非线性系统的Koopman可观测性与数据驱动的Koopman--Luenberger观测器
Koopman Observability of Nonlinear Systems and Data-Driven Koopman--Luenberger Observer
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中文总结 AI 辅助
本文提出Koopman可观测性与可检测性概念，基于RKHS建立指数可检测条件，利用Yosida近似和数据驱动方法将观测器增益求解化为半定规划，保证观测误差均方有界。
中文摘要 AI 辅助
虽然非线性系统通常可以通过其Koopman半群描述为无限维线性系统，但基于学习得到的Koopman算子模型，以类似于有限维线性系统的方式直接综合状态观测器，仍然是一个开放问题。本文提出了非线性系统的Koopman可观测性（精确和近似）、Koopman可检测性（指数和强）以及Koopman--Luenberger观测器的概念。通过在再生核希尔伯特空间（RKHS）上定义Koopman半群，建立了Koopman指数可检测性的条件，这涉及求解一个算子Lyapunov不等式，该不等式仅隐式地涉及Koopman半群的无穷小生成元。通过使用Koopman生成元的有限时域Yosida近似和经验采样算子，观测器增益的数据驱动求解可以简化为由样本指定的有限维子空间，从而通过凸半定规划实现。在采样算子及其经验算子的闭环耗散条件下，状态观测误差被保证在均方意义上有界。
英文摘要
While nonlinear systems can be generally described as infinite-dimensional linear systems via their Koopman semigroups, the direct synthesis of state observers on the premise of learned Koopman operator models, in an analogous manner to finite-dimensional linear systems, remains an open problem. In this paper, the concepts of Koopman observability (exact and approximate), Koopman detectability (exponential and strong), and Koopman--Luenberger observer of nonlinear systems are provided. By defining the Koopman semigroup on a reproducing kernel Hilbert space (RKHS), the conditions for Koopman exponential detectability are established, which involves solving an operator Lyapunov inequality that only implicitly involves the the infinitesimal generator of the Koopman semigroup. Through the use of a finite-horizon Yosida approximation of the Koopman generator and an empirical sampling operator, the data-driven solution of the observer gain can be reduced to a finite-dimensional subspace specified by the sample and hence fulfilled by convex semidefinite programming. The state observation error is guaranteed to be bounded in mean squares, under closed-loop dissipation conditions on the sampling operator and its empirical operator.
发表机构
- North Carolina State University（北卡罗来纳州立大学）
机构由 AI 辅助整理，请以论文原文为准。
