Chin-Tzong Pang

A note on the sequence of consecutive powers of a nonnegative matrix in max algebra

Abstract In this note, we shall consider the sequence of consecutive powers in max algebra of a nonnegative matrix with each element less than or equal to 1. The notion of asymptotic period is defined to study the `limiting behavior of this sequence. A simple and effective characterization for the asymptotic period of the sequence is given.

On the Stability of Takagi–Sugeno Fuzzy Systems With Time-Varying Uncertainties

This paper studies the problems of stability analysis of Takagi-Sugeno free fuzzy systems with time-varying uncertainties. In our prior study, we represented the time-varying uncertainty incurred in characteristic interval matrices in terms of the stability of Takagi-Sugeno free fuzzy systems with consequent parameter uncertainties. Based on Mayers convergent theorem for powers of single interval matrix and its generalization, we further proposed some sufficient conditions for the Takagi-Sugeno free fuzzy system with time-varying uncertainties to be globally asymptotically stable. In this paper, we propose the notion of simultaneously nilpotent interval matrices to investigate the Takagi-Sugeno free fuzzy system with time-varying uncertainties to be strongly stable within steps, where relates to the dimension of interval matrices. Moreover, a unique situation for the deterministic Takagi-Sugeno free fuzzy system to be strongly stable within steps is derived as well, where relates to the dimension of characteristic matrices for the deterministic Takagi-Sugeno free fuzzy system.

On nilpotent fuzzy matrices

Abstract In this paper, we study the issue of nilpotent fuzzy matrices. We first provide some properties of nilpotent fuzzy matrices in terms of eigenvalues. When a finite number of fuzzy matrices are simultaneously considered, we establish some characterizations of the simultaneous nilpotence for a finite number of fuzzy matrices. On the other hand, it is well known that the max–min algebra is one kind of lattice. We shall extend the results on simultaneous nilpotence to matrices on a bounded distributive lattice.

Sufficient conditions for the stability of linear Takagi-Sugeno free fuzzy systems

In this paper, we will study the stability issues of the linear Takagi-Sugeno (T-S) free fuzzy systems. Based on matrix norm, we propose a new sufficient condition for the linear T-S free fuzzy system to be globally asymptotically stable. We then study the stability analysis in the case of systems with consequent parameter uncertainty. Based on Mayers convergent theorem, we propose a sufficient condition, which is easily implemented, for the systems with consequent parameter uncertainty to be globally asymptotically stable.

On the sequence of consecutive powers of a fuzzy matrix with max-Archimedean-t-norms

In the literature, the limiting behavior of powers of a fuzzy matrix has been studied with max-product composition. Pang and Guu proposed a simple and effective characterization for the limiting behavior with the notion of asymptotic period. In this paper, we shall extend Pang and Guus work to the realm with the max-Archimedean-t-norm composition, to which the max-product belongs. Moreover, some sufficient conditions for the powers to converge in finitely many steps will be established.

AN INEQUALITY FOR THE SPECTRAL RADIUS OF AN INTERVAL MATRIX

Abstract For an n × n interval matrix A = (Aij), we say that A is majorized by the point matrix A = (a ij ) if aij = |Aij| when the jth column of A has the property that there exists a power A m containing in the same jth column at least one interval not degenerated to a point interval, and aij = Aij otherwise. Denoting the generalized spectral radius (in the sense of Daubechies and Lagarias) of A by ϱ( A ), and the usual spectral radius of A by ϱ(A), it is proved that if A is majorized by A then ϱ(A) ⩽ ϱ( A ) . This inequality sheds light on the asymptotic stability theory of discrete-time linear interval systems.

On simultaneously nilpotent fuzzy matrices

Abstract Nilpotent fuzzy matrices play a crucial role in the study of fuzzy matrices. In this paper, we shall extend the nilpotence to the notion of simultaneous nilpotence for a finite set of fuzzy matrices. The notion of simultaneous nilpotence relates to the infinite products of a finite number of fuzzy matrices which converge to the zero matrix. Properties of the simultaneous nilpotence will be established. In the study of consecutive powers of a fuzzy matrix, a controllable fuzzy matrix can be characterized by an associated nilpotent fuzzy matrix. In this paper, we propose the notion of simultaneously controllable fuzzy matrices which can be thought of as a generalization of the notion of controllable fuzzy matrices. Similar to the nilpotent characterization for a controllable fuzzy matrix, the simultaneously controllable fuzzy matrices can be characterized by a finite set of associated simultaneously nilpotent fuzzy matrices.

Convergence of products of fuzzy matrices

Convergence of powers of a fuzzy matrix has been studied in the literature. In this paper, we shall extend the scope to the products of a finite number of fuzzy matrices. We shall demonstrate that the outcomes of products of fuzzy matrices will be more complicated than the cases of powers of a fuzzy matrix. Two notions of convergence are established: weak convergence and strong convergence. The main results to guarantee the weak convergence of products of fuzzy matrices are due to the concepts of transitive and compact properties of fuzzy matrices, which can be viewed as a generalization of the transitive and compact properties of a fuzzy matrix. Sufficient conditions similar to the pinching theorem in Calculus are given as well for strong convergence.

On Infinite Products of Fuzzy Matrices

In this paper, we study the convergence of infinite products of a finite number of fuzzy matrices, where the operations involved are max-min algebra. Two types of convergences in this context will be discussed: the weak convergence and strong convergence. Since any given fuzzy matrix can be decomposed of the sum of its associated Boolean matrices, we shall show that the weak convergence of infinite products of a finite number of fuzzy matrices is equivalent to the weak convergence of infinite products of a finite number of the associated Boolean matrices. Further characterizations regarding the strong convergence will be established. On the other hand, sufficient conditions for the weak convergence of infinite products of fuzzy matrices are proposed. A necessary condition for the weak convergence of infinite products of fuzzy matrices is presented as well.

A new proof of Mayer's theorem☆

This paper gives a new proof of Mayers theorem concerning the convergence of powers of an interval matrix.