From semi-tensor product of matrices to logical control systems

Abstract

The logical control system theory (LCST) based on semi-tensor product (STP) of matrices has become a new direction in control field. The purpose of this paper is to give a comprehensive introduction to the logical control system, from the basic mathematical tools and analysis methods to the main results. First we introduce the STP of matrices. Meanwhile, a digging discussion is made to reveal some of its insides. Then we explain how to use STP to convert a logic expression into algebraic form. Then an itemized introduction to the main achievements in analysing and control design of LCSs in last few years is presented, which consists of the topological structure of logical systems, controllability, observability, and stabilization, the stability of disturbance decoupling, optimal control, identification of LCSs. Next, the extension from classical Boolean logic to $k$-valued logic, to multi-valued logic, and to mixed-valued logic is explained. All these results form the theoretical foundation of this new discipline. In addition, the concepts, notations, and the fundamental method/tool have been uniformly normalized. Then some current developments of LCST and its applications or potential applications have been discussed. Some open problems and possible research directions have also been presented. Finally, a personal prediction about the development of control theory is given as: The combination of logic control systems and dynamic control systems is expected to generate a ``information-control system, it is what Tsien Hsueshen predicted ``theorical control theory, and is also the next generation of control theory proposed by American discipline development report.