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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.

A kind of algebra, called numerical algebra, is proposed and investigated. As its opponent, non-numerical algebra is also defined. The numeralization and dis-numeralization, which convert non-numerical algebra to numerical algebra and vise versa, are considered. Product structure matrix (PSM) of a finite dimensional algebra is constructed. Using PSM, some fundamental properties of finite dimensional algebras are obtained. Then a necessary and sufficient condition for a numerical algebra to be a field is presented. Finally, the invertibility of Segre (commutative) quaternion and some related properties of matrices over Segre quaternion are investigated.

In this paper, the set stabilization problem of Markovian jump Boolean control networks (MJBCNs) is investigated via semi-tensor product of matrices. First, the conception of set stabilization is proposed for MJBCNs. Then based on the algebraic expression of MJBCN, a necessary and sufficient condition for set stabilization is provided by a linear programming problem, which is simple to solve. Moreover, by solving this linear programming problem, an algorithm for designing a state feedback controller is developed. Finally, two examples are presented to illustrate the feasibility of the obtained results.

Considered the stimulation of tumor necrosis factor as an impulsive control, an apoptosis network is modeled as a state-dependent impulsive Boolean network (SDIBN). Making cell death normally means driving the trajectory of an apoptosis network out of states that indicate cell survival. To achieve the goal, this article focuses on the pinning controller design for set reachability of SDIBNs. To begin with, the definitions of reachability and set reachability are introduced, and their relation is illustrated. For judging whether the trajectory of an SDIBN leaves undesirable states, a necessary and sufficient condition is presented according to the criteria for the set reachability. In addition, a series of algorithms is provided to find all possible sets of pinning nodes for the set reachability. Note that attractors containing in all undesirable states are studied to make SDIBNs set reachable via controlling the smallest states. For the purpose of determining pinning nodes for one-step set reachability, the Hamming distance is presented under scalar forms of states. Pinning nodes with the smallest cardinality for the set reachability are derived by deleting some redundant nodes. Compared with the existing results, the state feedback gain can be obtained without solving logical matrix equations. The computation complexity of the proposed approach is lower than that of the existing methods. Moreover, the method of designing pinning controllers is used to discuss apoptosis networks. The experimental result shows that apoptosis networks depart from undesirable states by controlling only one node.

This paper considers the set controllability of Boolean control networks with impulsive effects (BCNs-IE). To investigate this problem, set controllability matrices are proposed via semi-tensor product of matrices. As applications, two control problems, which are the controllability of BCNs-IE with mix-type controls and the output controllability of BCNs-IE, are solved by set controllability approach.

This paper investigates the network-based modeling and distributed active disturbance rejection control (ADRC) to address the Ackermann steering problem of a four-in-wheel motor drive electric vehicle with deception attacks on controller area networks (CAN). The distributed ADRC can achieve the independent steering, ensure a small steering radius and improve the stability and robustness of the vehicle under unknown tyre longitudinal forces and network attacks. Using an independent driving strategy and Ackermann steering geometry, a state-space model for rotational velocity tracking of each wheel is established, where a virtual external disturbance that consists of the tyre longitudinal force and the expected rotational velocity is imposed on the model. Considering the effect of deception attacks on measurement outputs, sampled-data-driven extended state observers are designed to estimate the tracking error and the external disturbance. To capture the interactions among four wheels, a distributed controller based on ADRC is proposed and the resulting system is formulated as a stochastic linear system with input delay and composite disturbance, where the composite disturbance is composed of false signals, a discretized disturbance error and the derivative of the longitudinal forces. A lemma is obtained to prove the discretized disturbance error to be energy-limited. Some stochastic stability conditions with H ∞ performance are derived by constructing a new discontinuous augmented Lyapunov–Krasovskii functional, and a design algorithm of the observer gain and the controller gain is presented. The effectiveness of the results is exemplified by two examples.

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.

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.

We investigate the solution and stability of continuous-time cross-dimensional linear systems (CCDLSs) with dimension bounded by V-addition and V-product. Using the integral iteration method, the solution to CCDLSs can be obtained. Based on the new algebraic expression of the solution and the Jordan decomposition method of matrix, a necessary and sufficient condition is derived for judging whether a CCDLS is asymptotically stable with a given initial state. This condition demonstrates a method for finding the domain of attraction and its relationships. Then, all the initial states that can be stabilized are studied, and a method for designing the corresponding controller is proposed. Two examples are presented to illustrate the validity of the theoretical results.

In actual economic activities, different economic systems usually interact with each other, and the coupled networked evolutionary game is an important theoretical tool to describe and analyse such...

This letter investigates the variance of state dimensions of linear discrete dimension-unbounded systems, which are a class of cross-dimensional systems. First, the dimensional relationship of states, initial values and system matrices of some special cases is discussed. Then, for the general case, to directly calculate the transition time and state dimensions after the transition time, an algorithm is established on the basis of the iterative time, dimensions of initial values and system matrices. Finally, there is an illustrative example to show the effectiveness of these obtained results.

Finite-time stabilization problem of Markovian jump Boolean networks is investigated via pinning control in this paper. Firstly, on the basis of converting Markovian jump Boolean networks into Markov chains, a necessary and sufficient condition for the finite-time stabilization of Markovian jump Boolean networks is put forward. Secondly, an algorithm for calculating minimum number of pinning nodes is devised by altering the transition probability matrix of Markov chains. Furthermore, after the pinning nodes and the stable transition matrix are determined, a method for designing all the required pinning state feedback controllers is proposed. Finally, a biological example is presented to illustrate the feasibility of theoretical results.

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 reconstr...

This paper investigates a minimal family of sub-bases. First, the concept of a minimal family of sub-bases is presented and its properties are studied. Then the relationship between reducts in covering information systems and minimal families of sub-bases is discussed. Based on Boolean matrices, an approach is provided to derive a minimal family of sub-bases. Finally, experiments are conducted to illustrate the effectiveness of the proposed approach.

In this study, the output tracking of delayed logical control networks (DLCNs) with state and control constraints is further investigated. Compared with other delays, state-dependent delay updates its value depending on the current state values and a pseudo-logical function. Multiple constraints mean that state values are constrained in a nonempty set and the design of the controller is conditioned. Using the semi-tensor product of matrices, dynamical equations of DLCNs are converted into an algebraic description, and an equivalent augmented system is constructed. Based on the augmented system, the output tracking problem is transformed into a set stabilization problem. A deformation of the state transition matrix is computed, and a necessary and sufficient condition is derived for the output tracking of a DLCN with multi-constraint. This condition is easily verified by mathematical software. In addition, the admissible state-feedback controller is designed to enable the outputs of the DLCN to track the reference signal. Finally, theoretical results are illustrated by an example.

This paper investigates the identification problem on the delay parameter for time-delay Boolean networks (TBNs). First, the observability of TBNs is proposed, and some observation conditions are obtained. Then the relation between observability and identifiability is discussed. Using the results of observability, some criteria for the identifiability of the delay parameter are derived. Finally, the paper shows an important result that the delay parameter can be identified if structure matrices are known. Moreover, two examples are given to show the feasibility of the proposed method.

This paper investigates controllability and reachability of k-valued logical control networks (k-VLCNs) with time-invariant delays in states, which can be transformed into the algebraic forms based on Cheng product. First, some concepts of controllability and reachability of k-VLCNs with time-invariant delays in states are proposed. Second, some necessary and sufficient conditions for controllability and reachability are derived. Finally, an example is introduced to show the efficiency of the main results.

This paper focuses on admissibility analysis and stabilization for neutral descriptor hybrid systems (NDHSs) with time-varying delays. The main object is to design mode dependent state feedback controller (SFC) and realize admissibility and stabilization of the NDHSs. Sufficient conditions are provided in terms of linear matrix inequalities (LMIs) via constructing integrated L-K functional. Two simulation examples including a partial element equivalent circuit (PEEC) model are used to testify the validity and practicality of the proposed method.

The concept of synchronizability exponent is proposed to define a quantitative characteristic of switching topology. Meanwhile, the Lyapunov exponent is introduced as a quantitative characteristic of agent dynamics. A quantitative relation that the Lyapunov exponent is less than the synchronizability exponent is presented as a new synchronization condition. This condition describes the relationship between agent dynamics and switching topology, which does not seem to have been explicitly considered so far. Under the new quantitative condition, we design state feedback and output feedback protocols to achieve synchronization of linear multiagent systems. We further show that the output feedback protocols can be constructed by separately designing state feedback protocols and state observers, namely the separation principle holds.

This paper devotes to introducing and investigating the detectability of probabilistic Boolean networks (PBNs). Two types of detectability, weak detectability and strong detectability, are proposed to deal with different situations. Firstly, by resorting to the semi-tensor product (STP) of matrices, the equivalent algebraic expression of the PBN is converted into another kind of expression, which is called the data form. This data form can estimate the current state distribution from the corresponding output sequence. Then, using this data form constructs the detection form, based on which, two necessary and sufficient conditions are derived for checking weak detectability and strong detectability. Secondly, combining the STP and this detection form, a nice algorithm is established for checking wether a PBN is weak detectability. Finally, for better presentation, two numerical examples are shown to demonstrate the effectiveness of the theoretical results, and the diagnosis of tuberculosis is briefly analyzed.

To deal with a class of nonlinear systems with unknown control directions, a command filter-based adaptive tracking controller is designed in this paper. In the design process, fuzzy logic system is required to handle nonlinear functions, command filter is employed to settle the explosion of complexity problem and Nussbaum function is introduced to compensate the influence of unknown directions problem. Finally, the proposed control approach guarantees that error signals converge into bounded compact sets around the origin and all closed-loop signals are bounded. The effectiveness of the presented scheme is illustrated by a simulation example.

This paper investigates minimal bases and minimal sub-bases for topological spaces. First, a necessary and sufficient condition is derived for the existence of minimal base for a general topological space. Then the concept of minimal sub-base for a topological space is proposed and its properties are discussed. Finally, for Alexandroff spaces, some special results with respect to minimal bases and minimal sub-bases are illustrated.

In this paper, the application of Ledley antecedence solutions in designing state feedback stabilizers of generic logic systems has been proposed. To make the method feasible, two modifications are made to the original Ledley antecedence solution theory: (i) the preassigned logical functions have been extended from being a set of equations to an admissible set; (ii) the domain of arguments has been extended from the whole state space to a restricted subset. In the proposed method, state feedback controls are considered as a set of extended Ledley antecedence solutions for a designed iterative admissible sets over their corresponding restricted subsets. Based on this, an algorithm has been proposed to verify the solvability, and simultaneously to provide all possible state feedback stabilizers when the problem is solvable. All stabilizers are optimal, which stabilize the logic systems from any initial state to the destination state/state set in the shortest time. The method is firstly demonstrated on Boolean control networks to achieve point stabilization. Then, with some minor modifications, the proposed method is also proven to be applicable to set stabilization problems. Finally, it is shown that in $k$-valued and mix-valued logical systems, the proposed method remains effective., Comment: 10 pages, 3 figures

In this paper, the detectability of mix-valued logical networks (MLNs) is investigated based on the semi-tensor product of matrices. Firstly, weak detectability and strong detectability are introduced into MLNs. Secondly, based on the data form of MLNs, the necessary and sufficient conditions are established to check the two types of detectability of MLNs. Furthermore, an algorithm is proposed to check the detectability of MLNs. Finally, two examples are given to show the feasibility of the theoretical results.

The degeneracy problem of nonlinear feedback shift registers (NFSRs) is investigated in this paper. A new method for finding sub-families included in a family of NFSR sequences is proposed. First, the algebraic form of Fibonacci NFSR is proposed by viewing the NFSR as a Boolean network. Some properties of the algebraic expression are provided. Then, the observability of NFSRs is defined and the observability matrix is constructed, based on some explicit forms two algorithms for obtaining the observability matrix are obtained. A way to find sub-families of the given NFSR is derived and some sufficient and necessary conditions are obtained. Finally, an example is presented to depict the theoretical results.

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. This work was partially supported by NNSF of China (61773371,61877036).

This technical note presents analytical investigations on detectability of Boolean network with pinning control and injection mode (BNPCIM). Detectability represents the property to uniquely determine the current state of the system according to known input-output sequences. Using Cheng product of matrices, BNPCIM can be converted into a special algebraic form of BCNs with mix-valued logical control. Based on different research requirements, three types of detectability for BNPCIM are proposed: weak detectability, detectability and strong detectability. Under free and networked input conditions, a sequence of matrices are constructed to reflect output and state information by explicit forms. Then by using the established matrices, several necessary and sufficient conditions for three types of detectability are derived. Moreover, to avoid unnecessary calculations, the maximum steps to achieve different detectability are gained. Finally, two illustrative examples are given to demonstrate the effectiveness of the obtained results.

Stabilizability analysis and stabilizable switching signals design of switched Boolean networks (SBNs) are further investigated via semi-tensor product of matrices in this paper. First, two kinds of stabilizability of SBNs, stabilizability under arbitrary switching signals and pointwise stabilizability, are discussed under the framework of an improved approach. Second, more general stable structures, kernel attractors, are defined and studied. Subsequently, several necessary and sufficient conditions are derived to ascertain that an SBN is stabilizable under arbitrary switching signals or pointwise stabilizable to a kernel attractor. Then a constructive method is presented to determine all stabilizable switching signals dependent on states (SSDSs), which drive the considered network stabilizable to a fixed point, a kernel attractor or a set. Finally, two examples are given to show the effectiveness of the obtained results.

Two types of controllability of dynamic-algebraic Boolean control networks are first presented. The necessary and sufficient conditions of such controllability are derived. A similar concept of coo...

Derived from a simplified intelligent traffic control system, sampled-data controllability and stabilizability of Boolean control networks are considered. Compared with the existing case of uniform (periodic) sampling in Boolean control networks, the nonuniform one is more general. Using linear span with integral coefficients, the distribution of sampling points can be obtained. Then by constructing novel systems, some necessary and sufficient conditions are proposed to determine sampled-data controllability and stabilizability. Finally, two illustrative examples, which are on apoptosis networks and traffic control systems, respectively, are worked out to show the effectiveness of the obtained results.