Asymptotical Feedback Set Stabilization of Probabilistic Boolean Control Networks

Long-Run Behavior Estimation of Temporal Boolean Networks With Multiple Data Losses

This brief devotes to investigating the long-run behavior estimation of temporal Boolean networks (TBNs) with multiple data losses, especially the asymptotical stability. The information transmission is modeled by Bernoulli variables, based on which an augmented system is constructed to facilitate the analysis. A theorem guarantees that the asymptotical stability of the original system can be converted to that of the augmented system. Subsequently, one necessary and sufficient condition is obtained for asymptotical stability. Furthermore, an auxiliary system is derived to study the synchronization issue of the ideal TBNs with normal data transmission and TBNs with multiple data losses, as well as an effective criterion for verifying synchronization. Finally, numerical examples are given to illustrate the validity of the theoretical results.

Set stabilization of logical control networks: A minimum node control approach

In network systems, control using minimum nodes or pinning control can be effectively used for stabilization problems to cut down the cost of control. In this paper, we investigate the set stabilization problem of logical control networks. In particular, we study the set stabilization problem of probabilistic Boolean networks (PBNs) and probabilistic Boolean control networks (PBCNs) via controlling minimal nodes. Firstly, an algorithm is given to search for the minimum index set of pinning nodes. Then, based on the analysis of its high computational complexity, we present optimized algorithms with lower computational complexity to ascertain the network control using minimum node sets. Moreover, some sufficient and necessary conditions are proposed to ensure the feasibility and effectiveness of the proposed algorithms. Furthermore, a theorem is presented for PBCNs to devise all state-feedback controllers corresponding to the set of pinning nodes. Finally, two models of gene regulatory networks are considered to show the efficacy of obtained results.

Pinning Controller Design for Set Reachability of State-Dependent Impulsive Boolean Networks

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.

Pinning Stabilizer Design for Probabilistic Boolean Control Networks via Condensation Digraph

This article investigates the design of pinning controllers for state feedback stabilization of probabilistic Boolean control networks (PBCNs), based on the condensation digraph method. First, two effective algorithms are presented to achieve state feedback stabilization of the considered system from the perspective of condensation digraph. One is to find the desired matrix, and the other is to search for the minimum number of pinned nodes and specific pinned nodes. Then, all the mode-independent pinning controllers can be designed based on the desired matrix and pinned nodes. Several examples are delineated to illustrate the validity of the main results.

Feedback Stabilization of Boolean Control Networks With Missing Data

Data loss is often random and unavoidable in realistic networks due to transmission failure or node faults. When it comes to Boolean control networks (BCNs), the model actually becomes a delayed system with unbounded time delays. It is difficult to find a suitable way to model it and transform it into a familiar form, so there have been no available results so far. In this article, the stabilization of BCNs is studied with Bernoulli-distributed missing data. First, an augmented probabilistic BCN (PBCN) is constructed to estimate the appearance of data loss items in the model form. Based on this model, some necessary and sufficient conditions are proposed based on the construction of reachable matrices and one-step state transition probability matrices. Moreover, algorithms are proposed to complete the state feedback stabilizability analysis. In addition, a constructive method is developed to design all feasible state feedback controllers. Finally, illustrative examples are given to show the effectiveness of the proposed results.

Multi-Sensor Fusion Boolean Bayesian Filtering for Stochastic Boolean Networks

Stochastic Boolean networks (SBNs) take process noise into account, so it is better to fit the actual situation and has a wider application background than Boolean networks (BNs). However, the presence of noise influences us to estimate the real state of the system. To minimize the inaccuracies caused by the presence of noise, an optimal state estimation problem is studied in this article. The multi-sensor fusion Boolean Bayesian filtering is proposed and a recursive algorithm is provided to calculate the prior and posterior belief of system state by fusing multi-sensor measurements based on the algebraic form of the SBN and Bayesian law. Then, the optimal state estimator is obtained, which minimizes the mean-square estimation error. Finally, a simulation example is carried out to demonstrate the performance of the proposed methodology. It has been shown through the simulation experiment that it increases the confidence level of the state estimation and improves the estimation performance using multi-sensor fusion compared with using single sensor.

Safe control of logical control networks with random impulses

Under the framework of a hybrid-index model, this paper investigates safe control problems of state-dependent random impulsive logical control networks (RILCNs) on both finite and infinite horizons, respectively. By using the ξ-domain method and the constructed transition probability matrix, the necessary and sufficient conditions for the solvability of safe control problems have been established. Further, based on the technique of state-space partition, two algorithms are proposed to design feedback controllers such that RILCNs can achieve the goal of safe control. Finally, two examples are shared to demonstrate the main results.

Weak Stabilization of Boolean Networks Under State-Flipped Control

In this brief, stabilization of Boolean networks (BNs) by flipping a subset of nodes is considered, here we call such action state-flipped control. The state-flipped control implies that the logical variables of certain nodes are flipped from 1 to 0 or 0 to 1 as time flows. Under state-flipped control on certain nodes, a state-flipped-transition matrix is defined to describe the impact on the state transition space. Weak stabilization is first defined and then some criteria are presented to judge the same. An algorithm is proposed to find a stabilizing kernel such that BNs can achieve weak stabilization to the desired state with in-degree more than 0. By defining a reachable set, another approach is proposed to verify weak stabilization, and an algorithm is given to obtain a flip sequence steering an initial state to a given target state. Subsequently, the issue of finding flip sequences to steer BNs from weak stabilization to global stabilization is addressed. In addition, a model-free reinforcement algorithm, namely the Q -learning ( [Formula: see text]) algorithm, is developed to find flip sequences to achieve global stabilization. Finally, several numerical examples are given to illustrate the obtained theoretical results.

Self-Triggered Scheduling for Boolean Control Networks

It has been shown that self-triggered control has the ability to deal with cases with constrained resources by properly setting up the rules for updating the system control when necessary. In this article, self-triggered stabilization of the Boolean control networks (BCNs), including the deterministic BCNs, probabilistic BCNs, and Markovian switching BCNs, is first investigated via the semitensor product of matrices and the Lyapunov theory of the Boolean networks. The self-triggered mechanism with the aim to determine when the controller should be updated is provided by the decrease of the corresponding Lyapunov functions between two consecutive samplings. Rigorous theoretical analysis is presented to prove that the designed self-triggered control strategy for BCNs is well defined and can make the controlled BCNs be stabilized at the equilibrium point.

State Estimation for Probabilistic Boolean Networks via Outputs Observation

This article studies the state estimation for probabilistic Boolean networks via observing output sequences. Detectability describes the ability of an observer to uniquely estimate system states. By defining the probability of an observed output sequence, a new concept called detectability measure is proposed. The detectability measure is defined as the limit of the sum of probabilities of all detectable output sequences when the length of output sequences goes to infinity, and it can be regarded as a quantitative assessment of state estimation. A stochastic state estimator is designed by defining a corresponding nondeterministic stochastic finite automaton, which combines the information of state estimation and probability of output sequences. The proposed concept of detectability measure further performs the quantitative analysis on detectability. Furthermore, by defining a Markov chain, the calculation of detectability measure is converted to the calculation of the sum of probabilities of certain specific states in Markov chain. Finally, numerical examples are given to illustrate the obtained theoretical results.