Disturbance decoupling of Boolean networks via robust indistinguishability method

Nominal Boolean network is a kind of Boolean network, which is obtained from a Boolean control network with all controls disconnected. In this paper, the relationship between the transition matrix of the original Boolean control network and the transition matrix of the corresponding nominal Boolean control network is investigated. Due to the fact that transition matrices can be derived from structure matrices, the relationship between structure matrices of the original Boolean control network and structure matrices of the nominal Boolean network is studied. Via constant values corresponding to disconnected controls, structure matrices of the nominal Boolean network can be derived from structure matrices of the original Boolean control network. Finally, a concrete algorithm is presented to derive transition matrices of nominal Boolean networks.

The paper deals with problems on the stability and stabilization of Boolean networks with respect to a given part of variables characterizing the Boolean network. Using the semi-tensor product of matrices and the matrix expression of logic, the dynamics of a Boolean (control) network can be converted to discrete time linear (bilinear) dynamics called the algebraic form of the Boolean (control) network. The two main results are as follows: (i) From the algebraic form of a Boolean (control) network, two partial stability definitions are introduced. One is partial stability with respect to another variable set, and the other is partial stability with respect to arbitrary initial states. (ii) Based on the algebraic form, necessary and sufficient conditions for the stability of global partial states with respect to the variable set and arbitrary initial states are obtained. Examples are provided to illustrate the effectiveness of our results.

Controllability and observability of Boolean control networks subject to stuck-at fault

We investigate a type of disturbance decoupling problem (DDP) of Boolean control networks. Using semi-tensor product of matrices, the dynamics of Boolean control network is expressed in its algebraic form. All the necessary arguments of the functions of outputs are called output-friendly coordinates. In order to estimate the solvability of DDP, we give a necessary and sufficient condition of the output-friendly coordinates being always in a known invariant subspace. Then it is computationally feasible to construct all the valid feedback control matrices. The logical function of each controller can be recovered from the obtained feedback control matrix. We further discuss the constraints of the selection of the invariant subspace. Examples are provided to show the effectiveness of the proposed method.

The stability of Boolean networks and the stabilization of Boolean control networks are investigated. Using semi-tensor product of matrices and the matrix expression of logic, the dynamics of a Boolean (control) network can be converted to a discrete time linear (bilinear) dynamics, called the algebraic form of the Boolean (control) network. They provide a framework for this study. Main results consist of two parts: (i) Using logic coordinate transformation, the known sufficient condition based on incidence matrix has been improved. It is also used in stabilizer design. (ii)Based on algebraic form, necessary and sufficient conditions for stability of Boolean networks and stabilization of Boolean control networks respectively are obtained.

On the ensemble controllability of Boolean control networks using STP method

Serval existing definitions for observability of Boolean control networks (BCNs) are revisited and a new definition termed observability via output feedback is proposed. This definition restricts the input sequences used to distinguish distinct initial states to be generated by output feedbacks. A technique called parallel extension is proposed for observability analysis, the main idea of which is to interconnect a BCN with its duplication in parallel and transfer the observability problem of the original BCN to the set reachability problem of the interconnected BCN. Novel necessary and sufficient conditions for observability and observability under arbitrary inputs have been given, respectively. In addition, by restricting the output feedback to be static, a sufficient condition for observability via output feedback is proposed. An example is given to illustrate the main results.

State feedback controller design for synchronization of master–slave Boolean networks based on core input-state cycles

Minimum energy control and optimal-satisfactory control of Boolean control network

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.

The paper studies the restabilization problem of Boolean control networks (BCNs) under the function perturbation. First, an important concept‐absorbable attractor is proposed for BCNs. Then, by using this concept, a necessary and sufficient restabilizability criterion is established. This criterion can be used to check whether the perturbed BCN can be restabilized to its original state by modifying the minimum number of parameters in the old controller. Finally, a constructive fine‐tuning method is given to modify the old controller. Compared with the existing results, which can only solve the problem of single column function perturbation, our results are more powerful since they are applicable not only to the single column function perturbation problem but also to the case of multiple columns. Finally, two examples are employed to show the effectiveness of our results.

This paper mainly investigates the reachability of a kind of Boolean control networks with delays in both state and control. By a system transformation, the reachability problem of the delayed Boolean control network becomes that of a Boolean control network without delays. The relationship between the transformed network and the original network is revealed. A necessary and sufficient condition for the reachability of the delayed Boolean control network is obtained. An example is given to illustrate the effectiveness of the obtained theoretical result.

Input observability of Boolean control networks

In this article, the controllability of asynchronous Boolean control networks (ABCNs) with time-invariant delay is studied. Using semi-tensor product, Boolean control networks under Harvey’s asynchronous update with time delay are converted into linear representation, and a general formula of control-depending network transition matrices is achieved. Firstly, a necessary and sufficient condition is proposed to verify that only control-depending fixed points of ABCNs can be deterministically controlled. Secondly, respectively for three kinds of controls, the controllability of the non-deterministic logical dynamics is investigated, and formulas are given to show: a) the reachable sets at time s under a group of specified controls; b) the reachable sets at time s under arbitrary controls and c) the probability values from a given initial state to different target states. Moreover, we also discuss how to find the particular control sequence to reach a specified target state with the maximum probability. Examples are shown to illustrate the feasibility of the proposed scheme.

We show some new results on the observability of Boolean control networks (BCNs). First, to study the observability, we combine two BCNs with the same transition matrix into a new BCN. Then, we propose the concept of a reachable set that results in a given set of initial states, and we derive four additional necessary and sufficient conditions for the observability of BCNs. In addition, we present an algoriTheoRemark and construct an observability graph to determine the observability of BCNs. Finally, we illustrate the obtained results using three numerical examples.