Finding Cut-Edges and the Minimum Spanning Tree via Semi-Tensor Product Approach

ABSTRACTIn this paper, the static output feedback stabilisation of discrete event dynamic systems (DEDSs) is investigated via the semi-tensor product (STP) of matrices. Firstly, the dynamics of DEDSs modelled by deterministic Moore-type finite automata are converted into an matrix expression in the STP frame. Secondly, necessary and sufficient conditions for the existence of a static output feedback control pattern, stabilising the controlled discrete event dynamic systems to some equilibrium point, are given, and constructive algorithms to seek the static output feedback control pattern including an effective specific solution algorithm and an analytic solution algorithm are proposed. Thirdly, the equilibrium-based static output feedback stabilisation of DEDSs is extended to the set-based static output feedback stabilisation and necessary and sufficient conditions and constructive algorithms to seek the corresponding control pattern are provided. Finally, some examples are presented to illustrate the effectiveness of the proposed approach.

In this paper, we address the problem of calculating basis siphons in Petri nets (PNs) by resorting to the semi-tensor product (STP) of matrices. The proposed appraoch is based on our previous results on the calculation of siphons and minimal siphons in PNs, whose key notion is that of the siphon equation (SE) of PNs. First, we discuss the properties of the SE of PNs in the framework of STP. Second, an efficient recursion algorithm is proposed, which can be applied to computing all basis siphons for any PNs. Some results on the computational complexity of the proposed algorithm, in this paper, are also provided, as well as comparison with the existing approach. Last, an example is presented to illustrate the theoretical results. The proposed approach proved highly effective in calculating all basis siphons of PNs and only requires matrix manipulations.

This paper proposes an improved algorithm for enumerating siphons and minimal siphons of Petri nets (PNs) in the framework of [18]. First, the logical equation of each transition is converted into a matrix equation by using the semi-tensor product (STP) of matrices, and then the matrix equation group of the PNs is obtained. Second, an improved algorithm is proposed to calculate the siphons and minimal siphons, respectively. Finally, two illustrative examples are presented to show the efficiency and application of the improved algorithm.

In this study, we investigate the control problem of asynchronous sequential machine with adversarial input by using the semi-tensor product (STP) of matrices. First, by applying the STP of matrices, a new algebraic expression of asynchronous sequential machines with an adversarial input is deduced. Next, by investigating the control process, the controller with the minimal state set is designed, and two algorithms for assigning the values to the state transition structure matrix and the output structure matrix of the controller are proposed. Then, in the framework of the algebraic expression of the asynchronous sequential machines with adversarial input and two proposed algorithms, an algebra expression of the closed-loop system dynamic is established; the theoretical results are verified by using this algebra expression. Finally, an example is presented to illustrate the validity and application of the proposed approach.

The logical control system theory (LCST) based on semi-tensor product (STP) of matrices has become a new direction in control field. The purpose of this paper is to give a comprehensive introduction to the logical control system, from the basic mathematical tools and analysis methods to the main results. First we introduce the STP of matrices. Meanwhile, a digging discussion is made to reveal some of its insides. Then we explain how to use STP to convert a logic expression into algebraic form. Then an itemized introduction to the main achievements in analysing and control design of LCSs in last few years is presented, which consists of the topological structure of logical systems, controllability, observability, and stabilization, the stability of disturbance decoupling, optimal control, identification of LCSs. Next, the extension from classical Boolean logic to $k$-valued logic, to multi-valued logic, and to mixed-valued logic is explained. All these results form the theoretical foundation of this new discipline. In addition, the concepts, notations, and the fundamental method/tool have been uniformly normalized. Then some current developments of LCST and its applications or potential applications have been discussed. Some open problems and possible research directions have also been presented. Finally, a personal prediction about the development of control theory is given as: The combination of logic control systems and dynamic control systems is expected to generate a ``information-control system, it is what Tsien Hsueshen predicted ``theorical control theory, and is also the next generation of control theory proposed by American discipline development report.

This paper studies the structural monostability and structural cycle‐stability of Boolean networks (BNs). Firstly, the structural‐equivalent Boolean networks are converted to the algebraic forms by using the semitensor product of matrices. Secondly, the concepts of structural monostability and structural cycle‐stability for Boolean networks are proposed. On the basis of the algebraic forms of structural‐equivalent Boolean networks, some necessary and sufficient conditions are presented for the structural monostability and structural cycle‐stability of Boolean networks. Finally, an illustrative example is worked out to show the effectiveness of the obtained results.

On stability of switched homogeneous nonlinear systems

This paper investigates the output controllability problem of temporal Boolean networks with inputs (control nodes) and outputs (controlled nodes). A temporal Boolean network is a logical dynamic system describing cellular networks with time delays. Using semi-tensor product of matrices, the temporal Boolean networks can be converted into discrete time linear dynamic systems. Some necessary and sufficient conditions on the output controllability via two kinds of inputs are obtained by providing corresponding reachable sets. Two examples are given to illustrate the obtained results.

This paper studies the stability and stabilization of multi-valued logical networks with respect to part of the system's states. By semi-tensor product of matrices, we first convert multi-valued logical networks into a linear discrete-time system. Then a necessary and sufficient condition for partial stability of k-valued logical networks is given. Finally, the stabilization of multi-valued logical control networks by state feedback control is investigated and the corresponding necessary and sufficient condition is obtained. An example is provided to illustrate the efficiency of the obtained results.

This note investigates the output regulation of Boolean control networks (BCNs) by using the semi-tensor product of matrices, and presents a number of new results. Firstly, based on an algebraic state-space representation of BCNs, a necessary and sufficient condition is presented for the solvability of the output regulation problem. Secondly, an effective method is proposed for the control design of the output regulation problem by using an augmented system approach. The study of an illustrative example shows that the obtained new results are effective in dealing with the output regulation of BCNs.

The concepts of k-degree and k-balance control sets of graphs are abstracted from some real-world problems such as the location of cameras for saving resources in street security monitoring and the arrangement of major sensors for balancing load in wireless sensor networks. Then using a new matrix analysis tool, semi-tensor product of matrices (STP) developed in recent years, such two kinds of control sets are formulated by establishing a necessary and sufficient conditions that determine whether or not a vertex subset of a graph is a k-degree control set, and a necessary condition that facilitates finding k-balance control sets of a graph. Finally the effectiveness of the idea and the correctness of the results are verified by studying security monitoring of a street.

In this paper, the stability of finite evolutionary games (EGs) with time-varying payoffs is studied by using the Lyapunov-based technique. First, the profile dynamics of EGs with time-varying payoffs is expressed into an algebraic form via the semi-tensor product of matrices. Second, a sufficient condition is obtained for the global stability of EGs with time-varying payoffs, and a common Lyapunov function is explored. By using the potential equation, a potential-based formula is proposed to construct a common Lyapunov function for an EG based on a near potential game. Finally, an example is given to illustrate the theoretical results.

This paper investigates the modeling and stability of a class of finite evolutionary games with time delays in strategies. First, the evolutionary dynamics of a sequence of strategy profiles, named as the profile trajectory, is proposed to describe the strategy updating process of the evolutionary games with time delays. Using the semi-tensor product of matrices, the profile trajectory dynamics with two kinds of time delays are converted into their algebraic forms respectively. Then a sufficient condition is obtained to assure the stability of the delayed evolutionary potential games at a pure Nash equilibrium.

Further results on feedback stabilization control design of Boolean control networks

Stabilizability analysis and switching signals design of switched Boolean networks