Partial Synchronization of Interconnected Boolean Networks

Necessary and Sufficient Conditions for Event-Triggered Set Stabilizability of Markovian Jump Logical Networks

This technical paper utilizes the Lyapunov theory to characterize the event-triggered set stabilizability of Markovian jump logical control networks (MJLCNs). Whereas the existing result for checking the set stabilizability of MJLCNs is only sufficient, this technical paper further establishes its necessary and sufficient condition. First, the Lyapunov function is established to describe the set stabilizability of MJLCNs necessarily and sufficiently by combining recurrent switching modes and desired state set. Then, the triggering condition and the input updating mechanism are designed regarding the value change of the Lyapunov function. Finally, the effectiveness of theoretical results is demonstrated by a biological example concerning the lac operon in Escherichia coli.

Minimal Pinning Control for Oscillatority of Boolean Networks

In this article, minimal pinning control for oscillatority (i.e., instability) of Boolean networks (BNs) under algebraic state space representations method is studied. First, two criteria for oscillatority of BNs are obtained from the aspects of state transition matrix (STM) and network structure (NS) of BNs, respectively. A distributed pinning control (DPC) from these two aspects is proposed: one is called STM-based DPC and the other one is called NS-based DPC, both of which are only dependent on local in-neighbors. As for STM-based DPC, one arbitrary node can be chosen to be controlled, based on certain solvability of several equations, meanwhile a hybrid pinning control (HPC) combining DPC and conventional pinning control (CPC) is also proposed. In addition, as for NS-based DPC, pinning control nodes (PCNs) can be found using the information of NS, which efficiently reduces the high computational complexity. The proposed STM-based DPC and NS-based DPC in this article are shown to be simple and concise, which provide a new direction to dramatically reduce control costs and computational complexity. Finally, gene networks are simulated to discuss the effectiveness of theoretical results.

Stabilization of Markovian Jump Boolean Control Networks via Sampled-Data Control

In this article, we study the finite-time stabilization and the asymptotic stabilization with probability one of Markovian jump Boolean control networks (MJBCNs) by sampled-data state feedback controls (SDSFCs). Based on the semi-tensor product (STP), we introduce an augmented variable multiplied by the vector form of the switching signal and the state of MJBCN. We find that under SDSFC, the sequence of the states of the augmented variable at sampling instants satisfies the Markov property. Based on the convergences of the switching signal and the augmented variable, we obtain the sufficient and necessary criteria for the finite-time stabilization and the asymptotic stabilization of MJBCNs by SDSFCs, respectively. Moreover, for the two kinds of stabilization, the feedback matrices of SDSFCs are constructed, respectively. Finally, the obtained results are applied to an apoptosis network and a model of the lactose operon in the Escherichia Coli.

Set Stability and Set Stabilization of Boolean Control Networks Avoiding Undesirable Set

The traditional set stability of Boolean networks (BNs) refers to whether all the states can converge to a given state subset. Different from the existing results, the set stability investigated in this paper is whether all states in a given initial set can converge to a given destination set. This paper studies the set stability and set stabilization avoiding undesirable sets of BNs and Boolean control networks (BCNs), respectively. First, by virtue of the semi-tensor product (STP) of matrices, the dynamics of BNs avoiding a given undesirable set are established. Then, the set reachability and set stability of BNs from the initial set to destination set avoiding an undesirable set are investigated, respectively. Furthermore, the set stabilization of BCNs from the initial set to destination set avoiding a given undesirable set are investigated. Finally, a design method for finding the time optimal set stabilizer is proposed, and an example is provided to illustrate the effectiveness of the results.

Pinning Controllability for a Boolean Network With Arbitrary Disturbance Inputs

In this paper, pinning controllability for a Boolean network (BN) under arbitrary disturbance inputs is considered. By selecting a fraction of nodes as the pinning nodes and injecting controllers that depend on the values of the disturbance inputs of the previous time instant, the controllability can be guaranteed for any BNs under arbitrary disturbance inputs. First, based on the necessary and sufficient conditions for the controllability of a BN with arbitrary disturbance inputs obtained in this paper, a constructive method for designing the transition matrix of the BN is presented, which further provides a method for selecting the pinning nodes. Second, a logical relationship between the control input nodes and system nodes is provided through solving logical matrix equations. Third, the control input sequence algorithm is also given. Finally, an example is presented to show the effectiveness of the proposed results.

Event-Triggered Sampled Feedback Synchronization in an Array of Output-Coupled Boolean Control Networks

In this article, synchronization problem in an array of output-coupled Boolean control networks (BCNs) is studied by using event-triggered sampled feedback control. Algebraic forms of an array of output-coupled BCNs are presented via the semitensor product (STP) of matrices. Based on the algebraic forms, a necessary and sufficient condition is obtained for the synchronization of an array of output-coupled BCNs. Furthermore, an algorithm is proposed to design event-triggered sampled feedback controllers. Finally, the obtained results are well illustrated by numerical examples.

Necessary and Sufficient Conditions on Pinning Stabilization for Stochastic Boolean Networks

In this paper, the stabilization problem of the Boolean network (BN) with stochastic disturbances via pinning control has been investigated. The necessary and sufficient conditions are given for robust stabilization of a BN with stochastic disturbances. Then, pinning control is considered to stabilize a BN with stochastic disturbances. An algorithm is given to obtain a new stable system and the pinning control, including the pinned nodes, control design, and control adding, is also solved. Finding the minimal number of pinned nodes is further analyzed. The necessary and sufficient conditions are obtained for the solvability of pinning control, based on which some matrices sets are constructed which leads to the necessary and sufficient conditions of pinning t nodes. Furthermore, an algorithm is introduced to search the minimal number of pinned nodes and what exactly they are, which will reduce the computational burden. Examples are given to illustrate the efficiency of the obtained results.

Set Stabilization of Probabilistic Boolean Control Networks: A Sampled-Data Control Approach

This article investigates the set stabilization of probabilistic Boolean control networks (PBCNs) under sampled-data (SD) state-feedback control within finite and infinite time, respectively. First, the algorithms are, respectively, proposed to find the sampled point set and the largest sampled point control invariant set (SPCIS) of PBCNs by SD state-feedback control. Based on this, a necessary and sufficient criterion is proposed for the global set stabilization of PBCNs by SD state-feedback control within finite time. Moreover, the time-optimal SD state-feedback controller is designed. It is interesting that if the sampled period (SP) is changed, the time of global set stabilization of PBCNs may also change or even the PBCNs cannot achieve set stabilization. Second, a criterion for the global set stabilization of PBCNs by SD state-feedback control within infinite time is obtained. Furthermore, all possible SD state-feedback controllers are obtained by using all the complete families of reachable sets. Finally, three examples are presented to illustrate the effectiveness of the obtained results.

Stabilization of Mode-Dependent Impulsive Hybrid Systems Driven by DFA With Mixed-Mode Effects

This paper is concerned with mode-dependent impulsive hybrid systems driven by deterministic finite automaton (DFA) with mixed-mode effects. In the hybrid systems, a complex phenomenon called mixed mode, caused in time-varying delay switching systems, is considered explicitly. Furthermore, mode-dependent impulses, which can exist not only at the instants coinciding with mode switching but also at the instants when there is no system switching, are also taken into consideration. First, we establish a rigorous mathematical equation expression of this class of hybrid systems. Then, several criteria of stabilization of this class of hybrid systems are presented based on semi-tensor product (STP) techniques, multiple Lyapunov-Krasovskii functionals, as well as the average dwell time approach. Finally, an example is simulated to illustrate the effectiveness of the obtained results.

Optimal State Estimation of Boolean Control Networks With Stochastic Disturbances

This paper presents an investigation of the optimal estimation of state for Boolean control networks subject to stochastic disturbances. The disturbances are modeled as independently and identically distributed processes that are assumed to be both mutually independent and independent of the current and the historical states. An iterative algorithm is proposed to calculate the conditional probability distribution of the state given the output measurements. This algorithm is applied to the problems of minimum mismatching estimation and maximum posterior estimation of the state. An example is provided to illustrate the proposed results.

Local Synchronization of Interconnected Boolean Networks With Stochastic Disturbances

This paper is concerned with the local synchronization problem for the interconnected Boolean networks (BNs) without and with stochastic disturbances. For the case without stochastic disturbances, first, the limit set and the transient period of the interconnected BNs are discussed by resorting to the properties of the reachable set for the global initial states set. Second, in terms of logical submatrices of a certain Boolean vector, a compact algebraic expression is presented for the limit set of the given initial states set. Based on it, several necessary and sufficient conditions are derived assuring the local synchronization of the interconnected BNs. Subsequently, an efficient algorithm is developed to calculate the largest domain of attraction. As for the interconnected BNs with stochastic disturbances, first, mutually independent two-valued random logical variables are introduced to describe the stochastic disturbances. Then, the corresponding local synchronization criteria are also established, and the algorithm to calculate the largest domain of attraction is designed. Finally, numerical examples are employed to illustrate the effectiveness of the obtained results/ algorithms.

Pinning Controllers for Activation Output Tracking of Boolean Network Under One-Bit Perturbation

This paper studies pinning controllers for activation output tracking (AOT) of Boolean network under one-bit perturbation, based on the semitensor product of matrices. First, the definition of AOT with respect to an activation number is presented, where the activation number means the number of active outputs whose logical variables are 1 s. Then, several criteria are established for AOT issue. Further, the impact of one-bit perturbation on AOT is studied, where one-bit perturbation means that only one logical function has one-bit change of its truth table by flipping the value from 1 to 0 or 0 to 1. In addition, if a one-bit perturbation is a valid perturbation on AOT, an output feedback pinning control is designed to recover AOT. The obtained results are effectively illustrated by a D. melanogaster segmentation polarity gene network and a reduced signal transduction network.

Leader-Follower Consensus of Multiagent Systems With Time Delays Over Finite Fields

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.

Sampled-Data Control for the Synchronization of Boolean Control Networks

In this paper, we investigate the sampled-data state feedback control (SDSFC) for the synchronization of Boolean control networks (BCNs) under the configuration of drive-response coupling. Necessary and sufficient conditions for the complete synchronization of BCNs are obtained by the algebraic representations of logical dynamics. Based on the analysis of the sampling periods, we establish an algorithm to guarantee the synchronization of drive-response coupled BCNs by SDSFC. An example is given to illustrate the significance of the obtained results.

H∞ State Estimation for Discrete-Time Nonlinear Singularly Perturbed Complex Networks Under the Round-Robin Protocol

This paper investigates the H_∞ state estimation problem for a class of discrete-time nonlinear singularly perturbed complex networks (SPCNs) under the Round-Robin (RR) protocol. A discrete-time nonlinear SPCN model is first devised on two time scales with their discrepancies reflected by a singular perturbation parameter (SPP). The network measurement outputs are transmitted via a communication network where the data transmissions are scheduled by the RR protocol with hope to avoid the undesired data collision. The error dynamics of the state estimation is governed by a switched system with a periodic switching parameter. A novel Lyapunov function is constructed that is dependent on both the transmission order and the SPP. By establishing a key lemma specifically tackling the SPP, sufficient conditions are obtained such that, for any SPP less than or equal to a predefined upper bound, the error dynamics of the state estimation is asymptotically stable and satisfies a prescribed H_∞ performance requirement. Furthermore, the explicit parameterization of the desired state estimator is given by means of the solution to a set of matrix inequalities, and the upper bound of the SPP is then evaluated in the feasibility of these matrix inequalities. Moreover, the corresponding results for linear discrete-time SPCNs are derived as corollaries. A numerical example is given to illustrate the effectiveness of the proposed state estimator design scheme.

Event-Triggered Control for the Disturbance Decoupling Problem of Boolean Control Networks

This paper investigates the disturbance decoupling problem (DDP) of Boolean control networks (BCNs) by event-triggered control. Using the semi-tensor product of matrices, algebraic forms of BCNs can be achieved, based on which, event-triggered controllers are designed to solve the DDP of BCNs. In addition, the DDP of Boolean partial control networks is also derived by event-triggered control. Finally, two illustrative examples demonstrate the effectiveness of proposed methods.

Synchronization Control for a Class of Discrete-Time Dynamical Networks With Packet Dropouts: A Coding-Decoding-Based Approach

The synchronization control problem is investigated for a class of discrete-time dynamical networks with packet dropouts via a coding-decoding-based approach. The data is transmitted through digital communication channels and only the sequence of finite coded signals is sent to the controller. A series of mutually independent Bernoulli distributed random variables is utilized to model the packet dropout phenomenon occurring in the transmissions of coded signals. The purpose of the addressed synchronization control problem is to design a suitable coding-decoding procedure for each node, based on which an efficient decoder-based control protocol is developed to guarantee that the closed-loop network achieves the desired synchronization performance. By applying a modified uniform quantization approach and the Kronecker product technique, criteria for ensuring the detectability of the dynamical network are established by means of the size of the coding alphabet, the coding period and the probability information of packet dropouts. Subsequently, by resorting to the input-to-state stability theory, the desired controller parameter is obtained in terms of the solutions to a certain set of inequality constraints which can be solved effectively via available software packages. Finally, two simulation examples are provided to demonstrate the effectiveness of the obtained results.