The calculation of the synchronization problem is a very important topic in the research of Boolean networks (BNs). This article focuses on the study of state-flipped control on the node-set synchronization problem of general BNs and incorporates reinforcement learning algorithms to enhance the calculation efficiency of the synchronization problem. First, the synchronized nodes are algebraically represented by the semi-tensor product (STP) of matrices, and the analysis of the BN for node-set synchronization under state-flipped control is given. Second, the sufficient and necessary conditions that can enable general BNs to realize node-set synchronization are derived. Meanwhile, an algorithm for finding the flip sequence is designed. In addition, this article develops a comprehensive calculation scheme connecting the recursive algorithm to theQ-learning for finding all the optimal paths. Finally, some cases are provided to verify the effectiveness of state-flipped control and the superiority of reinforcement learning in the calculation of synchronization problems. Numerical experiments demonstrate that the proposed reinforcement learning-based algorithm achieves up to 100% success rate in finding the optimal flip sequence, outperforming the traditional method (23.4% success rate), reducing the average flip action by 15%, and improving the scalability of large-scale networks.

Observability and reconstructibility are two fundamental issues in modern control theory, which are important in both state estimation and observer design. The existing results for verifying the observability and reconstructibility of probabilistic logical control networks (PLCNs) have exponential complexities. This article presents a new approach to verify the observability and reconstructibility of PLCNs, which can greatly reduce the computational complexity. Specifically, the problem is tackled in three different steps. Firstly, based on the division of the state pair space, an observability verification system is established. Secondly, the equivalence between the stabilization of the proposed observability verification system and the observability of PLCNs is revealed, and a new criterion is established to solve the observability of PLCNs. Under the framework, the computational complexity is discussed. Thirdly, the relationship between observability and reconstructibility of PLCNs is unveiled, and some new criteria are established to solve two kinds of reconstructibility problems for PLCNs. Finally, an example of a biological network, apoptosis network, is presented to demonstrate the feasibility of the methods proposed in this article.

This article explores the influence of stochastic function perturbations (SFPs) on the output tracking of Boolean control networks (BCNs). First, by establishing the transition probability matrix of BCNs subject to SFPs, a criterion is presented to verify whether or not the state transition of BCNs is affected by SFPs. Second, when the state transition of BCNs is influenced by SFPs, a perturbed parameterized set is constructed via the information of original BCNs and SFPs. Based on the perturbed parameterized set, several necessary and sufficient conditions are derived for the robust output tracking of BCNs subject to SFPs, including keeping finite-time output tracking and reducing to asymptotic output tracking. Finally, the obtained theoretical results are applied to a Boolean model of lac operon in the Escherichia coli.

This article addresses the disturbance decoupling problem (DDP) of probabilistic Boolean networks (PBNs). Existing research on the DDP of Boolean networks (BNs) has predominantly concentrated on deterministic BNs (DBNs), which inadequately model the complexities of real-world systems, particularly those exhibiting stochastic behavior and uncertainty. We introduce two definitions tailored to PBNs: disturbance decoupling in distribution and disturbance decoupling with probability one. Utilizing an algebraic representation framework, we derive algebraic conditions for both types of disturbance decoupling. Additionally, we employ a vertex partitioning approach to establish graphical conditions pertinent to these disturbance decoupling types. Our investigation further reveals the connections between DBNs and PBNs concerning disturbance decoupling. To demonstrate the effectiveness of our theoretical results, we present three examples.

An adaptive event-triggered control strategy for non-strict feedback nonlinear cyber-physical systems with time delays against deception attacks and actuator faults is presented. The most prominent difficulty lies in after the system signals are damaged by malicious deception attacks, all the exact states in the system are unavailable. Another design difficulty is the common existence of unknown nonlinearities and time-varying time delays, which makes it difficult to get the desired controller. Thus, to stabilize the studied cyber-physical systems, an unusual coordinate transformation is presented, where the attack gains and the problem of unavailability of states are considered simultaneously. Furthermore, the effect of the malicious deception attacks on the studied system is tackled by using Nussbaum technology and designing the Lyapunov functions with the attack gains. The unknown nonlinear functions, the mismatch problem for the control input caused by the nonlinearities of time-varying time delays are processed by neural networks technology and Lyapunov-Krasovskii functions. As a result, a controller based on event-triggered mechanism is constructed to reduce the waste of communication resources. The presented control strategy can ensure the signals in the closed-loop system are bounded. Finally, the Matlab simulation experiments are showed to verify the effectiveness of the developed strategy.Note to Practitioners—This paper studies the adaptive event-triggered control strategy for non-strict feedback nonlinear cyber-physical systems with time delay under deception attacks and actuator faults. Cyber-physical systems are widely used in power system, aviation system and other practical systems. However, due to the openness of network communication channels, attackers are prone to transmitting incorrect data to controllers and actuators, resulting in unusable system states, which is a challenging problem to ensure system stability by only using the compromised system states. In addition, the safety control problem of cyber-physical systems with time delays and actuator faults is investigated simultaneously, which poses significant challenges to the design of an desired controller. Therefore, by constructing Lyapunov functions and designing adaptive laws, not only the system under deception attacks is stabilized, but also the control strategy studied is made more practical.

This paper investigates the adaptive finite-time consensus tracking control problem for a class of nonlinear nonstrict-feedback multi-agent systems (MASs) with output constraints. Firstly, to deal with the output constraints problem, an improved tan-type nonlinear mapping (NM) function is presented for the first time. Then, the singularity problem that exists in finite-time control is successfully avoided by employing a novel switching function. A modified tracking error involving the dependent variable of the designed NM function is constructed to guarantee that the outputs of all agnets achieve consensus tracking simultaneously and satisfy the constraints requirement. In addition, a switching threshold event-triggered control (ETC) strategy is applied to design controller for each agent, which combines the advantages of fixed threshold strategy and relative threshold strategy, and saves system communication resources well. The developed distributed adaptive finite-time control protocol ensures that all the signals in the closed-loop system are bounded and the consensus tracking control is achieved in finite time. Finally, the effectiveness of the designed control strategy is verified by a simulation experimentNote to Practitioners—Due to the wide application of nonlinear MASs in practice, such as unmanned aerial vehicles (UAVs) formation control, robots formation control and so on, the adaptive finite-time consensus tracking control problem for a class of nonlinear nonstrict-feedback MASs is studied in this paper. In practical applications, the practical fast finite-time strategy can effectively increase the system convergence, the NM method not only solves the output constraints problem well, but also overcomes the conservativeness of traditional barrier Lyapunov functions. Then, in order to reduce the communication burden of the nonlinear MASs, a switching threshold ETC is considered. In addition, the system model and backstepping technology used in this paper are general and practical.

This article develops a new edge removal mechanism for the global stabilizability of Boolean networks (BNs). In order to achieve the edge removal control, several control variables are properly placed into the dynamics of BNs based on the fundamental logical operators. On the basis of the new edge removal mechanism, several necessary and sufficient conditions are obtained for the global stabilizability and set stabilizability of BNs. Furthermore, a kind of stable edge removal control is proposed and achieved via the -learning algorithm to optimize the edge removal mechanism. As an application, the edge removal control is used to verify whether or not the mammalian cortical area development model can be made stabilizable to the expected stable states.

Edge perturbations (EPs) are often induced by disease-causing mutations, which include edge removal and edge sign switch. This article explores the robust stability of Boolean networks (BNs) with these two kinds of EPs. At first, both edge removal perturbation and edge sign switch perturbation are converted to multibit function perturbations (FPs). After that, the parameterized reachability matrix is constructed to verify the robust stability of BNs with multibit FP. As applications, several new criteria are established for the robust stability of BNs subject to edge removal perturbation and edge sign switch perturbation. Finally, the results are demonstrated by the Boolean models of p53 pathways network and Toll pathway of Drosophila signaling pathway network.

The law of conservation of mass, represented in Boolean networks (BNs) as total-activity conservation, is one of the typical properties of biological networks. This article analyzes the total-activity conservation of BNs based on the algebraic state-space representation (ASSR) approach. First, the total-activity-conservative matrix is defined and a matrix-based criterion is proposed to verify the total-activity conservation of BNs. Meanwhile, when function perturbation is considered, robust total-activity conservation is investigated. Second, by means of the pseudo-Boolean function generated by total-activity conservation, a constructive design procedure of the Boolean dynamics is given to achieve the total-activity conservation. Third, the total-activity conservation of switched Boolean networks (SBNs) under arbitrary switching signal is studied, which together with network aggregation achieves the total-activity conservation of large-scale BNs.

Hierarchical fuzzy systems (HFSs) are successfully applied to control systems and computer networks. Particularly, correction factor is introduced to regulate fuzzy rules and alleviate the problems of memory usage and real-time performance of systems. This article considers the multi-input single-output HFSs with correction factors, and proposes an algebraic framework by using the semitensor product of matrices. First, a kind of fuzzy correction factor matrix (FCFM) is constructed to realize the fuzzification of correction factor, based on which, an algebraic form is developed for two-input single-output fuzzy systems. Second, the structure diagram of HFSs is decomposed into several fuzzy logic units, which have the form of two-input single-output fuzzy systems with FCFM. Based on this operation, the algebraic formulation of serial, parallel and hybrid HFSs with correction factors is proposed. Finally, the effectiveness of HFSs with FCFM is verified by studying the on-ramp metering of freeway.

As an important branch of fuzzy systems, a hierarchical fuzzy system (HFS) has a wide range of applications in system science, medical science, and engineering. Using the semitensor product of matrices, this article investigates the universal approximation of multi-input single-output HFSs and develops the construction algorithms of universal approximators. Initially, based on the fuzzy relation matrices of fuzzy logic units with Mamdani-type fuzzy rules, the algebraic formulation of HFSs consisting of fuzzy logic units is proposed. After that, the universal approximation of HFSs is explored via the algebraic formulation, and some effective algorithms are presented to construct the universal approximators of HFSs with respect to different scenarios of approximated functions. Finally, the effectiveness of obtained results is verified by the on-ramp metering of freeway.

Robust controllability and stabilization are two fundamental issues in modern control theory. This article investigates the robust controllability and stabilization of Boolean control networks (BCNs) subject to stochastic function perturbations (SFPs). First, by defining the concept of the weighted path and constructing the perturbed position index matrix, the robustness analysis of reachability is divided into two cases. According to these two cases, several criteria are proposed for the robust reachability of BCNs with SFPs. Second, based on the robust reachability, the robust controllability and state-feedback stabilization of BCNs with SFPs are further studied. Finally, the validity of the obtained results is supported by examples.

In this paper, a novel constrained switching rule, called “time-triggered logical switching” (TTLS) is considered for switched logical dynamic systems (SLDSs). Compared with the general time-triggered switching, the activation mode of TTLS cannot be arbitrary and is pre-allocated according to the logic operation, which is more practical. The TTLS is described as an LDS, and according to the characteristics of LDS, the stability analysis of SLDSs with TTLS is converted into the stability analysis of SLDSs under the logical switching cycle sequences. Firstly, based on the equivalent algebraic form of SLDSs with TTLS, combining the Lyapunov theory of LDS with average dwell-time method, several sufficient conditions are put forward for ensuring the point stability of the considered SLDSs. Then, by defining the switching cycle invariant subset and constructing a new system, the set stability analysis of the original system is transformed into the point stability analysis of the new system, and further, the obtained results for the point stability analysis are applied to the set stability analysis. At last, the validity of obtained results is illustrated by simulation on gene and protein signaling activity patterns.

The optimal strategy estimation of random evolutionary Boolean games (REBGs) is discussed in this article. First, using the minimum mean square error criterion, the optimal strategy estimator is proposed for REBGs. Then, a matrix approach is developed to calculate the optimal strategy estimator by the aid of a semitensor product of matrices, which includes the prediction matrix, updating distribution, and strategy iterative formula. Finally, an elucidative example is included to show the obtained results are valid.

This article analyzes the function perturbation impact on the finite-time stability and stabilization of the probabilistic Boolean networks (PBNs). First, the concept of stability in the distribution of PBNs is divided into two disjoint concepts, that is, finite-time stability with probability one (FTSPO) and asymptotical stability with probability one (ASPO), and a new criterion is proposed for the verification of ASPO. Second, by constructing a parameterized set, it is shown that PBNs subject to function perturbation keep FTSPO if and only if the perturbed point does not belong to the parameterized set, while PBNs become ASPO if and only if the perturbed point belongs to the parameterized set. Third, as an application of perturbed stability analysis, the robust state-feedback stabilization is discussed for probabilistic Boolean control networks (PBCNs) with function perturbation. Finally, the obtained results are applied to a WNT5A network and lac operon in the Escherichia coli, respectively.

Observability is a fundamental concept for the synthesis of both linear systems and nonlinear systems. This article devotes to discussing the robustness of observability for multivalued logical networks (MVLNs) subject to function perturbation and establishing a graph-based framework. First, based on the transition graph of undistinguishable pairs of states, a new graph-based criterion is presented for the observability of MVLNs. Second, a candidate set consisting of all suspicious undistinguishable pairs of states is defined, based on the cardinality of which and the graph-based condition, a series of effective criteria are proposed for the robustness of observability subject to function perturbations. Finally, the obtained results are applied to the robust observability analysis of the p53-MDM2 negative feedback regulatory loop.

The control design problem plays a fundamental role in the study of logical control networks (LCNs). This paper presents a detailed survey on new developments in control design techniques of LCNs. First, some preliminary results on the semi-tensor product method and LCNs are reviewed. Then, we move on to some new developments for control design techniques of LCNs, including the reachable set approach, the pinning control technique, the control Lyapunov function approach, the event-triggered control technique, and the sampled-data control technique. Finally, an illustrative example is given to demonstrate the effectiveness of these techniques.

In this paper, the problem of adaptive neural tracking control for a type of uncertain switched nonlinear nonlower-triangular system is considered. The innovations of this paper are summarized as follows: 1) input to state stability of unmodeled dynamics is removed, which is an indispensable assumption for the design of nonswitched unmodeled dynamic systems; 2) the design difficulties caused by the nonlower-triangular structure is handled by applying the universal approximation ability of radial basis function neural networks and the inherent properties of Gaussian functions, which avoids the restriction that the monotonously increasing bounding functions of the nonlower-triangular system functions must exist; and 3) multiple Lyapunov functions are utilized to develop a backstepping-like recursive design procedure such that the solvability of the adaptive neural tracking control issue of all subsystems is unnecessary. Based on the proposed controller design methods, it can be obtained that all signals in the closed-loop switched system remain bounded and the tracking error can eventually converge to a small neighborhood of the origin. In the simulation study, two examples are supplied to prove the practicability and feasibility of the developed design schemes.

This paper studies the leader-follower consensus of multiagent systems with time delays and switching topology over finite fields. First, an equivalent algebraic form is established for leader-follower multiagent systems with time delays over finite fields. Second, based on the algebraic form, a necessary and sufficient condition is presented for the finite-field leader-follower consensus with time delays. Third, the switching topology case is considered, and a new criterion is presented for the finite-field leader-follower consensus with time delays and switching topology. Finally, an example is worked out to illustrate the obtained results.

Function perturbation analysis of Boolean networks is an important topic in the study of gene regulation due to gene mutation or immeasurable variables. This brief studies the function perturbation impact on feedback stabilization of Boolean control networks (BCNs) by using the algebraic state space representation approach. First, the state feedback stabilization control design of BCNs is recalled and the function perturbation problem is formulated. Second, given a state feedback stabilizer, it is robust to the considered function perturbation if one of the following three cases happens: 1) the block where the function perturbation occurs is different from the block which is affected by the state feedback stabilizer (Case 1); 2) when Case 1 does not happen, the perturbed column converges to the equilibrium faster than the original column (Case 2); 3) when Cases 1 and 2 do not happen, the perturbed column does not belong to the reachable set of the original column (Case 3). Third, when the perturbed column belongs to the reachable set of the original column, a constructive procedure is proposed to modify the given state feedback stabilizer to be robust to the function perturbation. Finally, the obtained new results are applied to the function perturbation analysis of lactose operon in Escherichia coli. The main novelty of this brief is to develop a new theoretical framework for the robustness of feedback controllers of BCNs with respect to function perturbation, which is not solved in the existing literature.

This paper investigates the latticized linear programming that is subject to the fuzzy-relation inequality (FRI) constraints with the max-min composition by using the semi-tensor product method, and proposes a matrix approach to this problem. First, the resolution of the FRI is studied, and it is proved that all the minimal solutions and the unique maximum solution are within the finite parameter set solutions. Based on this and using the semi-tensor product, solving FRIs is converted to solving a set of algebraic inequalities, and some new results on the resolution of FRIs are presented. Second, the latticized linear programming that is subject to the FRI constraints is solved by taking the following two key steps: 1) the optimal value is obtained by calculating the minimum value of the objective function among all the minimal solutions to the FRI constraints; and 2) the optimal solution set is obtained by solving the fuzzy-relation equation that is generated by letting the objective function equal to the optimal value. The study of illustrative examples shows that the new results that are obtained in this paper are very effective in solving the latticized linear programming subject to the FRI constraints.