Abstract
In this paper, the robust invariant set (RIS) of Boolean (control) networks with disturbances is investigated. First, for a given fixed point, consider a special set called immediate neighborhoods of the fixed point; then a discrete derivative of Boolean functions at the fixed point is used to analyze the robust invariance, based on which a sufficient condition is obtained. Second, for more general sets, the robust output control invariant set (ROCIS) of Boolean control networks (BCNs) is investigated by semitensor product (STP) of matrices. Then, under a given output feedback controller, we obtain a necessary and sufficient condition to check whether a given set is robust control invariant set (RCIS). Furthermore, output feedback controllers are designed to make a set to be a RCIS. Finally, the proposed methods are illustrated by a reduced model of the lac operon in E. coli.
Highlights
In 1969, Kauffman [1] firstly used Boolean networks (BNs), which are a kind of logical networks to study genetic regulatory networks
We will design output feedback controllers such that the trajectories of Boolean control networks (BCNs) starting from some initial states in a given set will never leave the given set, which is called robust output control invariant set (ROCIS)
Giriv≤en2nn,oannedmwpetyasneatlySze=th{δe2i1fno, lδlo2i2nw, .in. .g, δ2irn } two problems: (i) Problem 1: For a given output feedback matrix G ∈ L2m×2p, analyze whether the set S is a ROCIS of system (20) under control system u(t) = Gy(t)
Summary
In 1969, Kauffman [1] firstly used Boolean networks (BNs), which are a kind of logical networks to study genetic regulatory networks. In detail, [28] investigated the evolutionarily stable strategy of finite evolutionary networked games by STP and designed event-triggered controllers such that systems could converge globally. Designing controllers is of great importance such that the set of desirable cellular states of BCNs with disturbance inputs is robust. Complexity set, which is called robust control invariant set (RCIS) [33], those trajectories will never leave the set no matter what disturbances are. In [33], Li et al used STP to study the RCIS of BCNs and presented an effective procedure to design state feedback controllers. We will design output feedback controllers such that the trajectories of BCNs starting from some initial states in a given set will never leave the given set, which is called robust output control invariant set (ROCIS).
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