Realization of Boolean Control Networks

Abstract

Based on the linear expression of the dynamics of Boolean networks, the coordinate transformation of Boolean variables is defined. It follows that the state space coordinate transformation for the dynamics of Boolean networks is revealed. Using it, the invariant subspace for a Boolean control network is defined. Then the structure of a Boolean control network is analyzed, and the controllable and observable normal forms and the Kalman decomposition form are presented. Finally the realization problem, including minimum realization, of Boolean control networks is investigated.

Invertibility and nonsingularity of Boolean control networks

“…In the framework of STP, many control-theoretic problems, e.g., controllability, observability, realization, disturbance decoupling, identification and reconstructibility problems of BCNs were solved (cf. Cheng, 2011, Cheng, Li, & Qi, 2010, Cheng & Qi, 2009, Cheng & Zhao, 2011, Hochma, Margaliot, Fornasini, & Valcher, 2013, Laschov et al, 2013, Li & Wang, 2012, Zhang & Zhang, 2014, Zhao et al, 2010. In Zhang and Zhang (2014), we for the first time solved how to determine observability of BCNs completely that was open for five years.…”

Minimum‐Time State Feedback Stabilization of Constrained Boolean Control Networks

“…Using this tool, one can convert a Boolean (control) network into a linear (bilinear) discrete-time system, and then conveniently analyze Boolean (control) networks by using the classical control theory. In [7], using the linear expression of Boolean networks, Cheng et al presented the controllable and observable normal forms of Boolean networks. Laschov and Margaliot [14] developed the maximum principle for the Mayer-type optimal control of Boolean networks based on the bilinear form of BCNs.…”

Cycles of periodically time-variant Boolean networks

“…With this new technique, the topological structures of time-invariant Boolean networks (TIBNs) can be revealed completely . Furthermore, many classical problems of control theory have been extended to Boolean control networks (BCNs) such as controllability, observability (Cheng & Qi, 2009), stabilization (Cheng, Qi, Li, & Liu, 2011), Kalman decomposition (Cheng, Li, & Qi, 2010), disturbance decoupling (Cheng, 2011) and optimal control (Zhao, Li, ✩ Li and Wang (2012), reachability and controllability are generalized to switched BNs, which are actually a special kind of time-variant systems. In Zhang and Zhang (2013), for general time-variant BCNs, a necessary and sufficient condition for the controllability is obtained and a control design algorithm is presented.…”