(?)₁-induced Performance Analysis Of 2-D Positive System And Its Application

Two-dimensional(2-D)systems have always been an interesting and challenging problems due to their significant applications,such as circuits analysis,digital image processing and thermal power engineering.Therefore,they have received considerable attention and a variety of theoretical contribution to this field have appeared in the literatures over the past decades.However,up to now,a large number of research results of positive system appear in one-dimensional system.There is a lack of research on two-dimensional positive system,and the results are relatively limited.Therefore,it is very important to study and obtain the relevant basic theory,which is the original motivation of this paper.The main contents of this paper are as follows:In the first part,the analysis of the (?)₁-induced norm has been devoted and a novel characterization has proposed for 2-D positive systems.As we know that the most popular models of 2-D linear systems were introduced by Roesser,Fornasini and Marchesini.For those two models,an analytical method to compute the exact value of (?)₁-induced norm is first presented.Then,on the basis of these results,a novel characterization for stability and (?)₁-induced performance is proposed for 2-D positive systems.Those characterization are formulated in the form of linear programming(LP).Finally,the theoretical results are illustrated through a numerical example.In the second part,the analysis of the (?)₁-induced norm has been devoted for two-dimensional discrete positive system based on T-S fuzzy model and the corresponding feedback controller has been designed.Through the T–S fuzzy model,the 2D nonlinear systems are formulated into a framework consisting of a set of local models smoothly connected by some membership functions,and the Lyapunov function which depends on the fuzzy basis function is selected to analyze the system performance under the interference input.An (?)₁-induced performance index is explicitly presented for2-D T–S fuzzy positive systems and analytically characterized under a linear Lyapunov function framework.Based on this condition,Sufficient conditions are established in terms of linear programming(LP)problems to design the desired feedback controller.Finally,the theoretical results are illustrated through a numerical example.In the finally part,the problem of positive filtering is investigated for positive two-dimensional(2D)Roesser T–S fuzzy systems under (?)₁-induced performance.Error-bounding positive filters are constructed to estimate the output of positive 2D T–S fuzzy systems.Firstly,a novel characterisation for asymptotic stability of the filtering error system with (?)₁-induced performance is derived.Then,based on that characterisation,sufficient conditions are derived to design the required filters expressed by linear programming problems.Finally,the theoretical results are illustrated through a numerical example.