Finite-Time Controllability and Set Controllability of Impulsive Probabilistic Boolean Control Networks

Abstract

This paper addresses the finite-time controllability and set controllability of impulsive probabilistic Boolean control networks (IPBCNs). Firstly, using the algebraic state space representation (ASSR) method, the transition probability matrix of IPBCNs is established. Secondly, a kind of finite step reachability matrix with probability one is constructed, based on which, several effective criteria are proposed for the finite-time controllability with probability one of IPBCNs. Thirdly, a necessary and sufficient condition is presented for the finite-time set controllability with probability one of IPBCNs by constructing the set controllability probability distribution vector. Finally, the obtained results are extended to switching topology case.

Robust fuzzy model predictive control for voltage regulation in islanded microgrids

“…In other words, this matrix should be a full rank matrix and rows should be independent of each other. As was mentioned in [35], if matrices $${{\bf A}_{{{inv,i}}}}$$ and $${{{\bf B}}_{{{inv,i}}}}$$ are time independent, the pair ($${{\bf A}_{{{inv,i}}}}$$, $${{{\bf B}}_{{{inv,i}}}}$$) is controllable. Therefore, according to Equation (8), it can be easily concluded that this system is controllable.…”

“…Remark 1. (Controllability) [35]: System (8) is controllable if for any initial state x 0 and any final state x f , there exists an input sequence, which transfers x 0 to x f in a finite time interval. Otherwise, the system is uncontrollable.…”

“…(y(k + 1) − y r (k + 1)) T P e−1 (y(k + 1) − y r (k + 1)) ≤ 𝜉 e−1 (35) The above inequality is developed to consider noises and disturbances. By taking into account the disturbances w T (k)w(k) ≤ 𝜂 2 , inequality (35) holds if and only if:…”

Finite-time observability of probabilistic Boolean multiplex control networks

“…Through the BNs, the macro-behavior and micro-mechanism of complex systems are combined, which not only gives us methodological enlightenment to study complex systems, but also presents a novel approach to address the complexity issues in the real world, specifically in systems biology [15,1], chemistry [12], engineering [17], social networks [31], etc. Recently, the introduction of the semi-tensor product (STP) has led to significant advancements in addressing various theoretical challenges associated with BNs, such as disturbance decoupling [3], stability and stabilization [2,32,43,29,36], reachability [45,28], controllability [30,19,35], synchronization [48], optimal control [34,39,40] and other related problems [37,7,10].…”