Theory Of Semi-tensor Product Research Articles

Optimal control of Boolean control networks with state-triggered impulses

In the real world, abrupt changes caused by external environment and system interaction can exhibit a pattern resembling an “impulse” response. A novel model, known as the hybrid-index model, has recently been introduced for Boolean control networks (BCNs) to characterize such impulsive responses. The optimal control problem is a fundamental issue in control theory, and many control problems can be classified as special cases of optimal control problems. Nevertheless, the optimal control problem for the hybrid-index model remains unresolved. This study delves into the optimal control problem of impulsive Boolean control networks (IBCNs) under the hybrid-index model framework. Forward completeness serves as a crucial condition to avert Zeno behavior in the hybrid-index model of IBCNs. In contrast to prior studies on the hybrid-index model of IBCNs, the study shows that the optimal control problem of IBCNs can be addressed even in cases where the system is not forward complete. Leveraging the theory of semi-tensor product, both finite-horizon and infinite-horizon optimal control problems of IBCNs are tackled using the quotient mapping method and the recursive method for optimal control of Boolean control networks. The study yields the optimal cost and optimal control law through state feedback. To illustrate the efficacy of the proposed approach, a mathematical example and a biological example known as the λ switch are presented.

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.

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.

EFR-CSTP: Encryption for face recognition based on the chaos and semi-tensor product theory

Face recognition is a relatively common method humans use to confirm their identity. Protecting the safe transmission of face images has become a hot issue. This paper proposes an encryption method for face images, EFR-CSTP. The EFR-CSTP is very efficient because it only encrypts the facial in the face image. Firstly, the secret keys are generated by Hash. They are the parameters and the initial values of the two-dimensional logistic tent modular map (2D-LTMM), the chaotic sequence required for encryption by 2D-LTMM. Secondly, facial information is recognized by the histogram of oriented gradients (HOG). Note that the entire algorithm only encrypts this facial image. Finally, the facial image is used as the input of the EFR-CSTP, and the ciphertext image is generated by scrambling, STP combined with the XOR diffusion method. We evaluated our algorithm in the IMDB-WIKI dataset. We also evaluate the EFR-CSTP and other algorithms on the same datasets. The evaluation results show that the EFR-CSTP has excellent performance and efficiency.

Image encryption algorithm based on semi-tensor product theory

An image encryption algorithm is proposed by combining chaotic system with semi-tensor product theory. First, the four-way diffusion is performed on the plaintext image with a random interpolation. To further destroy or eliminate the image structure features, the image pixels are split into the octal plane and confused by Josephs scrambling with a variable step. Finally, the ciphertext image is obtained by performing a semi-tensor product operation on an encryption matrix. In this algorithm, the selection of the positions in the chaotic sequence is highly correlated with the plaintext image, which ensures the security of the algorithm without affecting the encryption efficiency.

Stability analysis of time-delay differential systems with impulsive effect suffered by logic choice

In the field of artificial intelligence, nonlinear systems with logic function are everywhere. This paper investigates a time-delay differential system with impulsive effects suffered by logic choice. By the semi-tensor product theory of matrices and constructing Lyapunov function, we analyze the stability of the new system and obtain some stability criteria. The numerical simulations illustrate the validity of these criteria.

A matrix analytic approach for Bayesian network modeling and inference of a manufacturing system

A shared digital twin of a manufacturing system is valuable to provide maintenance services in a collaborative manner. An accurate analytical model of the Bayesian network is crucial to depict endogenous failure mechanisms in the digital twin. Nodes’ connections in Bayesian networks correspond to a range of linear, bilinear or multilinear mappings over finite-state variables. The conditional probability table (CPT) of a child node can be represented as a k-dimensional tensor if it has (k-1) parent nodes. A new matrix analytic approach is proposed for Bayesian network inference based on the theory of semi-tensor product. The matrix representation of probabilistic networks is firstly studied, and a specific parameter training algorithm is constructed based on the matrix model. Bayesian network inference algorithms, including both forward and backward reasoning, are then presented for reliability analysis and fault diagnosis. A real manufacturing system is applied to verify the proposed approach. This matrix analytic approach helps to study the Bayesian network’s mathematical properties, and it is proved to be convenient and efficient in probability network modeling and inference.

A chaotic image encryption algorithm based on a counting system and the semi-tensor product

Based on the n-ary counting system, combined with the matrix semi-tensor product theory and Hilbert curve, a chaotic image encryption algorithm is designed. Different from the traditional encryption method, the algorithm proposed in this paper is an encryption algorithm with scrambling and diffusion at the same time. First, the pixel value is converted from decimal to n-ary. In the n-ary counting system, the plaintext image is randomly divided into some groups, and the Hilbert curve is used for scrambling to each group. The blocks are converted into scrambled images, so that the scrambling and diffusion can be carried out at the same time. Then, in order to improve the security of the algorithm, another round of diffusion is carried out based on matrix semi-tensor product mechanism. Chaotic sequence is generated by Chen system. This chaotic sequence performs matrix semi-tensor product operation with the first round of encrypted image, and generate second encrypted images. Finally, this encryption method is applied to color image encryption. Compared with some representative algorithms, the experimental results show that the algorithm proposed in this paper is secure and it can resist common attacks.

Analyses and Encryption Implementation of a New Chaotic System Based on Semitensor Product

Semitensor product theory can deal with matrices multiplication with different numbers of columns and rows. Therefore, a new chaotic system for different high dimensions can be created by employing a semitensor product of chaotic systems with different dimensions, so that more channels can be selected for encryption. This paper proposes a new chaotic system generated by semitensor product applied on Qi and Lorenz systems. The corresponding dynamic characteristics of the new system are discussed in this paper to verify the existences of different attractors. The detailed algorithms are illustrated in this paper. The FPGA hardware encryption implementations are also elaborated and conducted. Correspondingly, the randomness tests are realized as well, and compared to that of the individual Qi system and Lorenz system, the proposed system in this paper owns the better randomness characteristic. The statistical analyses and differential and correlation analyses are also discussed.

Application of matrix semi-tensor product in chaotic image encryption

This paper proposes a new image encryption method based on matrix semi-tensor product theory. Using Hyperchaotic Lorenz system to generate chaotic sequences, and then using this chaotic sequence to generate two fixed scrambling matrices. The plain image is scrambled by chaotic positioning sort scrambled method. Analogous to a chemical reaction, the plain image scrambled image is one of the reactant, then add a matrix which is related to the plain image information and extracted from the chaotic sequence (this matrix can be differ in size from the plain image matrix, different matrices can be intercepted and generated according to different plain image). This matrix is scrambled as another reactant, Apply the method of semi-tensor product to carry on the reaction (diffusion), finally get the product (encryption image). This method breaks the shackles of the traditional matrix operation and makes the reaction matrix have more forms. Compared with other experimental results, the proposed algorithm is more secure and available.

Image encryption algorithm for synchronously updating Boolean networks based on matrix semi-tensor product theory

This paper studies chaotic image encryption technology and an application of matrix semi-tensor product theory, and a Boolean network encryption algorithm for a synchronous update process is proposed. A 2D-LASM chaotic system is used to generate a random key stream. First, a Boolean network is coded, and a Boolean matrix is generated. If necessary, the Boolean network matrix is diffused in one round so that the Boolean matrix can be saved in the form of an image. Then, three random position scramblings are used to scramble the plaintext image. Finally, using a matrix semi-tensor product technique to generate an encrypted image in a second round of diffusion, a new Boolean network can be generated by encoding the encrypted image. In secure communications, users can choose to implement an image encryption transmission or a Boolean network encryption transmission according to their own needs. Compared with other algorithms, this algorithm exhibits good security characteristics.

Three matrix conditions for the reduction of finite automata based on the theory of semi-tensor product of matrices

Three matrix conditions for the reduction of finite automata based on the theory of semi-tensor product of matrices

Game‐theoretic demand‐side management and closed‐loop control for a class of networked smart grid

A new systematic scheme is proposed for demand-side management and control of a class of networked smart grid. The networked smart grid is modelled into an evolutionary game, and the theory of semi-tensor product is applied to analyse its dynamic behaviours. By using a newly defined stacked form of structural matrix, a closed-loop feedback control is designed to shape the structural matrix of the evolutionary game, such that the Nash equilibrium (with respect to minimum total cost) can be reached and maintained. Based on the proposed feedback control, a robust control can be designed for the smart grid to reach the Nash equilibrium in case of model uncertainties resulted from fake behaviours of some users. Examples are provided to illustrate the theoretical results.

Algebraic state space approach to model and control combined automata

A new modeling tool, algebraic state space approach to logical dynamic systems, which is developed recently based on the theory of semi-tensor product of matrices (STP), is applied to the automata field. Using the STP, this paper investigates the modeling and controlling problems of combined automata constructed in the ways of parallel, serial and feedback. By representing the states, input and output symbols in vector forms, the transition and output functions are expressed as algebraic equations of the states and inputs. Based on such algebraic descriptions, the control problems of combined automata, including output control and state control, are considered, and two necessary and sufficient conditions are presented for the controllability, by which two algorithms are established to find out all the control strings that make a combined automaton go to a target state or produce a desired output. The results are quite different from existing methods and provide a new angle and means to understand and analyze the dynamics of combined automata.

Principle for performing attractor transits with single control in Boolean networks

We present an algebraic approach to reveal attractor transitions in Boolean networks under single control based on the recently developed matrix semitensor product theory. In this setting, the reachability of attractors is estimated by the state transition matrices. We then propose procedures that compute the shortest control sequence and the result of each step of input (control) exactly. The general derivation is exemplified by numerical simulations for two kinds of gene regulation networks, the protein-nucleic acid interactions network and the cAMP receptor of Dictyostelium discoideum network.

SYNCHRONIZATION ANALYSIS FOR MULTIVALUED LOGICAL NETWORKS

The synchronization for two k-valued logical networks of the same dimensions is studied in this paper. First, based on the theory of semi-tensor product of matrices, the master-slave systems (two k-valued logical networks) are converted into discrete-time systems. Second, both open-loop control and feedback control are provided to make the slave network synchronize with the master k-valued logical network. Finally, examples are provided to illustrate the efficiency of the obtained results.

Attractor Transformation by Impulsive Control in Boolean Control Network

Boolean control networks have recently been attracting considerable interests as computational models for genetic regulatory networks. In this paper, we present an approach of impulsive control for attractor transitions in Boolean control networks based on the recent developed matrix semitensor product theory. The reachability of attractors is estimated, and the controller is also obtained. The general derivation proposed here is exemplified with a kind of gene model, which is the protein-nucleic acid interactions network, on numerical simulations.

Controllability and optimal control of a temporal Boolean network

This paper investigates the controllability and optimal control of a temporal Boolean network, where the time delays are time variant. First, using the theory of semi-tensor product of matrices, the logical systems can be converted into a discrete time variant system. Second, necessary and sufficient conditions for the controllability via two types of controls are provided respectively. Third, optimal control design algorithms are presented. Finally, examples are given to illustrate the proposed results.

Evaluation of attractors and basins of asynchronous random Boolean networks

We present an algebraic approach for determining the attractors and their basins of random Boolean networks under an asynchronous stochastic update based on the recently developed matrix semitensor product theory, which allows for converting the logical dynamics of a Boolean network into a standard iterative dynamics. In this setting, all attractors and basins are specified by the network transition matrices. We then devise procedures that can find all attractors and their basins exactly. We also discuss the issue of overlapping basins in asynchronous random Boolean networks, and we propose methods to compute the weight of each attractor and the basin entropy of the systems.

CJ89-51. The application of matrix flowing process in the foundry technology CAD : By Chen Jinshui and Hou Fusheng (Tianjin University, Tianjin, China). Chinese J. Mech. Engng25, No. 4 (Dec. 1989), pp. 26–32

This paper proposes a new image encryption method based on matrix semi-tensor product theory. Using Hyperchaotic Lorenz system to generate chaotic sequences, and then using this chaotic sequence to generate two fixed scrambling matrices. The plain image is scrambled by chaotic positioning sort scrambled method. Analogous to a chemical reaction, the plain image scrambled image is one of the reactant, then add a matrix which is related to the plain image information and extracted from the chaotic sequence (this matrix can be differ in size from the plain image matrix, different matrices can be intercepted and generated according to different plain image). This matrix is scrambled as another reactant, Apply the method of semi-tensor product to carry on the reaction (diffusion), finally get the product (encryption image). This method breaks the shackles of the traditional matrix operation and makes the reaction matrix have more forms. Compared with other experimental results, the proposed algorithm is more secure and available.