Boolean Control Networks Research Articles

Reduced-Order Observer Design for Boolean Control Networks

In this paper, we propose an approach to design reduced-order state observer for Boolean control networks by applying the semi-tensor product. At first, an approach is given to find reducible state variables. After that, we introduce an approach for reduced-order observer design based on the reducible state variables. Then, it is shown that similar to the Luenberger-like observer, the state estimate provided by the reduced-order observer also converges to real state at time not later than the minimal reconstructibility index. Additionally, the reduced-order observer requires lower computational effort, which facilitates an online implementation.

Double Deep-Q Learning-Based Output Tracking of Probabilistic Boolean Control Networks

In this article, a reinforcement learning (RL)-based scalable technique is presented to control the probabilistic Boolean control networks (PBCNs). In particular, a double deep-Q network (DDQN) approach is firstly proposed to address the output tracking problem of PBCNs, and optimal state feedback controllers are obtained such that the output of PBCNs tracks a constant as well as a time-varying reference signal. The presented method is model-free and offers scalability, thereby provides an efficient way to control large-scale PBCNs that are a natural choice to model gene regulatory networks (GRNs). Finally, three PBCN models of GRNs including a 16-gene and 28-gene networks are considered to verify the presented results.

Finite-Time Controllability and Set Controllability of Impulsive Probabilistic Boolean Control Networks

This paper addresses the finite-time controllability and set controllability of impulsive probabilistic Boolean control networks (IPBCNs). Firstly, using the algebraic state space representation (ASSR) method, the transition probability matrix of IPBCNs is established. Secondly, a kind of finite step reachability matrix with probability one is constructed, based on which, several effective criteria are proposed for the finite-time controllability with probability one of IPBCNs. Thirdly, a necessary and sufficient condition is presented for the finite-time set controllability with probability one of IPBCNs by constructing the set controllability probability distribution vector. Finally, the obtained results are extended to switching topology case.

Reinforcement Learning Approach to Feedback Stabilization Problem of Probabilistic Boolean Control Networks

In this letter, we study the control of probabilistic Boolean control networks (PBCNs) by leveraging a model-free reinforcement learning (RL) technique. In particular, we propose a Q-learning (QL) based approach to address the feedback stabilization problem of PBCNs, and we design optimal state feedback controllers such that the PBCN is stabilized at a given equilibrium point. The optimal controllers are designed for both finite-time stability and asymptotic stability of PBCNs. In order to verify the convergence of the proposed QL algorithm, the obtained optimal policy is compared with the optimal solutions of model-based techniques, namely value iteration (VI) and semi-tensor product (STP) methods. Finally, some PBCN models of gene regulatory networks (GRNs) are considered to verify the obtained 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.

Cascading decomposition of Boolean control networks: a graph-theoretical method

Two types of cascading decomposition problems of Boolean control networks are investigated using a graph-theoretical method. A new graphic concept called nested perfect equal vertex partition (NPEVP) is proposed. Based on NPEVP, the necessary and sufficient graphic conditions for solvability of the cascading decomposition problems are obtained. Given the proposed graphic conditions, the logical coordinate transformations are constructively obtained to realize the corresponding cascading decomposition forms. Finally, two illustrative examples are provided to validate the results.

Switching-based stabilization of aperiodic sampled-data Boolean control networks with all subsystems unstable

We aim to further study the global stability of Boolean control networks (BCNs) under aperiodic sampled-data control (ASDC). According to our previous work, it is known that a BCN under ASDC can be transformed into a switched Boolean network (SBN), and further global stability of the BCN under ASDC can be obtained by studying the global stability of the transformed SBN. Unfortunately, since the major idea of our previous work is to use stable subsystems to offset the state divergence caused by unstable subsystems, the SBN considered has at least one stable subsystem. The central thought in this paper is that switching behavior also has good stabilization; i.e., the SBN can also be stable with appropriate switching laws designed, even if all subsystems are unstable. This is completely different from that in our previous work. Specifically, for this case, the dwell time (DT) should be limited within a pair of upper and lower bounds. By means of the discretized Lyapunov function and DT, a sufficient condition for global stability is obtained. Finally, the above results are demonstrated by a biological example.

Mayer-Type Optimal Control of Probabilistic Boolean Control Network With Uncertain Selection Probabilities.

This article considers a Mayer-type optimal control problem of probabilistic Boolean control networks (PBCNs) with uncertainty on selection probabilities which obey Beta probabilistic distributions. The expectation with respect to both the selection probabilities and the transitions of state variables is set as a cost function, and it deduces an equivalent formulation as a multistage decision problem. Furthermore, the dynamic programming technique is applied to solve the problem and performs a novel optimization algorithm in the fashion of semitensor product. A numerical example of a biological model of apoptosis protein demonstrates the effectiveness and feasibility of the proposed framework and algorithms.

Controllability and Observability of Boolean Control Networks via Sampled-Data Control

In this paper, the controllability and observability of sampled-data Boolean control networks (SDBCNs) are investigated. New phenomena are observed in the study of the controllability and observability of SDBCNs. We routinely convert SDBCNs into linear discrete-time systems by the semitensor product of matrices. Necessary and sufficient conditions are derived for the controllability of SDBCNs. After that, we combine two SDBCNs with the same transition matrix into a new SDBCN to study the observability. Using an iterative algorithm, a stable row vector ${\mathcal {U}}^*$ , called the observability row vector, in finite iterations, is obtained. It is proved that an SDBCN is observable, if and only if $\Vert {\mathcal {U}}^*\Vert _1=N^2-N$ with $N:=2^n$ , where $n$ is the number of state-variables of BNs. Moreover, based on graph theory, a more effective algorithm is given to determine the observability of SDBCNs. Its complexity is not related to the length of the sampling period. In addition, some equivalent necessary and sufficient conditions are put forward for the observability of SDBCNs. Numerical examples are given to demonstrate the effectiveness of the obtained results.

Lyapunov Functions for the Set Stability and the Synchronization of Boolean Control Networks

We propose Lyapunov functions for the set stability of Boolean networks (BNs) and control Lyapunov functions for the feedback set stabilization and the synchronization of Boolean control networks (BCNs) in this brief. Meanwhile, some propositions of these Lyapunov functions are put forward. Sufficient and necessary conditions for the set stability of BNs, the feedback set stabilization of BCNs and the synchronization of drive-response coupled BCNs in the form of Lyapunov functions are presented successively. Moreover, an algorithm is given to construct control Lyapunov functions and state feedback controllers simultaneously for the synchronization of coupled BCNs. Two examples are given to illustrate the obtained results.

An average optimal control approach to the set stabilization problem for boolean control networks

AbstractThis paper addresses the set stabilization problem for deterministic Boolean control networks (BCNs). An optimal control approach is investigated to solve the problems by using the semi‐tensor product of matrices, where a policy iteration algorithm for the set stabilization problem is deduced. Finally, the intervention problem of a cAMP receptor protein is addressed in the framework of the set stabilization problem. The problem is solved to validate the effectiveness of the proposed policy iteration approach for a practical application.

An improved multiple‐state observer of Boolean control networks

AbstractThis paper explores the issue of state estimation for Boolean control networks (BCNs), and a kind of improved multiple‐state observer is proposed. The improved multiple‐state observer can be described by means of a specific BCN that overcomes the difficulty of the existing multiple state observers where it is difficult to find a general expression for the observer gain matrix. Next, based on the states that can possibly generate the output and those that are observed by the designed observer in current time step, an adaptive algorithm that completes the design of the multiple‐state observer is provided to update the observer states, and which makes the state estimation of Boolean control networks feasible. Finally, an example is presented to illustrate the effectiveness of the obtained results.

Output Tracking of Boolean Control Networks

The output tracking (regulation) problem of Boolean control networks is considered. First, formulas are given for calculating the set of control attractors and the (maximum) control invariant subset within a given subspace. Next, by constructing an auxiliary system and using the set controllability method, an easily verifiable necessary and sufficient condition is presented. In addition, an effective algorithm for designing open-loop control sequences are provided. Closed-loop controls are also designed, which is optimal in tracking time. Finally, an illustrative example is given.

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.

On pinning reachability of probabilistic Boolean control networks

On pinning reachability of probabilistic Boolean control networks

Controllability and observability of state‐dependent switched Boolean control networks with input constraints

AbstractThis paper investigates the controllability and observability of state‐dependent switched Boolean control networks (SDSBCNs) with input constraints. First, the algebraic form of SDSBCNs is derived via the semi‐tensor product of matrices. Then constrained model‐input‐state (CMIS) matrices are introduced for SDSBCNs with input constraints. On the basis of CMIS matrices, two necessary and sufficient conditions are provided for the controllability and observability. Finally, two examples are given to show the effectiveness of the main results.

Event-Based Robust Synchronization of Boolean Control Networks

In this brief, we mainly concentrate on the robust synchronization of boolean control networks (BCNs) under event-triggered control (ETC). Different from the previous designed event-triggered controller for BCNs, the proposed event-triggered scheme (ETS) ensures the control input is not updated over a certain period of time. The robust synchronization problem in this brief is firstly converted into the robust set stabilization problem. Then, an optimal algorithm which realizes the minimum of the event-triggered number is proposed to design state feedback controllers for robust synchronization of BCNs. Finally, a numerical example is given to show the effectiveness of the theoretical results.

Synchronization of drive-response Boolean control networks with impulsive disturbances

In this paper, we focus on the topic of synchronization of Boolean control networks (BCNs) with drive-response structure with impulsive disturbances. By transforming BCNs with impulsive disturbances into its algebraic expression, free control sequence for synchronization of BCNs with impulsive disturbances is designed. Besides, some necessary and sufficient conditions are derived for synchronization of BCNs with impulsive disturbances under feedback control. Moreover, an algorithm is given to design feedback controllers. Lastly, one example is used to better illustrate the derived results.

Sampled-Data State-Feedback Stabilization of Probabilistic Boolean Control Networks: A Control Lyapunov Function Approach.

This article investigates the partial stabilization problem of probabilistic Boolean control networks (PBCNs) under sample-data state-feedback control (SDSFC) with a control Lyapunov function (CLF) approach. First, the probability structure matrix of the considered PBCN is represented by a Boolean matrix, based on which, a new algebraic form of the system is obtained. Second, we convert the partial stabilization problem of PBCNs into the global set stabilization one. Third, we define CLF and its structural matrix under SDSFC. It is found that the existence of a CLF is equivalent to that of SDSFC. Then, a necessary and sufficient condition is obtained for the existence of CLF under SDSFC, based on which, all possible sample-data state-feedback controllers and corresponding structural matrices of CLF are designed by two different methods. Finally, examples are given to illustrate the efficiency of the obtained results.

Output Tracking Control Design of Switched Boolean Control Networks

In this letter, the output tracking control design of switched Boolean control networks (SBCNs) is investigated via state feedback and output feedback control. The algebraic state-space representation method which resorts to the semi-tensor product (STP) of matrices is utilized; necessary and sufficient conditions for the solvability of the output tracking control problem are presented. A constructive procedure is given to obtain all possible switching signal-dependent state feedback and output feedback controllers under arbitrary switching signals such that the output of SBCNs tracks a time-varying reference trajectory. Finally, a SBCN model of the lactose regulation in the Escherichia Coli bacteria is considered to illustrate the effectiveness of the proposed results.