Approximately bisimilar symbolic models for nonlinear control systems
Approximately bisimilar symbolic models for nonlinear control systems
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DOI：
10.1016/j.automatica.2008.02.021
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发表时间：
2007-06
期刊：
影响因子：
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通讯作者：
G. Pola;A. Girard;P. Tabuada
中科院分区：
文献类型：
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作者：
G. Pola;A. Girard;P. Tabuada
控制系统通常由描述物理现象如何受到某些控制参数或输入影响的微分方程建模。虽然这些模型在处理物理现象时非常强大，但它们不太适合描述与物理世界接口的软件和硬件。由于这个原因，人们越来越关注通过符号模型来描述控制系统，符号模型是对连续动态的抽象描述，其中每个“符号”对应于连续模型中状态的“集合”。由于这些符号模型与计算机科学中用于描述软件和硬件的模型具有相同的性质，因此它们为研究软件和硬件与物理世界相互作用的控制问题提供了统一的语言。此外，符号模型的使用使人们能够利用监督控制技术和控制器合成目的的博弈论算法。在本文中，我们证明了每一个增量全局渐近稳定的非线性控制系统是近似等价（双相似）的符号模型。近似误差是符号模型构造中的设计参数，并且可以呈现为期望的小。此外，如果控制系统的状态空间是有界的，所得到的符号模型是有限的。对于数字控制系统，在更强的增量输入-状态稳定性假设下，可以通过适当的量化输入来构造符号模型。
Control systems are usually modeled by differential equations describing how physical phenomena can be influenced by certain control parameters or inputs. Although these models are very powerful when dealing with physical phenomena, they are less suited to describe software and hardware interfacing with the physical world. For this reason there is a growing interest in describing control systems through symbolic models that are abstract descriptions of the continuous dynamics, where each “symbol” corresponds to an “aggregate” of states in the continuous model. Since these symbolic models are of the same nature of the models used in computer science to describe software and hardware, they provide a unified language to study problems of control in which software and hardware interact with the physical world. Furthermore, the use of symbolic models enables one to leverage techniques from supervisory control and algorithms from game theory for controller synthesis purposes. In this paper we show that every incrementally globally asymptotically stable nonlinear control system is approximately equivalent (bisimilar) to a symbolic model. The approximation error is a design parameter in the construction of the symbolic model and can be rendered as small as desired. Furthermore, if the state space of the control system is bounded, the obtained symbolic model is finite. For digital control systems, and under the stronger assumption of incremental input-to-state stability, symbolic models can be constructed through a suitable quantization of the inputs.