This article addresses the fundamental issues of solvability and normalization in singular Boolean networks (SBNs) from a new perspective based on the admissible and normal initial state sets. It presents novel results and removes the restrictive conditions found in existing literature. First, the state transition matrix of SBNs is constructed by defining a new operator, and the admissible initial state set with the normal initial state set, of SBNs is introduced. Second, the problems of the solvability and uniqueness of the solution to a general SBN are converted into computing its admissible and normal initial state sets, which can be analytically computed using the derived formulas. Third, based on the normal initial state set, a necessary and sufficient condition for solving the normalization problem of general SBNs is established for the first time, which removes the restrictions in the existing literature. Finally, the results obtained are compared with existing literature and illustrated with examples.

Multi-objective games (MOGs) have received much attention in recent years as a class of games with vector payoffs. Based on the semi-tensor product (STP), this paper discusses the MOG, including the existence, finite-step reachability, and finite-step controllability of Pareto equilibrium of this model, from both static and dynamic perspectives. First, the MOG concept is presented using multi-layer graphs, and STP is used to convert the payoff function into its algebraic form. Then, from the static perspective, two necessary and sufficient conditions are proposed to verify whether all players can meet their expectations and whether the strategy profile is a Pareto equilibrium, separately. Furthermore, from the dynamic perspective, a strategy updating rule is designed to investigate the finite-step reachability of the evolutionary MOG. Finally, the finite-step controllability of the evolutionary MOG is analyzed by adding pseudo-players, and a backward search algorithm is provided to find the shortest evolutionary process and control sequence.

In this paper, the distributed cooperative control problems of discrete-time second-order networks over finite fields are investigated, under which a new perspective is provided for discrete-time second-order networks of agents with limited capacities of memory and communication. Firstly, the discrete-time second-order multi-agent model with a given protocol is established over the finite field. Then the leaderless finite-field synchronization network is studied, some necessary and sufficient conditions along with the design principles of the synchronous protocol are given to guarantee the network synchronization. In addition, the finite-field consensus problems including leaderless and leader-following cases are studied in sequence, both of which prove to be unachievable under the distributed control protocol. Finally, an application on camera networks and numerical examples are provided to support the obtained resultsNote to Practitioners—The multi-agent system is a collection of multiple agents with computing and mobility capabilities, and its distributed cooperative control problem aims to achieve the global objective by neighbor-based protocols, mainly including consensus, synchronization, formation control, etc. Because of its strong fault tolerance, robustness and scalability, it is widely used in military, industrial and agricultural production, medicine, transportation and so on. These three factors that mainly affect the group behavior of agents are the dynamic evolution, the network topology (that is, the connection between agents), and the communication mechanism. However, in practical applications, due to limited computation, memory, and communication capabilities, the state of the network is constrainted, which directly affects the dynamic evolution of the system. In this paper, a new kind of second-order synchronization dynamics, which can be used in certain applications such as multi-agent systems with periodic boundary conditions on position and velocity, is proposed. The states and communication weights of agents can only attain values from a finite alphabet equipped with a prime number of integers, called finite fields. Such networks have a finite convergence time, and are therefore friendly to multi-agent systems with limited capacities.

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 presents a novel dynamic programming approach to determine the robust controllability of Boolean control networks (BCNs) subject to stochastic disturbances. By applying Bellman's optimality principle, we derive the recurrence relation for computing the optimal time matrix, a crucial concept characterizing robust reachability between two arbitrary states. We develop a finite-termination dynamic programming algorithm to calculate the optimal time matrix exactly and efficiently, with a rigorously certified iteration count. Sufficient and necessary conditions for robust controllability are then established based on the optimal time matrix. Furthermore, for any pair of reachable states, we construct time-optimal state feedback control laws to steer the system from the initial state to the target state, regardless of disturbances. Finally, extensive numerical experiments with biological networks validate the effectiveness of the proposed approach, showing significant improvements in computational efficiency. Additionally, we introduce a Q-learning-based algorithm and compare its performance, highlighting the advantages of our dynamic programming approach in terms of both efficiency and solution quality.

A pinning control strategy for the local synchronization of drive-response Boolean networks is proposed in this article. A necessary and sufficient condition of the local synchronization is given, along with the maximum initial state set guaranteeing the synchronization. A procedure for determining pinning nodes with the minimum number and designing a state feedback for each pinning node, leading to a local synchronous drive-response Boolean network, is provided. Finally, a gene regulation network for the D. melanogaster segmentation polarity is elaborated to illustrate the effectiveness of the proposed results.

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.

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.

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.

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.

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.

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.

Observability ensures that any two distinct initial states can be uniquely determined by their outputs, so the stream ciphers can avoid unobservable nonlinear feedback shift registers (NFSRs) to prevent the occurrence of equivalent keys. This paper discusses the observability of Galois NFSRs over finite fields. Galois NFSRs are treated as logical networks using the semi-tensor product. The vector form of the state transition matrix is introduced, by which a necessary and sufficient condition is proposed, as well as an algorithm for determining the observability of general Galois NFSRs. Moreover, a new observability matrix is defined, which can derive a matrix method with lower computation complexity. Furthermore, the observability of two special types of Galois NFSRs, a full-length Galois NFSR and a nonsingular Galois NFSR, is investigated. Two methods are proposed to determine the observability of these two special types of NFSRs, and some numerical examples are provided to support these results.

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

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.

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 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.

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 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.

The problem of solving max-min fuzzy relational equations is investigated. First, we show that if there is a solution, then there is a corresponding solution within the set of parameters [briefly, the parameter set solution (PSS)]. Then, the semitensor product of matrices is used to convert the logical equations into algebraic equations via the vector expression of logical variables. Under this form, every PSS can be obtained. It is proved that all the solutions can be revealed from their corresponding PSS. Some examples are presented to demonstrate the algorithm to solve fuzzy relational equations.

The static output feedback control problem is investigated in this paper for continuous descriptor systems. New necessary and sufficient conditions are established to ensure the admissibility of the continuous descriptor systems. Then, based on the new necessary and sufficient conditions, sufficient conditions are derived for admissibility of the closed-loop systems. The desired static output feedback controllers can be derived directly via the cone complementarity linearization techniques. Finally, an illustrative example is presented to demonstrate the effectiveness of the proposed approach in this paper.