Search

In this paper, the controllability of multi-agent systems over finite fields is studied based on the high-order fully actuated (HOFA) system approaches. The concept of HOFA systems over finite fields is introduced. Necessary and sufficient conditions for the controllability of linear systems over finite fields are established. It is proved that a leader–follower multi-agent system over finite fields is controllable if and only if it can be transformed into an HOFA system. Then the stabilisation control protocol is proposed based on the full-actuation property, and the upper bound of settling time for the system can be determined. The uncontrollable leader–follower multi-agent system over finite fields can be transformed into a controllable subsystem and an autonomous subsystem via structural decomposition, the conditions for the system to be stabilisable are provided. Finally, an illustrative example is shown to verify the validity of the main results of this paper. [ABSTRACT FROM AUTHOR]

In stream cipher designing, nonsingularity is a crucial requirement to ensure that the feedback shift registers (FSRs) do not produce keys that are equivalent to one another. This study uses a semi-tensor product to examine the nonsingularity of Trivium-like cascade FSRs over a finite field. The Trivium-like cascade FSRs are expressed algebraically using the semi-tensor product, allowing them to be viewed as logical networks and introducing a novel state transition matrix. Several necessary and sufficient conditions for the nonsingularity of FSRs and Trivium-like cascade FSRs over a finite field are established by dividing the structural matrices of feedback functions into different parts. These findings are also applicable to binary FSRs and binary Trivium-like cascade FSRs. [ABSTRACT FROM AUTHOR]

The simplification problem of logical functions is investigated via the matrix method. First, necessary and sufficient conditions are put forward for the decomposition of logical matrices. Based on this, several criteria are proposed for the simplification of logical functions. Furthermore, an algorithm, which can derive simpler logical forms, is developed, and illustrative examples are given to verify the effectiveness. Finally, the obtained theoretical results are applied to the simplification of electric circuits. [ABSTRACT FROM AUTHOR]

Nonlinear feedback shift registers (NFSRs) are the main components of stream ciphers and convolutional decoders. Recent years have seen an increase in the requirement for information security, which has sparked NFSR research. However, the NFSR study is very imperfect as a result of the lack of appropriate mathematical tools. Many scholars have discovered in recent years that the introduction of semi-tensor products (STP) of matrices can overcome this issue because STP can convert the NFSR into a quasi-linear form. As a result of STP, new NFSR research has emerged from a different angle. In view of this, in order to generalize the latest achievements of NFSRs based on STP and provide some directions for future development, the research results are summarized and sorted out, broadly including the modeling of NFSRs, the analysis of the structure of NFSRs, and the study of the properties of NFSRs. [ABSTRACT FROM AUTHOR]

This paper investigates the controllability of Markovian jump Boolean control networks (MJBCNs) by using graphical approach. To solve this problem, a deterministic directed graph is constructed to describe the stochastic evolutionary process of the considered MJBCN under the framework of semi-tensor product of matrices. Then, combining with breadth-first search, the special subgraph with breadth-first tree as the principal part is obtained. Based on this, the necessary and sufficient conditions to verify the controllability of MJBCNs are deduced, and the problem of determining the minimal controllable time is solved completely. Subsequently, two examples are given to illustrate the validity and correctness of the method. [ABSTRACT FROM AUTHOR]

This paper investigates the reconstructibility of switched Boolean control networks. Reconstructibility is the ability to determine the current state based on the known data. Four types of reconstructibility are proposed for the first time. Two methods for studying reconstructibility are provided. One method is to compare the periodic state-model-input trajectories with the same period and the other is to construct the distinguishing matrix, which can describe all initial states and their corresponding outputs driven by different model-input sequences. By virtue of them, several conditions to determine four types of reconstructibility are obtained. Finally, a state observer is designed and two examples are given to illustrate the effectiveness of obtained results. [ABSTRACT FROM AUTHOR]

This paper investigates the controllability and reachability of periodically time-variant mixed-valued logical control networks (PTMLCNs). The PTMLCN considered in this paper consists of several mixed-valued logical control networks with periodically switching signals, which circulates among different mixed-valued logical control networks. First, a PTMLCN is transformed into a discrete dynamic system by the semi-tensor product. Based on this algebraic expression, the time-dependent input-state incident matrix and the time-dependent state transition matrix are defined and the relationship between these two matrices is given. Secondly, the controllability and reachability of PTMLCNs are defined. Subsequently, by virtue of the proposed matrices, a series of necessary and sufficient conditions are given for checking controllability and reachability, and the algorithm for finding the optimal control sequence to reach the target state in the shortest time is designed. Finally, the effectiveness of the proposed method is verified by an example. [ABSTRACT FROM AUTHOR]

This study investigates the set reachability of Markovian jump Boolean networks (MJBNs). Firstly, through comprehensive analysis, two kinds of set reachability, strong set reachability and weak set reachability, are defined. By using semi‐tensor product of matrices, the considered MJBN is converted into a Markov chain which can not only describe the state transition of the MJBN synchronously, but also maintain its transition probability. Based on this, the set reachability problem of the MJBN is transformed into the reachability and convergence problems of the corresponding Markov chain. Then some necessary and sufficient conditions for set reachability of MJBNs are obtained. Finally, the observability and output stability problems of MJBNs are discussed as two applications of set reachability. [ABSTRACT FROM AUTHOR]

Three types of reconstructibility of singular Boolean control networks (SBCNs) are introduced and investigated in this paper. Two approaches are proposed to solve these three types of reconstructibility of SBCNs, one is the weighted set graph approach, the other is automata approach. Using the former, some criterions are obtained, while by the latter, several necessary and sufficient conditions are derived for the reconstructibility of SBCNs. Then an algorithm is designed to determine the reconstructible state and the computational complexity is analyzed. Finally, two numerical examples are shown to demonstrate the effectiveness of the theoretical results. [ABSTRACT FROM AUTHOR]

This paper focuses on the general decomposition problem of fuzzy relations, which can be regarded as an extension of the common one, modeled by fuzzy relational equation (FRE) X ∘ X = R. The considered decomposition problem is described by an FRE X ∘ Y = R. For the purpose of deriving all solutions to the decomposition problem, the structure of solutions to the corresponding FRE is investigated first, and a necessary and sufficient condition for the existence of solutions is obtained. Then, based on semi-tensor product (STP) of matrices, an algebraic approach is proposed. Furthermore, all solutions to the general decomposition problem can be computed via the proposed method. In order to demonstrate the effectiveness of the presented approach, two numerical examples are given. [ABSTRACT FROM AUTHOR]

This paper investigates model matching problem of switched asynchronous sequential machines (ASMs) via semi-tensor product (STP) of matrices. A switched ASM is composed of several input/state ASMs and it cut-overs between submachines in asynchronous mechanism. By using STP method, a switched ASM is depicted as a discrete-time equation form. Under this framework, stable transition matrix and admissible model-input-state transition matrix are defined. Then, two important problems, reachability and optimal control input design of switched ASMs, are discussed by these two new matrices, respectively. Finally, a necessary and sufficient condition for the existence of a controller for model matching is obtained and a practical example on asynchronous error accumulators is provided to show the feasibility of the proposed method. [ABSTRACT FROM AUTHOR]

This paper investigates the set controllability of switched Boolean control networks. Firstly, the set controllability matrix of switched Boolean control networks is presented via semi-tensor product of matrices and Warshall algorithm. Secondly, set stabilization problem and switched output regulation problem are transformed into set controllability problem. Based on it, necessary and sufficient conditions for the solvability of set stabilization problem and switched output regulation problem are obtained, respectively. Furthermore, the method is proposed for designing the switching signals driving the switched Boolean networks to be set controllable. Finally, examples are given to illustrate the effectiveness of the obtained results. [ABSTRACT FROM AUTHOR]

• The reduced state transition matrix of Boolean (control) networks is introduced to analyze the disturbance decoupling problem. • The relationship between three types of disturbance decoupling and robust indistinguishability is revealed. • Algebraic conditions with lower computational complexity are given for disturbance decoupling. Disturbances are ubiquitous and affect the normal operation of systems. This paper investigates the disturbance decoupling problem of Boolean networks (BNs) and Boolean control networks (BCNs) by a robust indistinguishability method. Utilizing a new method based on the reduced state transition matrix, the relationship between three types of disturbance decoupling and robust indistinguishability is revealed, which also builds a link between robust observability and disturbance decoupling. Based on a parameter extraction mapping, several feasible criteria are presented for original and weak disturbance decoupling of BNs and BCNs. Additionally, our approach is more concise and has a lower computational complexity than the existing methods. Finally, two examples are presented to illustrate the effectiveness of the theoretical results. [ABSTRACT FROM AUTHOR]

Composite asynchronous sequential machines (ASMs) are studied by a matrix‐based approach in this study. A composite ASM consists of several input/state asynchronous machines in a cascading fashion. By using the semi‐tensor product method, the algebraic expression of composite ASMs is presented, based on which reachability of composite ASMs is investigated. Then, a necessary and sufficient condition for the existence of a controller for model matching is obtained and an algorithm to design the corresponding controller is established. Finally, an example is proposed to illustrate the effectiveness of the obtained results. [ABSTRACT FROM AUTHOR]

In this study, implicit Boolean networks (IBNs), which are more general than classic BNs, are proposed for the first time motivated by the river‐crossing decision problem. By resorting to the admissible set, some necessary and sufficient conditions are established, under which IBNs can be equivalently converted into classic BNs or restricted BNs. Subsequently, an improved approach is presented to determine the topological structure of dynamic‐algebraic BNs (D‐ABNs), based on which transformation relations between IBNs and D‐ABNs are given. Finally, a biochemical oscillator example is used to show the application of the obtained results. [ABSTRACT FROM AUTHOR]

This paper investigates observability of Boolean control networks (BCNs) and gives some novel criteria on observability. Firstly, the observability, the strong observability and the super-strong observability are discussed via the augmented approach. Based on the constructed logical control system, these three kinds of observability are converted into the consistent and any-input sequence reachability. Subsequently, the minimum steps to determine corresponding observability are discussed. Then, by constructing finite-time convergent matrix sequences, a non-augmented approach, which can determine observability via the limit matrix in the minimum steps if the system is with corresponding observability, is presented to testify three types of observability. Finally, examples are worked out to illustrate the obtained results. [ABSTRACT FROM AUTHOR]

This note presents an iterative algorithm to solve the coupled Sylvester-transpose matrix equations (including the generalized coupled Sylvester matrix equations and Lyapunov matrix equations as special cases) over generalized centro-symmetric matrices. When the considered matrix equations are consistent, for any initial generalized centro-symmetric matrix group, a generalized centro-symmetric solution group can be obtained within finite iteration steps in the absence of roundoff errors. The least Frobenius norm generalized centro-symmetric solution group of the coupled Sylvester-transpose matrix equations can be derived when a suitable initial generalized centro-symmetric matrix group is chosen. In addition, for a given generalized centro-symmetric matrix group, the optimal approximation generalized centro-symmetric solution group can be obtained by finding the least Frobenius norm generalized centro-symmetric solution group of new coupled Sylvester-transpose matrix equations. Finally, a numerical example is given to demonstrate the efficiency of the introduced iterative algorithm. [ABSTRACT FROM AUTHOR]

The norm properties of Lyapunov mappings and their restrictions on symmetric and skew-symmetric subspaces are investigated. For non-negative, non-positive, and tridiagonal matrices, this paper gives an affirmative answer to an open conjecture which says that the norm of a Lyapunov mapping over the space of real square matrices equals that of its restricted map over the subspace of real symmetric matrices. In addition, matrix transformation and norm relationships on certain matrices arising from the Lyapunov mapping and its restrictions are provided. [ABSTRACT FROM AUTHOR]

ABSTRACT In this paper, we investigate observer‐based switching‐like adaptive‐triggered resilient state feedback control for discrete singular systems under deception‐injection (DI) attacks. Considering state variables are not completely measurable, a delayed state observer is engineered to reconstruct system states and the occurrence of DI attacks is described by Bernoulli random variables. The upper bound method is used to deal with the DI attacks with jumping patterns and imprecise occurrence probabilities. Taking into account the sporadic nature of DI attacks, switching‐like adaptive‐triggered resilient control method is presented, and the collaboratively designed triggered mechanism and resilient control strategy can not only automatically adjust data transmission based on the system states, but also save communication resources and network bandwidth, and enhance the tolerance to hostile attacks so as to improve safety and reliability of networked singular systems. By utilizing singular value decomposition technique, the noncausal behavior of the singular system is avoided and by applying linear matrix inequality (LMI) technology, the desired gains of controller and observer are achieved concurrently, then neoteric conditions about H∞$$ {H}_{\infty } $$ stochastic admissibility of closed‐loop discrete singular systems are provided. Ultimately, the validity of the proposed resilient coordination control approach is verified via a direct current (DC) motor model. [ABSTRACT FROM AUTHOR]

This paper mainly establishes the network-based evolution model for a class of hybrid dynamic games (HDG) systems by considering the fixed event-triggering mechanism. First, a general state-space model for HDG is introduced, and a seven-tuple representation for its normal form is established. Then we establish the directed network models, which include a tree network structure form and a multi-layer temporal model. Such network models present the advantages of evolution analysis on HDG. Finally, we use HDG example with the Lanchester equation to demonstrate the feasibility of the model and the reliability of its evolutionary process. [ABSTRACT FROM AUTHOR]

In the present paper, by using of the coefficients of the characteristic polynomial of matrix B and the so-called Leverrier algorithm, the explicit solutions to the Sylvester-conjugate matrix equation AX−X¯B=C (including the Lyapunov-conjugate matrix equation as special case) have been constructed. While one of the explicit solutions is stated as a polynomial of coefficient matrices of the matrix equation, one of the explicit solutions is expressed by the symmetric operator matrix, controllability matrix and observability matrix. Comparing to the existing results, there is no requirement on the coefficient matrices. At the end of this paper, one numerical example is shown to illustrate the effectiveness of the proposed method. [ABSTRACT FROM AUTHOR]

The general singular Boolean networks are proposed in this study, motivated by the algebraic form of dynamic‐algebraic Boolean networks via the semi‐tensor product of matrices. First, one of the most important problems for this kind of networks, solvability problem, is discussed. Then, in order to calculate the fixed points and cycles, the transition matrix of a singular Boolean network is defined, which contains all the state transferring information. At last, the general singular Boolean control networks are considered with their solvability and the disturbance decoupling problem is presented and solved by a constant control. Illustrative examples are given to show the feasibility of the results. [ABSTRACT FROM AUTHOR]

This study discusses the robust stability problem of Boolean networks (BNs) with data loss and disturbances, where data loss is appropriately described by random Bernoulli distribution sequences. Firstly, a BN with data loss and disturbances is converted into an algebraic form via the semi-tensor product (STP) technique. Accordingly, the original system is constructed as a probabilistic augmented system, based on which the problem of stability with probability one for the original system becomes a set stability with probability one for the augmented system. Subsequently, certain criteria are proposed for the robust stability of the systems. Moreover, an algorithm is developed to verify the robust set stability of the augmented system based on truth matrices. Finally, the validity of the obtained results is demonstrated by an illustrative example. [ABSTRACT FROM AUTHOR]

The formation control (FC) problem is investigated via a mix-valued logic-based approach. First, a trajectory-tracking algorithm of mix-valued logic control networks is proposed. Then, a new formulation of FC problems is established and a feedback control is designed to solve FC problems. The mathematical description of partial-formation control (PFC) problems is then designed as a structure of logical networks. An interesting practical example of PFC is also presented and discussed in detail. [ABSTRACT FROM PUBLISHER]

In this article, the set stabilization problem of Boolean control networks (BCNs) using output-feedback controllers is investigated comprehensively. First, a novel method for stabilizing BCNs, namely, the state dynamics matrix-based method, is proposed. Then, based on the above method, the time-invariant output-feedback (TIOF) and the time-variant output-feedback (TVOF) laws are derived. The necessary and sufficient conditions of both TIOF and TVOF stabilization of BCNs are also given. In addition, corresponding algorithms are provided such that all possible TIOF stabilizers and TVOF stabilizers for given BCNs can be designed. Finally, some illustrative examples are demonstrated, where the results are discussed and the effectiveness of the presented method is validated. [ABSTRACT FROM AUTHOR]

Abstract: We introduce two kinds of sums of Kronecker products, and their induced operators. We study the algebraic properties of these two kinds of matrices and their associated operators; the properties include their eigenvalues, their eigenvectors, and the relationships between their spectral radii or spectral abscissae. Furthermore, two projected matrices of these Kronecker products and their induced operators are also studied. [Copyright &y& Elsevier]

Abstract: This paper deals with the problem of stability analysis of neutral type neural networks with both discrete and unbounded distributed delays. In terms of linear matrix inequalities, delay-dependent conditions are obtained, which guarantee the considered delayed neural network to have a unique equilibrium point, which is globally asymptotically stable. Numerical examples are given to show the less conservatism and effectiveness of the proposed stability conditions. [Copyright &y& Elsevier]

Abstract: The problems of robust stabilization and robust control with maximal decay rate are investigated for discrete-time stochastic systems with time-varying norm-bounded parameter uncertainties. For the robust stabilization problem, an optimal state feedback controller is designed, which ensures that the closed-loop system is robustly stochastically stable with maximal decay rate, while for the robust control problem, the designed state feedback controller guarantees that the resulting closed-loop system is with a specified performance level, besides robust stochastic stability with maximal decay rate. The method based on the convex optimization and linear matrix inequality is developed to solve these problems. Two examples are provided to show the effectiveness of the proposed approach. [Copyright &y& Elsevier]

Abstract: This paper considers the problem of robust stability analysis of Cohen–Grossberg neural networks with time-varying delays and norm-bounded parameter uncertainties. The activation functions are assumed to be bounded and globally Lipschitz continuous. Both the monotonic increasing and non-monotonic increasing activations are considered. In terms of linear matrix inequalities (LMIs), sufficient conditions are obtained by using the Lyapunov–Krasovskii method, which guarantee the existence, uniqueness and global robust asymptotic stability of the equilibrium point of the delayed Cohen–Grossberg neural network. It is theoretically established that the derived LMI conditions are less conservative than certain existing ones in the literature. Finally, numerical examples are provided to demonstrate the less conservatism and effectiveness of the proposed LMI conditions. [Copyright &y& Elsevier]