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Design of distributed pinning controllers for set stabilization of $ k $-valued logical control networks is investigated in this paper. Firstly, by analyzing the coupling correlations among nodes, pinned node set is determined. Secondly, based on the solvability of a class of matrix equations, sufficient conditions which resort to local information are put forward for the design of pinning controllers. Furthermore, an algorithm for designing pinning feedback controllers is presented, where the designed controllers are related to part of state variables instead of all variables. Finally, two illustrative examples are presented to verify the effectiveness of the main results.

This paper investigates observability of Boolean control networks(BCNs) and probabilistic Boolean networks(PBNs). First, weak observability of BCNs is discussed via the non-augmented approach. The obtained result is then applied to determine (asymptotic) observability of PBNs. Finally, complexity of algorithms based on new criteria is analyzed. Compared with existing ones, time and space complexity do not get worse, even are improved under some mild conditions.

This paper investigates the observability of singular Boolean control networks (SBCNs) with state delays. First, via Cheng product, the considered system is converted into its algebraic form. Then, under the uniqueness of the solution, two methods are proposed to transform the system into a general Boolean control network (BCN). Therefore, the considered system’s observability is equivalent to the BCN’s observability. Subsequently, the restricted input-state incidence matrix is introduced and some necessary and sufficient conditions are presented for the observability of BCNs with state delays. Finally, two illustrative examples are presented to show that the main results obtained are effective in analyzing the observability of SBCNs with state delays.

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.

In this paper, the finite horizon tracking control problem of probabilistic Boolean control networks (PBCNs) is studied. For a given reference output trajectory, two trackability definitions are introduced according to whether the tracking probability is 1. Under the framework of the semi-tensor product, some necessary and sufficient conditions are obtained to determine whether the reference output trajectory is trackable with probability (probability one) by a PBCN starting from a given initial state. Based on this, two algorithms are proposed to determine the maximum tracking probability and the corresponding optimal control policy sequence. By determining the tracking error of the reference output trajectory, two related optimal control problems are considered: one is to minimize the expected value of the total tracking error, and the other is to minimize the maximum tracking error. Inspired by dynamic programming, corresponding algorithms are given to solve these two problems. Finally, two examples are given to verify the validity and correctness of the results.

This article considers admissibility analysis and stabilization for implicit Markovian jump systems (IMJSs) with retarded discrete-distributed delays. Admissibility analysis is investigated for the unforced delay IMJSs by virtue of reciprocally convex integral inequality technique and Barbalat’s lemma. State feedback controller is designed via the matrix transformation technique to realize the stabilization of the delayed closed-loop IMJSs. By selecting comprehensive L-K functional with modes and delays information, admissibilization conditions are presented in terms of LMIs. Two illustrative examples including an inverted pendulum controlled by a direct current motor (DCMCIP) system are utilized to certify the effectiveness and practicality of the admissibilization technique.

Searching a condensed adequate set of k -valued logic is a challenging problem. Using algebraic expression of logical functions, a normal form for k -valued logic was presented in Cheng et al. [3]. Starting from the normal form, searching adequate set of k -valued logic is converted to searching set of generators of their isomorphic matrix group/semigroup. Two adequate sets of k -valued logic are revealed, which consist of only 4 logical operators: one binary and three unary operators. These two adequate sets are uniformly defined for all k ≥ 3 .

This paper investigates the reconstructibility of switched Boolean control networks (SBCNs). Several new types of definition about reconstructibility are proposed where the existence and arbitrariness of the switching signal and the input sequence are inconsistent. A weighted pair graph describing all pairs of states which are indistinguished is defined. A deterministic finite automata method is applied into the reconstructibility analysis. Accordingly, several algorithms are designed to determine these types of reconstructibility. Moreover, for a given reconstructible SBCN, an algorithm for determining the current state is provided. Finally, an example is provided to demonstrate the effectiveness of the proposed algorithms and results.

In this article, pinning stabilization of Boolean networks is further investigated based on semi-tensor product of matrices. First, two novel algorithms are devised to obtain a globally stable Boolean network from new perspectives. One of the two algorithms is to find a globally stable Boolean network by altering the columns of the transition matrix, and the other one aims to apply a perturbation method to find the smallest number of pinning nodes, on which the controls are placed to ensure the global stability of the controlled networks. Then, all the required pinning controllers can be designed. Examples are given to illustrate the effectiveness of the obtained results.

This paper mainly investigates some properties of Cheng projection, which was proposed recently by Prof. Cheng to reduce the dimension of vector. As a linear transformation from the original vector space to the target vector space, the matrix representation of Cheng projection is given. Then, the structure matrix of Cheng projection, called the Cheng projection matrix, is obtained. Algebraic properties of Cheng projection are derived via its projection matrix. Furthermore, the product and norm of Cheng projection matrices are concerned.

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.