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Inspired by the cyber-physical systems (CPS) of numerical methods for stochastic differential equations, we present a CPS model of sampled-data control systems (typically a synonym for computer control systems), which regards the intersection of the physical and the cyber (the key feature of CPS). As a theoretic foundation, we develop by the Lyapunov method a stability theory for a general class of stochastic impulsive differential equations (SiDE) which is formulated as a canonical form for CPS that may work in feedback loops and thus include those of sampled-data control systems. Applying the fundamental theory, we study stability of the CPS, which implies that of the sampled-data control system. By our CPS approach, we not only obtain stability criteria for the CPS of sampled-data control systems but also reveal the equivalence and intrinsic relationship between the two main approaches (viz. controller emulation and discrete-time approximation) in the literature. As the applications of our CPS theory, we propose a control design method for feedback stabilization of the CPS of sampled-data stochastic systems. Illustrative examples are conducted to verify that our method significantly improves the existing results. In this paper, we initiate the study of a systems science of design for CPS. This provokes many open and interesting problems.

This paper presents the cyber-physical system (CPS) of a numerical method (the widely-used Euler-Maruyama method) and establishes a foundational theory of the CPSs of numerical methods for stochastic differential equations (SDEs), which transforms the way we understand the relationship between the numerical method and the underlying SDE. The CPS is a seamless integration of the SDE and the numerical method, unlike in the literature where they are treated as separate systems linked by inequalities. We formulate a new and general class of stochastic impulsive differential equations (SiDEs) that can serve as a canonic form of the CPSs and establish a Lyapunov stability theory as a theoretic foundation for our class of SiDEs. By the CPS approach, we show the equivalence and intrinsic relationship between the stability of the SDE and the stability of the numerical method. As application of our proposed results, we develop the CPS theory for linear systems and present the CPS Lyapunov inequality that is the necessary and sufficient condition for mean-square stability of the CPS of the Euler-Maruyama method for linear SDEs. Our proposed CPS theory initiates the study of systems numerics and provokes many open and interesting problems for future work.

Projective non-negative matrix factorization (PNMF) as a variant of NMF has received considerable attention. However, the existing PNMF methods can be further improved from two aspects. On the one hand, the square loss function that is intended to measure the reconstruction error is sensitive to noise. On the other hand, it is non-trivial to estimate the intrinsic manifold of the feature space in a principal manner. So current paper is an attempt that has proposed a new method named as robust ensemble manifold projective non-negative matrix factorization (REPNMF) for image representation. Specifically, REPNMF not only assesses the influence of noise by imposing a spare noise matrix for image reconstruction, but it also assumes that the intrinsic manifold exists in a convex hull of certain pre-given manifold candidates. We aim to remove noise from the data and find the optimized combination of candidate manifolds to approximate the intrinsic manifold simultaneously. We develop iterative multiplicative updating rules for the optimization of REPNMF along with its convergence proof. The experimental results on four image datasets verify that REPNMF is superior as compare to other related state-of-the-art methods.