State Feedback Stabilization of Generic Logic Systems via Ledley Antecedence Solution

TL;DR

This paper introduces a modified Ledley antecedence solution method for designing optimal state feedback stabilizers in generic logic systems, including Boolean, k-valued, and mix-valued systems, ensuring shortest-time stabilization.

Contribution

The paper extends Ledley antecedence solutions to admissible sets and restricted domains, providing an algorithm for verifying solvability and constructing all optimal stabilizers.

Findings

Method successfully stabilizes Boolean control networks in shortest time.

Applicable to k-valued and mix-valued logical systems with minor modifications.

Provides a systematic approach for verifying and designing stabilizers in logic systems.

Abstract

In this paper, the application of Ledley antecedence solutions in designing state feedback stabilizers of generic logic systems has been proposed. To make the method feasible, two modifications are made to the original Ledley antecedence solution theory: (i) the preassigned logical functions have been extended from being a set of equations to an admissible set; (ii) the domain of arguments has been extended from the whole state space to a restricted subset. In the proposed method, state feedback controls are considered as a set of extended Ledley antecedence solutions for a designed iterative admissible sets over their corresponding restricted subsets. Based on this, an algorithm has been proposed to verify the solvability, and simultaneously to provide all possible state feedback stabilizers when the problem is solvable. All stabilizers are optimal, which stabilize the logic systems from any initial state to the destination state/state set in the shortest time. The method is firstly demonstrated on Boolean control networks to achieve point stabilization. Then, with some minor modifications, the proposed method is also proven to be applicable to set stabilization problems. Finally, it is shown that in $k$-valued and mix-valued logical systems, the proposed method remains effective.