The goal of this paper is to present a systematic method to compute reference dependent positively invariant sets for systems subject to constraints. To this end, we first characterize these sets as level sets of reference dependent Lyapunov functions. Based on this characterization and using Sum of Squares (SOS) theory, we provide a polynomial certificate for the existence of such sets. Subsequently, through some algebraic manipulations, we express this certificate in terms of a Semi-Definite Programming (SDP) problem which maximizes the size of the resulting reference dependent invariant sets. We then present the results of implementing the proposed method to an example system and propose some variations of the proposed method that may help in reducing the numerical issues of the method. Finally, the proposed method is employed in the Model Predictive Control (MPC) for Tracking scheme to compute the terminal set, and in the Explicit Reference Governor (ERG) scheme to compute the so-called Dynamic Safety Margin (DSM). The effectiveness of the proposed method in each of the schemes is demonstrated through a simulation study.

中文翻译：

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参考相关不变集：基于平方和的计算及其在约束控制中的应用

本文的目标是提出一种系统方法来计算受约束系统的参考依赖正不变集。为此，我们首先将这些集合表征为依赖于参考的李雅普诺夫函数的水平集。基于此特征并使用平方和 (SOS) 理论，我们为此类集合的存在提供多项式证明。随后，通过一些代数操作，我们根据半确定规划 (SDP) 问题来表达该证明，该问题最大化所得参考相关不变量集的大小。然后，我们将实施所提出方法的结果呈现给示例系统，并提出所提出方法的一些变体，这些变体可能有助于减少该方法的数值问题。最后，所提出的方法在用于跟踪的模型预测控制 (MPC) 方案中用于计算终端集，并在显式参考调控器 (ERG) 方案中用于计算所谓的动态安全裕度 (DSM)。通过仿真研究证明了所提出的方法在每个方案中的有效性。