Observability of Boolean control networks

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.

Input observability of Boolean control networks

The observability of Boolean control networks is investigated. The pairs of states are classified into three classes: (i)diagonal, (ii) h-distinguishable, and h-indistinguishable. For h-indistinguishable pairs, we construct a matrix W called the transferable matrix, which indicates the control-transferability among h-indistinguishable pairs. Modifying W yields a matrix U0, which is used as the initial matrix for an iterative algorithm. After finite iterations a stable Uk∗ is reached, which is called the observability matrix. It is proved that a Boolean control network is observable, if and only if, the last column for all rows of Uk∗ are one. Some numerical examples are presented.

A note on observability of Boolean control networks

Observability of Boolean networks via set controllability approach

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.

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 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

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

Disturbance decoupling of Boolean networks via robust indistinguishability method

Controllability and observability of singular Boolean control networks are proposed in this paper from its algebraic form via semi-tensor product. First, the input-state incidence matrix of singular Boolean control networks is presented based on which the generalized input-state incidence matrix is also defined. Second, a necessary and sufficient condition for the controllability of singular Boolean control networks is given, as well as an algorithm to design controls which drive a state to a reachable state. At last, similar to Boolean control networks, a sufficient condition of observability of singular Boolean control networks is obtained.

The concept of bisimulation has been recently introduced in the study of Boolean control networks (BCNs). The notion provides a perspective that explains how a state trajectory of one BCN can be paired with a state trajectory of another BCN, and vice versa. The present paper further pursues the notion of bisimulation in BCNs. It first gives necessary and sufficient conditions for checking this notion. The conditions are more general and neater than those previously published; more importantly, they allow us to go further to consider reducing a BCN via bisimulation. A bisimulation-based methodology is then proposed to reduce the size of the state space of a BCN. Applying the reduction methodology allows one to infer control properties of a BCN from those of the corresponding reduced system. Possibilities of applications of the results are illustrated on several Boolean models of real biomolecular systems.

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.

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