Semi-tensor Product Method Research Articles

One-Bit Function Perturbation Impact on Robust Set Stability of Boolean Networks with Disturbances

This paper investigates the one-bit function perturbation (OBFP) impact on the robust set stability of Boolean networks with disturbances (DBNs). Firstly, the dynamics of these networks are converted into the algebraic forms utilizing the semi-tensor product (STP) method. Secondly, OBFP’s impact on the robust set stability of DBNs is divided into two situations. Then, by constructing a state set and defining an index vector, several necessary and sufficient conditions to guarantee that a DBN under OBFP can stay robust set stable unchanged are provided. Finally, a biological example is proposed to demonstrate the effectiveness of the obtained theoretical results.

Stability analysis of large-scale Boolean networks via compositional method

This paper formulates a compositional method to alleviate the high computational cost of studying large-scale Boolean networks (BNs). The algebraic expression of conjunctive composition is firstly presented through the semi-tensor product method. Accordingly, the relationship of state transitions between component BNs and conjunctive compositional BN is revealed. To describe the merging process during the conjunctive composition of state transitions, the single-node compositional equation is constructed, based on which a new criterion is built to analyze the stability of conjunctive compositional BNs. The idea of compositional equation is used to further explore the multi-node composition. Finally, the application to the BN model of Yeast Apoptosis demonstrates the effectiveness of the compositional method.

Interval Type-2 Fuzzy Relation Matrix Model via Semitensor Product

A new modeling idea of interval type-2 fuzzy logic system (IT2 FLS) is proposed on the basis of matrix semi-tensor product (STP) fuzzy representation technique, to construct an interval type-2 fuzzy relationship matrix (IT2 FRM) model. The model naturally inherits the advantages of STP theory, which enables the fuzzy logic reasoning process to be translated into algebraic form for computation. In most cases, the fuzzy rules in the IT2 FRM are generally unknown. To address the above situation, the parameters of this model can be trained with observed data pairs. Firstly, the fuzzy rules are generated by the space division method, and the initial parameters of the model are determined. Secondly, the parameters in the model are trained using the steepest descent algorithm. Finally, the IT2 FRM model is validated by simulation, and the results demonstrate that the IT2 FLS can be designed using the STP method.

Fault Detectability of Asynchronous Switched Boolean Networks: A Set Reachability Approach.

This article addresses the fault detectability problem of asynchronous switched Boolean networks, which is focused on whether occurrence of the faults would have an impact on the outputs of the considered network. By applying the semi-tensor product method, the asynchronous switching scheme of the considered system is converted into multiple switching signals. Based on them, an augmented system is established to transform the fault detectability issue of the original system into a set reachability problem. Moreover, some criteria for the fault detectability of the asynchronous switched Boolean control networks are also proposed. Additionally, two algorithms are given to design feasible control sequences after explicitly analyzing the considered augmented system. Finally, an example is provided to illustrate the significance of the obtained theoretical results.

Research on the Dynamic Characteristics of Photovoltaic Power Production and Sales Based on Game Theory

This paper mainly studies the dynamic characteristics of production and sales of distributed photovoltaic power. This is beneficial for the construction and development of a green power system, and it promotes the sustainable development of the social economy. First, the production and sales laws of the photovoltaic power are analyzed, and the trade process between photovoltaic power producers (PVPs) and photovoltaic power demanders (PVDs) is modeled as a game. Different from the existing relevant literature, two kinds of payoff bimatrices are provided, which correspond to the PVD market and PVP market, respectively. Then, the dynamic characteristics of the model are revealed by using the semitensor product method. The results present a more reliable theoretical basis for the sustainable development of the green electricity market. Finally, through an illustrative example, it can be seen that the strategies of all participants are constantly changing to obtain more profits rather than stable at a traditional Nash equilibrium point. It is worth pointing out that the method and results are applicable to other distributed low-carbon energies, contributing to the development of sustainable energy systems.

Set Stability of Probabilistic Logical Networks and Its Application to State-Based Games

Set stability of probabilistic multi-valued logical networks (PMLNs) is investigated in this paper. Different from the existing concepts of set stability, the set stability considered in this paper is starting from a given initial set to a given destination set. Firstly, an algebraic expression of PMLNs is established by using the semi-tensor product method. Then, a matrix method is proposed to obtain the largest invariant subset contained in the given destination set. Based on the invariant subset, a necessary and sufficient condition for set stability from a given initial set to a destination set is presented. Finally, application to state-based games is considered, and the criterion proposed for PMLNs is applied to verify the stability of recurrent state equilibrium.

A novel method for driven stability of NFSRs via STP

The nonlinear feedback shift register (NFSR) is the main component of the convolutional decoder. This paper provides a novel method for investigating the driven stability of NFSRs with input via semi-tensor product (STP). By using the STP method, a backward state transition matrix, as well as a backward NFSR, is constructed. Backward NFSRs can help to find the predecessors of any states so that an algorithm for the global stability of autonomous NFSRs is designed. On this basis, two algorithms are proposed for the driven stability of NFSRs with input, and they have lower computational complexity than the existing methods. Meanwhile, some numerical examples are presented to support the results of this paper. Finally, the relationship between stability and connectedness is discussed fully, which derives some interesting results.

Asymptotical set stabilization of large-scale logical networks with stochastic delays and the application in finite-field networks

In this paper, asymptotical set stabilization (ASS) of k-valued logical networks (KVLNs) with stochastic delays is studied by designing network-structure-based (NS-based) distributed pinning controllers (DPCs). Firstly, based on the NS-based graph method, k-valued delayed logical networks (KVDLNs) are converted into KVLNs with virtual vertices to handle the effect of stochastic delays. In addition, KVLNs with virtual vertices are transformed into the corresponding algebraic form by the semi-tensor product (STP) method. Based on the system vertices classification technique and the concept of feedback arc set (FAS), the pinning nodes set (PNS) is determined. The mode-independent and mode-dependent DPCs are designed to realize ASS of KVDLNs based on the determined PNS. Next, the obtained theoretical results are further applied to the consensus of finite-field networks (FFNs) with stochastic delays. Finally, two examples are provided to validate the theoretical results of this paper.

Mean Square Exponential Stability of Stochastic Delay Differential Systems with Logic Impulses

This paper focuses on the mean square exponential stability of stochastic delay differential systems with logic impulses. Firstly, a class of nonlinear stochastic delay differential systems with logic impulses is constructed. Then, the logic impulses are transformed into an equivalent algebraic expression by using the semi-tensor product method. Thirdly, the mean square exponential stability criteria of nonlinear stochastic delay differential systems with logic impulses are given. Finally, two kinds of stochastic delay differential systems with logic impulses and uncertain parameters are discussed, and the coefficient conditions guaranteeing the mean square exponential stability of these systems are obtained.

Lyapunov-based sampled-data set stabilisation of boolean control networks with time delay and state constraint

Lyapunov-based approach to sampled-data set stabilisation of constrained delayed Boolean control networks (DBCNs) is investigated in this paper. The main mathematical tool is semi-tensor product (STP) of matrices. Since the sampling interval is selected from a finite set, the STP method is adopted to convert the dynamics of constrained DBCNs under nonuniform sampled-data control into a switched Boolean network (SBN). It is worth noting that the switches can only occur at the sampling instant. Using the techniques of Lyapunov function and average dwell time, several sufficient conditions are proposed for the global stability of SBN. Moreover, by virtue of reachable set approach, a procedure is established to design state feedback sampled-data stabilisers for constrained DBCNs. The obtained results are applied to the cell survival regulation of apoptosis networks.

Lebesgue sampling approach to robust stabilization of boolean control networks with external disturbances

This paper introduces the Lebesgue sampling approach to the robust stabilization of Boolean control networks (BCNs) with external disturbances. Given a Lebesgue sampling region and a feedback control, a time aggregated system is obtained via the semi-tensor product method. Then, a new criterion is presented for the robust stabilization of time aggregated system. Furthermore, given a signal of Lebesgue sampling, a sequence of the Lebesgue type robust reachable sets is constructed. Based on these reachable sets, several algorithms are presented to design both Lebesgue sampling region and sampled-data state feedback control for the robust stabilization of BCNs.

Survey of semi-tensor product method in robustness analysis on finite systems.

Recently, the theory of semi-tensor product (STP) method of matrices has received much attention from variety communities covering engineering, economics and industries, etc. This paper describes a detailed survey on some recent applications of the STP method in finite systems. First, some useful mathematical tools on the STP method are provided. Second, many recent developments about robustness analysis on the given finite systems are delineated, such as robust stable analysis of switched logical networks with time-delayed, robust set stabilization of Boolean control networks, event-triggered controller design for robust set stabilization of logical networks, stability analysis in distribution of probabilistic Boolean networks, and how to solve a disturbance decoupling problem by event triggered control for logical control networks. Finally, several research problems in future works are predicted.

I-S detectability of partially-observed discrete event systems: a novel matrix-based method

The state estimation problem is always considered to be critical in discrete event systems (DESs). In this paper, two methods are proposed for investigating the initial state estimation problem: one is the matrix-based dimension reduction tracking observation system, and the other is the matrix-based reversal observation system. In our paper, the initial state estimation is treated as the initial-state detectability (I-S detectability). Using the Boolean semi-tensor product method of matrices, the corresponding algebraic forms of the partially-observed DES are separately constructed, where the corresponding computational complexity is reduced to some degree. Based on the newly defined state transition output-event observation matrix, necessary and sufficient criteria are established to determine the I-S detectability of the addressed system. Illustrative examples are also given to show feasibility of the derived results.

Robust Output Tracking of Boolean Control Networks over Finite Time

With an increase in tracking time, the operating cost of the controller will increase accordingly. Considering the biological applications of Boolean control networks (BCNs), it is necessary to study the control problem of BCNs over finite time. In this paper, we study the output tracking problem of a BCN with disturbance inputs in a given finite time. First, the logical form of BCNs is transformed into an algebraic form using the semi-tensor product (STP) method. Then, the robust output tracking problems of a reference output trajectory and the outputs of a reference system over finite time are transformed into the robust reachability problem of the BCNs. Based on the truth matrix technique, two necessary and sufficient conditions are provided for the trackability of the reference outputs over finite time. Moreover, two algorithms are proposed to design the controllers in the case of the traceable outputs. It should be pointed out that the truth matrix method we used here has some unique features, including its simple computation and concise expression. Finally, two illustrative examples are presented to demonstrate the results in this paper.

Robust Stabilization of Impulsive Boolean Control Networks with Function Perturbation

This paper studies the robust stabilization of impulsive Boolean control networks (IBCNs) with function perturbation. A Boolean control network (BCN) with a state-dependent impulsive sequence is converted to an equivalent BCN by the semi-tensor product method. Based on the equivalence of stabilization between the IBCN and the corresponding BCN, several criteria are proposed for the robust stabilization of IBCNs. Furthermore, when the IBCN is not robustly stabilizable after the function perturbation, an algorithm is presented to modify the control or the impulse-triggered set. Finally, an example is given to verify the obtained results.

Survey on Mathematical Models and Methods of Complex Logical Dynamical Systems

Logical dynamical systems (LDSs) have wide applications in gene regulation, game theory, digital circuits, and so on. In LDSs, phenomena such as impulsive effect, time delays, and asynchronous behavior are not negligible, which generate complex LDSs. This paper presents a detailed survey on models and methods of investigating LDSs. Firstly, some preliminary results on LDSs and semi-tensor product (STP) method are presented. Secondly, some new developments on modeling complex LDSs are summarized, including switched LDSs, probabilistic LDSs, delayed LDSs, LDSs with impulsive effects, asynchronous LDSs, constrained LDSs, and implicit LDSs. Finally, the control design techniques of LDSs are reviewed, including reachable set approach, sampled-data control, event-triggered control, and control Lyapunov function method.

A Matrix Approach to the Modeling and Analysis of Network Evolutionary Games with Mixed Strategy Updating Rules

So far, most studies on networked evolutionary game have focused on a single update rule. This paper investigated the seeking of the Nash Equilibrium and strategy consensus of the evolutionary networked game with mixed updating rules. First, we construct the algebraic formulation for the dynamics of the evolutionary networked game by using the semi-tensor product method. Second, based on the algebraic form, the dynamic behavior of networked evolutionary games is discussed, and an algorithm to seek the Nash equilibrium is proposed. Third, we investigate the strategy consensus problem for a special networked evolutionary game. Finally, some illustrative examples are given to verify our conclusions.

Research on strategy of green electricity acquisition transaction of park-level energy internet by using STP

In order to save resources and reduce air pollution, human beings have begun to pay attention to the production and use of photovoltaic, wind power and other green power. Due to the difficulty of direct transaction between green power producers and power users, a park-level energy Internet has been proposed and used to connect all kinds of green electricity with power users. Then park users can effectively buy and use green electricity. Taking the park-level energy Internet as the scenario, this paper constructs a transaction model between green power operators and green power producers. The model is a dynamic game of complete and perfect information. The dynamic characteristics of this game model are analyzed by using semi-tensor product method, and corresponding strategies are provided for all players. From the results obtained, it is easy to find that in many cases, the strategy profile of all participants are constantly changing to obtain more profits, rather than stable at some traditional Nash equilibrium.

On robust control invariance and robust set stabilization of mix‐valued logical control networks

AbstractThis article investigates the robust control invariance and robust set stabilization problems based on a semi‐tensor product method for a class of mix‐valued logical control networks (MVLCNs) with disturbances. First, a calculation method for the largest robust control invariant subset contained in a given set is proposed. Second, based on the robust control invariant subset, the robust set stabilization of MVLCNs is discussed, and new results are presented. Furthermore, the design algorithm of time‐optimal state feedback stabilizers via antecedence solution technique is derived. The study of an illustrative example shows the effectiveness of the obtained new results.

The minimum number of discriminant equations for a symmetric game

The discriminant equations with the minimum number for a symmetric game are derived. By using a new version of the semi-tensor product method based on adjacent transpositions, the analysis of symmetric games is greatly simplified. Some new necessary and sufficient conditions for symmetric games are proposed, a basis of the symmetric game space is easily calculated, and the computing complexity is achieved. Finally, two examples are displayed to show the effectiveness of the proposed method.