Verification and synthesis of Cyber-Physical Systems (CPS) are challenging and still raise numerous issues so far. In this paper, an original framework with mixed sets defined as function images of symbol type domains is first proposed. Syntax and semantics are explicitly distinguished. Then, both continuous (interval) and discrete (signed, boolean) symbol types are used to model dependencies through linear and polynomial functions, so leading to mixed zonotopic and polynotopic sets. Polynotopes extend sparse polynomial zonotopes with typed symbols. Polynotopes can both propagate a mixed encoding of intervals and describe the behavior of logic gates. A functional completeness result is given, as well as an inclusion method for elementary nonlinear and switching functions. A Polynotopic Kalman Filter (PKF) is then proposed as a hybrid nonlinear extension of Zonotopic Kalman Filters (ZKF). Bridges with a stochastic uncertainty paradigm are outlined. Finally, several discrete, continuous and hybrid numerical examples including comparisons illustrate the effectiveness of the theoretical results.

中文翻译：

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带类型符号的功能集：用于混合非线性可达性和过滤的框架和混合 Polynotopes

网络物理系统 (CPS) 的验证和综合具有挑战性，并且迄今为止仍存在许多问题。在本文中，首次提出了一种原始框架，其混合集定义为符号类型域的函数图像。语法和语义是明确区分的。然后，连续（区间）和离散（有符号，布尔）符号类型都用于通过线性和多项式函数对依赖项进行建模，从而导致混合带域和多项式集。Polynotopes 使用类型符号扩展稀疏多项式zonotopes。多项式既可以传播区间的混合编码，又可以描述逻辑门的行为。给出了函数完备性结果，以及初等非线性和切换函数的包含方法。然后提出多核卡尔曼滤波器 (PKF) 作为带域卡尔曼滤波器 (ZKF) 的混合非线性扩展。概述了具有随机不确定性范式的桥梁。最后，包括比较在内的几个离散、连续和混合数值例子说明了理论结果的有效性。