Controllability and Observability of Singular Boolean Control Networks

Topological structure of implicit Boolean networks

“…There is a model in the cell cycle, biochemical oscillator, which controls the DNA replication and mitosis phases [29]. The model consists of three states, cyclins , cyclin‐dependent kinases , and ligases , which are denoted by , and , respectively.…”

“…Dynamic‐algebraic BNs (D‐ABNs) were proposed in [26], also called singular BNs [27], whose properties were detailedly investigated using STP [28]. Along this line, a great deal of the literature about D‐ABNs has gradually emerged such as analysis of attractors [29], controllability and observability [29], optimal control [30], and so on. Up to now, there exist many types of BNs including classic BNs, probabilistic BNs [14, 31], switched BNs [32–35] etc.…”

Controllability of dynamic‐algebraic Boolean networks based on a new normalisation approach

“…Therefore, (4) holds. □ Remark 2 The matrices used to judge the controllability are W in this paper and in [21], respectively. It is found that general form of singular Boolean control networks (SBCNs) was considered in [21].…”

“…□ Remark 2 The matrices used to judge the controllability are W in this paper and in [21], respectively. It is found that general form of singular Boolean control networks (SBCNs) was considered in [21]. However, the main result in [21] can not be used to judge the controllability of the considered system here, since the set given in [21] is empty in this paper.…”

“…DABNs with control, are called dynamic‐algebraic Boolean control networks (DABCNs). There are lots of works about DABNs and DABCNs, such as solvability [18], topological structure [19], disturbance decoupling problem [19, 20], and controllability [21]. The normalisation of DABNs is that states , which are determined by certain algebraic equations, can be expressed by functions of states , which satisfy Boolean dynamic models.…”