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This article proposes a dimensionality reduction approach to study the output regulation problem (ORP) of Boolean control networks (BCNs), which has much lower computational complexity than previous results. First, an auxiliary system which is much smaller in scale than the augmented system in previous approach is constructed. By analyzing the set stabilization of the auxiliary system as well as the original BCN, a necessary and sufficient condition to detect the solvability of the ORP is presented. Second, a method to design the state feedback controls for the ORP is proposed. Finally, two biological examples are given to demonstrate the effectiveness and advantage of the obtained new results.

This article investigates the set stabilization of networked evolutionary games (NEGs) with probabilistic time-varying delays (PTDs). Utilizing the algebraic state-space approach, a unified algebraic framework is constructed to investigate the evolutionary game with arbitrary network structures and probabilistic delays. Based on the algebraic formulation, the set stability of NEGs with PTDs is studied first, and a necessary and sufficient condition is proposed to characterize the set stability. Then, the set stabilization of NEGs with PTDs is investigated via the truth matrix method, including the derivation of verification conditions and the design of the controller. Finally, an example is carried out to illustrate the theoretical results.

This article investigates the output tracking problem of switched Boolean networks using self-triggered control. First, by virtue of the constructed auxiliary system, two necessary and sufficient conditions for output tracking are provided. Then, a definition for the Lyapunov function of output tracking is proposed. Based on these, two criteria for output tracking are derived from the perspective of the Lyapunov function, and a procedure is presented for calculating the Lyapunov function. Furthermore, a self-triggered control mechanism for output tracking is devised. By utilizing the established Lyapunov function, a design method for the self-triggered control of output tracking is provided. Finally, a biological example, known as metastatic melanoma, verifies that the proposed self-triggered control mechanism can effectively reduce the frequencies of control updates.

In this paper, the approximate synchronization of leader-follower multiagent systems (MASs) over finite fields is studied in regard to local and global synchronization. First, the approximately synchronous state set (ASSS) is obtained. Second, combined with ASSS and transient periods, some criteria for the local and global approximate synchronization of systems are given. Moreover, the algorithms for calculating the maximum approximately synchronous basin (MASB) and the maximum control invariant set (MCIS) are presented. Third, the global approximate synchronization of the system is achieved by designing the state feedback control, and a design algorithm of the controller using the truth matrix method is proposed. Moreover, the results of approximate synchronization are degenerated to complete synchronization. Last, two examples are shown to demonstrate the results of this paper.

This brief presents an adaptive fuzzy dynamic event-triggered tracking control scheme with guaranteed transient performance for flexible-joint (FJ) robot system. Unlike traditional backstepping control method, we approach the control problem for second-order systems based on fully actuated system approach, which greatly reduces the computational complexity. A dynamic event-triggered mechanism (DETM) is proposed that not only saves resources but also adjusts the number of triggers flexibly. In addition, we design a new time-varying smooth segmentation function and normalized function which removes the constraints on the performance function as well as achieving tracking accuracy in the desired time. At the same time, it is proved by Lyapunov stability theory that the proposed control strategy ensures that all signals of the closed-loop system are bounded and the Zeno behavior can be avoided. Finally, the proposed control strategy is applied to the 2-link FJ robot on hardware experiment platform and the feasibility and superiority of the scheme are verified.

This brief presents an efficient approach for minimum-time control of Boolean control networks (BCNs) based on graph theory. All three cases of the problem are considered depending on whether the initial state, desired state, or both are fixed. To facilitate the use of graph-theoretical tools, the dynamics of a BCN is first characterized by its state transition graph (STG). Subsequently, the equivalence between a time-optimal state trajectory and a shortest path in the STG is established. We then develop efficient algorithms based on the breath-first search of the STG to quickly identify the reachable set, count the distance of each reachable vertex, and collect time-optimal controls for each state. Finally, all state-feedback gain matrices for minimum-time control are characterized. Our approach is featured by its high versatility and superior time efficiency. The proposed graphical framework handles all three cases uniformly and can be easily extended to more variants with minor or even no algorithm modifications. This is in contrast to existing methods that only target specific cases. The time complexity of our approach is considerably lower than that of existing methods, contributing to higher scalability for relatively large networks.

In this article, a uniform method is provided to investigate the observability decomposition of Boolean control networks (BCNs) under different definitions of observability. Since the observability of BCNs is defined via the distinguishability of states, an algorithm is devised to identify and sort distinguishable states. To find the possible control sequences, a truth matrix reflecting the relationship between state pairs and control inputs is constructed. Using the truth matrix, necessary and sufficient conditions for several types of observability are given. Based on criteria for observability and some matrix discriminants, the realizability of observability decomposition is discussed. Notably, the proposed method is applicable to the observability decomposition under different definitions of observability, while the existing result is valid for only one type of observability. Furthermore, an example is introduced to illustrate the feasibility of the presented results.

Transition analysis is essential for analyzing and regulating logical control networks (LCNs). The algebraic state-space representation method, which relies on the semitensor product of matrices, has been shown to allow deterministic LCNs to be represented as linear-like systems. However, due to the inherent uncertainty, it is hard to obtain the algebraic expression of an stochastic LCN. A unified paradigm for the transition analysis of LCNs with stochastic and deterministic dynamics is provided in this research. First, the algebraic expression of LCN with deterministic dynamics is reviewed. Second, the algebraic expression of LCN with stochastic dynamics is considered, where the nonequivalence between the dispersed form and the integrated form is proposed. Then, the reason for the nonequivalence is provided. After that, a consistency condition is presented to bridge the gap between the independent model and the conditionally independent model. Finally, we specifically point out that probabilistic LCN satisfies the consistency criteria, allowing one to calculate the probabilistic LCN transition matrix by using a power-reducing operator.

In this paper, the demand response considering interactive decision making between residential users and utility companies is modelled and controlled using networked evolutionary game (NEG) theory. The NEG is achieved via a widely used mathematical tool, namely the semi-tensor product (STP) of matrices, through which, feedback controls can be implemented to dynamically regulate the game-based networks. Firstly, the dynamic interactions between utility companies who provide various energy consumption packs and residential users who can intelligently choose preferable packs are modelled in the state-space form, in which both sides are consistently pursuing their maximum payoffs. Then, a quantitive analysis of such system’s equilibria is conducted using rigorous mathematical derivations, followed by the design of a profile feedback controller when the existing equilibria are not satisfactory towards the required energy consumption. Finally, an illustrative example is demonstrated, where the dynamic gaming between utility companies and users is illustrated and the effectiveness of the proposed feedback control is validated. Note to Practitioners—This paper addresses a very practical engineering problem, which is the demand response in the modern smart grid. Different from general optimization approaches in the existing works, the method proposed by this paper improves the demand response via feedback control of networked systems, which is based on the rigorous matrix approaches via state space equations. In addition, the work in this paper considers very practical factors in actual engineering (i.e., users with different action choices and decision logics being all together), which also provides much value to practitioners than the relevant works with similar theoretical approaches. The method proposed in this paper provides an effective regulation of the demand response in practice and can be flexibly adjusted according to actual situations.

This article investigates the relationship between realization and observability of Boolean control networks (BCNs) and gives some invariants under minimum realization (MR), such as, decoupling, invertibility, etc. It is proved that the MR of a BCN exists uniquely up to coordinate transformations and a BCN is an MR of itself if and only if it is weakly observable. Based on these, the MR of a given BCN can be constructed. Moreover, the observability decomposition of BCNs is discussed and an equivalent condition is obtained to determine the decomposability, which guarantees the existence and uniqueness of the regular MR (RMR) that is a special type of MR. Then, the RMR of the given BCN can be obtained. Finally, an example is given to show the effectiveness of the obtained results.

Disturbance decoupling is one of the fundamental problems in control systems. This article addresses the original disturbance decoupling problem of singular Boolean networks (SBNs) from a new point of view. The robust solvability of SBNs is discussed by the truth matrix of algebraic constraints first. In order to cope with the multiple solutions, an SBN with disturbances is equivalently transformed into a switched restricted Boolean network (BN). Then utilizing a new approach based on the reduced state transition matrix, the relationship between original disturbance decoupling and robust indistinguishability is revealed, which also builds a link with robust observability. By virtue of a parameter extraction mapping, a verifiable criterion is presented for original disturbance decoupling. Moreover, our approach is still applicable to BNs and of lower computational complexity than the existing results. Finally, illustrative examples are worked out to show the effectiveness of the obtained results.

This brief studies the event-triggered synchronization problem of the multiagent systems (MASs) over finite fields. Firstly, the MASs over finite fields with event-triggered control (ETC) are modeled as algebraic form by the semi-tensor product (STP) method. Secondly, the leader-follower synchronization problem is transformed into the set stabilization problem about the systems with algebraic form. Moreover, to reduce the control execution times, the maximum invariant set or control invariant set with the smallest proportion of states to be controlled is selected as the target set rather than the maximum control invariant set. Thirdly, some necessary and sufficient conditions of the event-triggered synchronization of the MASs as well as the design method of the optimal state feedback event-triggered controller are given. Finally, an example is presented to illustrate the conclusion of this brief.

Set stabilization is one of the essential problems in engineering systems, and self-triggered control (STC) can save the storage space for interactive information, and can be successfully applied in networked control systems with limited communication resources. In this study, the set stabilization problem and STC design of Boolean control networks are investigated via the semi-tensor product technique. On the one hand, the largest control invariant subset is calculated in terms of the strongly connected components of the state transition graph, by which a graph-theoretical condition for set stabilization is derived. On the other hand, a characteristic function is exploited to determine the triggering mechanism and feasible controls. Based on this, the minimum-time and minimum-triggering open-loop, state-feedback and output-feedback STCs for set stabilization are designed, respectively. As classic applications of self-triggered set stabilization, self-triggered synchronization, self-triggered output tracking and self-triggered output regulation are discussed as well. Additionally, several practical examples are given to illustrate the effectiveness of theoretical 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 investigates the networked evolutionary games (NEGs) with profile-dependent delays, including modeling and stability analysis. Profile-dependent delay, which varies with the game profiles, slows the information transmission between participants. Firstly, the dynamics model is proposed for the profile-dependent delayed NEG, then the algebraic formulation is established using the algebraic state space approach. Secondly, the dynamic behavior of the game is discussed, involving general stability and evolutionarily stable profile analysis. Necessary and sufficient criteria are derived using the matrices, which can be easily verified by mathematical software. Finally, a numerical example is carried out to demonstrate the validity of the theoretical results.

A new modeling idea of interval type-2 fuzzy logic system (IT2 FLS) is proposed on the basis of matrix semitensor 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. First, the fuzzy rules are generated by the space division method, and the initial parameters of the model are determined. Second, 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.

This article investigates the synchronization problem of networks over finite fields. First, based on the finite-field synchronizable property of matrices, a necessary and sufficient condition is derived for finite-field synchronous networks. Then, the inverse recursion subspaces of networks over finite fields are studied. It is revealed that the synchronization of finite-field networks can be determined by a fraction of states. Finally, as an application, the obtained results are generalized to a class of Boolean networks.

This paper investigates the robust reachability of Boolean control networks (BCNs) with arbitrary disturbance inputs. First, in order to describe the dynamics of a BCN with arbitrary disturbance inputs, a new concept, possible reachability, is introduced. By exploring the relationship between the robust reachability and the transition matrix, a necessary and sufficient condition is derived for checking the robust reachability. Subsequently, the dynamic progress graph is introduced to describe the dynamics of BCNs and based on the breadth-first search (BFS), an algorithm is proposed to design the shortest control input sequence to ensure robust reachability.

Bisimulations are established forms of behavioral equivalences for discrete event systems. Recently, bisimulations have been introduced to fuzzy transition systems (FTSs) and developed quickly. Relational lifting by using weight functions is one of the most used methods in bisimulation research, which lifts a crisp relation on the state spaces to a crisp relation on the distributions of state spaces of FTSs. This article proposes a method, called fuzzy relational lifting method, via fuzzy similarity measures induced by residua (implications) in complete residuated lattices. In contrast with the existing method, fuzzy relational lifting method relaxes the conditions of lifting relation by using weight functions and is a generalization of the method. Some properties of fuzzy lifting operation are presented. Based on the fuzzy lifting operation, a fuzzy bisimulation for nondeterministic fuzzy transition systems is introduced, which is much more natural and robust than some behavioral measures, and can measure the degree of similarity between any two states, which are deemed to be equivalent intuitively but simply distinguished by other behavioral measures. Then some properties of fuzzy bisimulations are provided. Moreover, a fixed-point characterization of fuzzy bisimulations is given and the greatest fuzzy bisimulation is computed. Finally, a real-valued logic is introduced to characterize fuzzy bisimulations over Gödel algebra.

In this article, the finite-time disturbance decoupling problem (FTDDP) of Boolean networks (BNs) and Boolean control networks (BCNs) is studied. A novel necessary and sufficient condition is first provided to determine whether a given BN is finite-time disturbance decoupled. Then, an algorithm is devised to find the shortest time that guarantees that a given BN is finite-time disturbance decoupled. In addition, using a constructed truth matrix, a necessary and sufficient condition for solving the disturbance decoupling problem of BCNs is proposed, and all possible time optimal state feedback controllers are designed. Finally, two examples are introduced to illustrate the feasibility of the obtained results.

In this article, the observability of probabilistic Boolean networks (PBNs) is investigated. First, the distinguishable state pairs are obtained by constructing a matrix sequence, based on which a novel criterion for examining the observability of PBNs is given. Then, an algorithm is proposed to verify the observability of PBNs. Finally, some numerical examples are presented to illustrate the proposed method.

This article investigates the delay synchronization of drive Boolean network (DBN) and response BN (RBN) [response Boolean control network (RBCN)]. Synchronization and time delay are two common phenomena in real life. In aircraft formation, the latter's trajectory is consistent with the former's, but there exists a time gap when they arrive at the same position, which illustrates that time delay can happen in the process of synchronization. According to the delay step depending on initial states or not, two types of delay synchronization are proposed: 1) weak and 2) strong delay synchronization. Compared with synchronization, delay synchronization is only the theoretical generalization and has different application fields. Using the results of limit sets, several criteria for delay synchronization of the DBN and RBN are derived. This article provides an important result that delay synchronization of the DBN and RBCN can be equivalent to set stabilization of the augmented BCN, then the necessary and sufficient conditions are obtained. Moreover, two algorithms are established to design all time-optimal state feedback controllers. Finally, two examples are shown to demonstrate the effectiveness of the results.

This paper studies the strategy optimization for a type of evolutionary games on coupled networks under sampled-data state feedback controls (SDSFCs) with Lebesgue sampling, which is more economical than traditional state feedback controls. Firstly, using the semi-tensor product of matrices, the algebraic expression of a controlled evolutionary game on a two-layer coupled network is established. Secondly, for a given Lebesgue sampling region, a necessary and sufficient condition is presented to detect whether each player’s payoff can ultimately remain at or above its own threshold, and the corresponding SDSFCs are designed. Furthermore, for a given signal of Lebesgue sampling, an approach is provided to obtain a desired sampling region, under which each player’s payoff always meets their threshold condition after a certain time. Finally, an illustrative example is provided to support our new results.

Time delay is inevitable in information communication. Switched time delay, as a time-varying delay, plays an important role in information interaction between agents. In this article, networked evolutionary games (NEGs) with switched time delay are investigated, including modeling and dynamics analysis. First, the dynamic model of the delayed NEG is proposed, then the algebraic formulation is established by using the algebraic state-space approach. Second, dynamic behavior of the game is discussed, involving general convergence and evolutionarily stable profile analysis. Necessary and sufficient conditions are derived for convergence of the game and evolutionarily stable profile, respectively. Finally, two illustrative examples are employed to demonstrate the validity of the theoretical results.

An adequate set of k-valued logic is provided, which contains only two operators. It is also proved that this adequate set is of minimum size.

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.

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 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.

This paper investigates the approximate synchronization of singular logical networks (SLNs) using algebraic representations. Different from complete synchronization, which requires the state trajectories of the drive-response SLNs to be completely consistent after a finite time, approximate synchronization allows for admissible errors between the state trajectories of the drive-response SLNs. A definition of approximate synchronization for SLNs is proposed. By analyzing the constructed admissible matrices, the solvability of SLNs is discussed. A criterion is provided for the approximate synchronization of SLNs. Self-triggered control is then introduced to address the approximate synchronization of SLNs. Based on this, an algorithm is presented for designing the self-triggered state feedback control of approximate synchronization. The method presented in this paper can significantly reduce the updating frequencies of the controller. Finally, all the theoretical results are illustrated through a genetic regulatory network.

This paper discusses coordinated control problems for systems over finite fields via fully actuated approach. Novel types of consensus and synchronization, which are irrelevant to initial states, are proposed. For different coordinated control problems, suitable controllers are designed, which are able to drive the system to the corresponding trajectory in finite steps. Besides, the conclusions are applied to stabilization, and the convergence time is analyzed. As an application, a camera network example is given to illustrate the 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.

This paper investigates stability and guaranteed cost of time-triggered Boolean networks (BNs) based on the semitensor product of matrices. The time triggering is generated by mode-dependent average dwell-time switching signals in the BNs. With the help of the copositive Lyapunov function, a sufficient condition is derived to ensure that the considered network is globally stable under a designed average dwell-time switching signal. Subsequently, an infinite time cost function is further discussed and its bound is presented according to the obtained stability result. Numerical examples are finally given to show the feasibility of the theoretical results.

This study is concerned with the fault detectability of Boolean control networks (BCNs) by two nonaugmented methods. Firstly, the equivalent system-based approach is considered, and the equivalence of BCNs is applied to analyze weak active fault detectability. Further, an iterative matrix set-based approach is proposed, by which, several novel criteria for strong and weak active fault detectability are presented. Meanwhile, effective algorithms are designed to check strong and weak active fault detectability and generate all feasible input sequences with the minimum length. In comparison, our results reduce the computational complexity dramatically than the existing studies.

The α-exponential stability analysis and l1controller synthesis of positive Takagi–Sugeno fuzzy discrete-time systems with time-varying delays are discussed in this paper. A sufficient condition of α-exponential stability for autonomous positive fuzzy systems is first established by constructing a new Lyapunov–Krasovskii functional, which depends on a given scalar 0<α<1and the bound of time-varying delays. Then, a sufficient condition for controlled systems with l1gain constraint is also presented while imposing α-exponential stability, upon which, a constrained l1controller is designed for both single-input and multi-input cases. Finally, an example is given to show the feasibility of the theoretical results.

This paper is intended to study output feedback-based H∞admissibilization for singular Markovian jump time-varying delays systems via Moore–Penrose generalized inversion technique. The main aim is to design singular dynamic compensator based on output feedback control idea and Moore–Penrose generalized inversion technique, such that the time-varying delays singular Markovian jump system realizes not only stochastic stability, but also regularity, impulse-freeness (which are collectively known as stochastic admissibility) and achieves H∞interference level. Time delay-dependent and mode-dependent L–K candidate functional is constructed, Moore–Penrose generalized inversion technique is utilized, then the improved H∞admissibilization conditions for singular Markovian jump time-varying delays systems are provided by virtue of linear matrix inequalities. Simulation examples including a direct current motor-controlled inverted pendulum system are employed to confirm the effectiveness and practicability of the addressed singular dynamic compensation technique.

Algebraic state space theory (ASST) of logical systems, developed based on the semi-tensor product (STP) which is a new matrix analysis tool built in recent ten years, provides an algebraic analysis approach for many fields of science, such as logical dynamical systems, finite-valued systems, discrete event dynamic systems, and networked game systems. This study focuses on comprehensively surveying the applications of the ASST method to the field of finite state machines (FSMs). Some necessary preliminaries on the method are first reviewed. Then the applications of the method in the FSM field are reviewed, including deterministic FSMs, nondeterministic FSMs, probabilistic FSMs, networked FSMs, and controlled and combined FSMs. In addition, other applications related to both STP and FSMs are surveyed, such as the application of FSM to Boolean control networks and the application of graph theory to FSMs. Finally, some potential research directions with respect to the ASST method in the FSM field are predicted.

This paper introduces singular mix-valued logical networks and singular mix-valued logical control networks. Via semi-tensor product, a singular mix-valued logical network can be converted to an algebraic form. On this basis, the normalization problem and solvability are discussed. Then, fixed points and cycles of singular mix-valued logical networks are also studied. Furthermore, the optimal control problem of singular mix-valued logical control networks is presented and a necessary condition for the existence of the optimal control is supplied, based on which, an algorithm is provided. Illustrative examples are given to show the feasibility of the results.

In this paper, the problem of consensus for continuous time singular systems of multi-agent networks is considered. The definition of r-consensus is introduced for singular systems of multi-agent networks. Firstly, linear systems with algebraic constraints are considered, and the corresponding results about consensus and average-consensus are derived. Then r-consensus and consensus problems of singular systems are investigated. Sufficient conditions of r-consensus and consensus are obtained, respectively. Finally, an illustrative example is given to show the effectiveness of the proposed method.

This paper considers the impacts on attractors of function perturbations in Boolean networks via semi-tensor product of matrices. Two types of function perturbations: one-bit perturbation and modifications of update schedule, are investigated. First, the algebraic form of perturbed Boolean networks under one-bit perturbation is given, based on which several necessary and sufficient conditions of different kinds of effects on state transition and attractors are obtained. Besides, a Boolean network with update schedule is studied and the definition of its adjacent diagraph is first presented in this paper. Furthermore, transition matrix of the updated network is calculated to analyze the changes of fixed points and limit cycles. At last, identifying function perturbations with its application in a Drosophila melanogastersegmentation polarity gene network is shown to demonstrate the practicability and effectiveness of the theoretical results.