Convergence Of The Power Sequence Of A Fuzzy Matrix

In this article, we mainly study the convergence of the powers of a fuzzy matrix , to which much attention is paid by many scholars in the field of the theory on fuzzy control. The object of fuzzy control is fuzzy system, and one of the main purposes to control a plant is to make it perfect as much as we want. It is important to make the system to attain the steady state in finite time. While a fuzzy system is described by a fuzzy relational matrix corresponding to the fuzzy input, the fuzzy output and the fuzzy controller. So, to study the convergence of powers of a fuzzy matrix is one of keys to study the stability of a fuzzy system.We divide the article into three chapters, and study the convergence of powers of a fuzzy matrix from different aspects, i.e., we study the convergence of powers of a fuzzy matrix under the max-t-norm composition for different t-norms.The first chapter is preliminarlies, in which some definitions, such as fuzzy matrix, triangular norm, fuzzy operations on fuzzy matrix, convergence and oscillation of powers of a fuzzy matrix, graph theory of powers of a fuzzy matrix and so on, have been introducted.In the second chapter, convergence of the power sequence of an n n fuzzy matrix under the max-product composition has been studied. And the following results are proved(l)The powers of an nx n fuzzy matrix under the max-product composition either converge to an idempotent fuzzy matrix with a finite or infinite index, or oscillate with a finite or infinite period;(2)There exist seven cases on the convergence of an nx n fuzzy matrix, say A, under the max-product composition, i.e.,In the third chapter, firstly, because of one of the same characters on product and Zero t-norm, i.e.. aTa < a(0 < a < 1), where T is product or Zero t-norm, convergence of the power sequence of an nxn fuzzy matrix under the max-Zero t-norm (recorded in brief by max-to) composition has been studied. Since Lukasiewicz t-norm is one kind of Zero t-norm, convergence of the power sequence of an nxn fuzzy matrix under the max-Lukasiewicz t-norm (recorded in brief by max-Lu) composition has been studied, too. Although R0 is not a Zero t-norm, it takes on some better characters. Therefore, convergence of the power sequence of an n n fuzzy matrix under the max-R0 composition has been studied . In the end, we compare defferent results on convergence of powers of an nx n fuzzy matrix under these four compositions, respectively: max-min , max-H0, max-Lu , max-product. In this chapter, the following results are proved(l)the powers of an nx n fuzzy matrix under the max-to composition either converge to an idempotent fuzzy matrix with a finite or infinite index, or oscillate with a finite or infinite period;(2) the powers of an nx n fuzzy matrix under the max-Lu composition either converge to an idempotent fuzzy matrix with a finite index, or oscillate with a finite period;(3)the powers of an nx n fuzzy matrix under the max-R0 composition either converge to an idempotent fuzzy matrix with a finite index, or oscillate with a finite period;(4)the powers of an nx n fuzzy matrix, say A, converge under the max-min composition = converge under the max-R0 composition = converge under the max-Lu composition = converge under the max-product composition.