The control properties of discrete-time switched linear systems (SLSs) with switching signals generated by logical dynamical systems are studied using the semitensor product (STP) approach. With the algebraic state-space representation (ASSR), the linear modes and the logical generators are aggregated as a system with hybrid states, leading to the criteria of reachability, controllability, observability, and reconstructibility of the SLSs. Algorithms for checking these properties are given. Then, two kinds of realization problems concerning whether the logical dynamical systems can generate the desired switching signals are investigated, and necessary and sufficient conditions for the realizability of the desired switching signals are given with respect to the cases of fixed operating time (FOT) switching and finite reference signal switching.

It is a common belief that the order of a Boolean network is mainly determined by its attractors, including fixed points and cycles. Using the semi-tensor product (STP) of matrices and the algebraic state-space representation (ASSR) of the Boolean networks, this article reveals that in addition to this explicit order, there is a certain implicit or hidden order, which is determined by the fixed points and limit cycles of their dual networks. The structure and certain properties of dual networks are investigated. Instead of a trajectory, which describes the evolution of a state, the hidden order provides a global horizon to describe the evolution of the overall network. We conjecture that the order of networks is mainly determined by the dual attractors via their corresponding hidden orders. Then these results about the Boolean networks are further extended to the k -valued case.

In this article, a general approach for directed graph clustering and two new density-based clustering objectives are presented. First, using an equivalence between the clustering objective functions and a trace maximization expression, the directed graph clustering objectives are converted into the corresponding weighted kernel$k$-means problems. Then, a nonspectral algorithm, which covers both the direction and weight information of the directed graphs, is thus proposed. Next, with Rayleigh’s quotient, the upper and lower bounds of clustering objectives are obtained. After that, we introduce a new definition of weak links to characterize the effectiveness of clustering. Finally, illustrative examples are given to demonstrate effectiveness of the results. This article provides a glance at the potential connection between density-based and pattern-based clustering. Compared with other approaches for directed graph clustering, the method proposed in this article naturally avoids the loss of the nonsymmetric edge data because there is no need for any additional symmetrization.