On Observability of Fuzzy Dynamical Matrix Lyapunov Systems

. In this paper we generate a fuzzy dynamical matrix Lyapunov system andobtain a sucient condition for the observability of this system. 1. IntroductionThe importance of control theory in Applied mathematics and its occurrencein several problems such as mechanics, electromagnetic theory, thermodynamics,articial satellites etc., are well known. The observability condition assures theconstruction of the state from the output. This property is intrensic for systemsand play an important role in the theory of linear systems.The objective of this paper is to provide sucient condition for observability ofrst order matrix Lyapunov system described by(1.1) X 0 (t) = A(t)X(t)+X(t)B(t)+F(t)U(t), X(0) = X 0 , t>0,(1.2) Y(t) = C(t)X(t)+D(t)U(t),where U(t) is a n× nfuzzy input matrix called fuzzy control and Y(t) is a n× nfuzzy output matrix. Here A(t),B(t),F(t),C(t), and D(t) are matrices of ordern×n, whose elements are continuous functions of t on J= [0,T] ⊂ R(T>0).The problem of controllability and observability for systems of ordinary dier-ential equations has been studied by Barnett [2] and for matrix Lyapunov systemsby Murty, Rao and Suresh Kumar [8]. Recently the observability criteria for fuzzydynamical control systems was studied by Ding and Kandel [5].In section 2 we present some basic denitions and results relating to fuzzy setsand also some properties of Kronecker products and obtain general solution of thesystem (1.1), when U(t) is a crisp continuous matrix.In section 3 we generate a fuzzy dynamical matrix Lyapunov system and alsoobtain its solution set.