器。为此，我们为此类系统提出了一种稳健的递归线性二次调节器。我们通过在一个统一的框架中结合最小二乘和惩罚函数来定义二次最小-最大优化问题。我们设计了一个一步成本函数来完全包含每个模式的整个顶点集，同时保持其二次结构和问题的凸性。然后递归获得解决方案，不需要数值优化包。我们通过扩展递归解的矩阵结构来建立收敛和稳定性的条件。此外，