The Study Of Controllable Linear Systems’ Reachable Sets Under The Bounded Control Function

Over the years, control theory has been widely used in the aerospace, informationand network, nanotechnology, biology and so on. It has a significant impact on people’slife. Therefore, more and more people pay an attention to the study of control theory. Thelinear systems are considered to be the basic research objects in the field of control. Butmost of the problems we have encountered in real life are the nonlinear systems.Nonlinear system is very complicated to discuss, in order to facilitate our research anddiscussion, some not serious nonlinear systems which only work in the small areas nearthe working point, usually we study the nature of them by using approximate linearmodels.In control theory, the classical theories are controllability and observability. For theclassical control theory, control function is equivalent to observable function. In therange of the reachable set the linear system is controllable. The study of reachable set hasa significant impact on the controllability. In the practical applications, the ability ofcontrol is always limited, so studying the reachable set under bounded control functionhas a very significant impact. By the premise of the controlled system, this paper we willdiscuss the range of the reachable set under the bounded control function.In this paper, the main work includes: for the two-dimensional varying-time linearsystem and high-dimensional varying-time linear system, we will discuss the range ofreachable set by using the “low energy” function which is derived by the mapping fromthe introducing. For the three-dimension linear continuous-time systems, we mainlystudy the relationship among the reachable set and the system matrix and the controlmatrix and under the bounded control function and single input, and give a series of theexpressions of the range of the reachable set by using eigenvalues of the system matrix.We will see some studies of low-dimension linear discrete-time systems’ reachable setsunder the bounded control function. But for the linear continuous-time systems, theresearch of this aspect is very scanty. So it does in varying-time linear system, someconclusions are not rigorous.