Stability, Inertia, and Robust Stability

Flooding and Overflow Mitigation Using Deep Reinforcement Learning Based on Koopman Operator of Urban Drainage Systems

“…One of the important aspects of a given system is whether a linear structure behind the dynamics can be found (Bieker et al., 2020; Bistrian & Navon, 2015; Brunton et al., 2016; Krener & Isidori, 1983; Page & Kerswell, 2018). This is because, many properties of the system, such as stability, etc., can be directly analyzed using many effective methodologies designed for linear structures (Anderson et al., 2013; DATTA, 2004; De Santis et al., 2008; Kozlov & Furta, 1996; Rowley et al., 2009). Therefore, a large number of studies in the context of mathematics, cybernetics, and machine learning have considered the existence of a linear structure as a reflection of interpretability (Liu et al., 2021; Mardt et al., 2018; Montavon et al., 2017; Samek et al., 2021; Tu et al., 2014; Zhang et al., 2021).…”

“…We take the asymptotically stability as an example. According to the Lyapunov stability theorem, a discrete‐time system like DLEDMD (Equation ) is asymptotically stable if and only if all the eigenvalues of Σ are inside the unit circle (DATTA, 2004). In this study, the eigenvalues of DLEDMD (in Figure 11) were easily obtained with the help of the Koopman matrix.…”

“…Many properties of a given dynamic system can be easily analyzed through its linear structure with the help of many effective methodologies (DATTA, 2004; De Santis et al., 2008; Kozlov & Furta, 1996). Therefore, the existence of a linear structure is a reflection of better interpretability in the context of mathematics, cybernetics, and machine learning (Liu et al., 2021; Mardt et al., 2018; Montavon et al., 2017; Tu et al., 2014; Zhang et al., 2021).…”

Simulation and Dynamic Properties Analysis of the Anaerobic–Anoxic–Oxic Process in a Wastewater Treatment PLANT Based on Koopman Operator and Deep Learning

“…If the eigenvalues and eigenvectors of the Koopman matrix cannot be obtained, the linear structure can also be given through the singular values and singular vectors of the matrix K [40]. Through the eigenvalues, the dynamic properties of A2O process can be directly analyzed and evaluated using the theorems of existing dynamic system and linear system, including the Lyapunov stability and convergence of the dynamics [16]. This can determine the ability of a biochemical reactor to withstand external disturbances and its long-term behavioral characteristics during operation.…”

“…The Koopman operator provides a solution to the above problem. For a given nonlinear dynamic system, obtaining its Koopman operator can help us find its eigenvalues, thereby revealing its linear structure and analyzing its dynamic properties mathematically [15][16][17][18][19][20][21]. The theory and correlated methods have been widely used in model prediction [22], data fusion [23], fluid dynamics [24,25], and system control [26][27][28].…”