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In this article, a general notion of common diagonal Lyapunov matrix is formulated for a collection of n × n matrices A 1, … , A s , and cones k 1, … , k s in ℝ n . Necessary and sufficient conditions are derived for the existence of a common diagonal Lyapunov matrix in this setting. The conditions are similar to and extend the well-known criteria for the case s = 1, k 1 = ℝ n . [ABSTRACT FROM AUTHOR]

Abstract: An important problem in system theory concerns determining whether or not a given LTI system is diagonally stable. More precisely, this problem is concerned with determining conditions on a matrix such that there exists a diagonal matrix with positive diagonal entries (i.e. a positive diagonal matrix), satisfying . While this problem has attracted much attention over the past half century, two results of note stand out: (i) a result based on Theorems of the Alternative derived by Barker, Berman and Plemmons; and (ii) algebraic conditions derived by Redheffer. This paper is concerned with the second of these conditions. Our principal contribution is to show that Redheffer’s result can be obtained from the Kalman–Yacubovich–Popov lemma. We then show that this method of proof leads to natural generalisations of Redheffer’s result and we use these results to derive new conditions for diagonal and Hurwitz stability for special classes of matrices. [Copyright &y& Elsevier]

Abstract: In this paper we study communication networks that employ drop-tail queueing and additive-increase multiplicative-decrease (AIMD) congestion control algorithms. We show that the theory of non-negative matrices may be employed to model such networks and to derive basic theorems concerning their behaviour. [Copyright &y& Elsevier]

We study communication networks that employ drop-tail queueing and Additive-Increase Multiplicative-Decrease (AIMD) congestion control algorithms. It is shown that the theory of nonnegative matrices may be employed to model such networks. In particular, important network properties, such as: 1) fairness; 2) rate of convergence; and 3) throughput, can be characterized by certain nonnegative matrices. We demonstrate that these results can be used to develop tools for analyzing the behavior of AIMD communication networks. The accuracy of the models is demonstrated by several NS studies. [ABSTRACT FROM AUTHOR]

We study the spectrum of a positive matrix that arises in the study of certain communication networks. Bounds are given on the rate of convergence of the networks to the equilibrium condition. [Copyright &y& Elsevier]

The Inverse Frobenius–Perron Problem (IFPP) concerns the creation of discrete chaotic mappings with arbitrary invariant densities. In this Letter, we present a new and elegant solution to the IFPP, based on positive matrix theory. Our method allows chaotic maps with arbitrary piecewise-constant invariant densities, and with arbitrary mixing properties, to be synthesized. [Copyright &y& Elsevier]

A sufficient condition for the existence of a Lyapunov function of the form V(x) = xT xP, P = PT > 0, P ∈ Rn×n, for the stable linear time invariant systems x˙ = iA x, iA ∈ Rn×n, iA ∈ A ≜ {A1, …, mA}, is that the matrices iA are Hurwitz, and that a non‐singular matrix T exists, such that iT A T−1, i ∈ {1, …, m}, is upper triangular [Mori, Y., Mori, T., & Kuroe, Y., Proceedings of Electronic Information and Systems Conference (1996); Proceedings of 36th Conference on Decision and Control, (1997); Liberzon, D., Hespanha, J.P., & Morse, S., Technical Report, Laboratory for Control Science and Engineering, Yale University, (1998); Shorten, R. & Narendra, K., Proceedings of Conference on Decision and Control, (1998)]. The existence of such a function, referred to as a common quadratic Lyapunov function (CQLF), is sufficient to guarantee the exponential stability of the switching system x˙ = A(t)x, A(t) ∈ A. In this paper we investigate the stability properties of a related class of switching systems. We consider sets of matrices A, where no single matrix T exists that simultaneously transforms each iA ∈ A to upper triangular form, but where a set of non‐singular matrices i jT exist such that the matrices {i jiT A i jT−1, i jjT A i jT−1}, i, j ∈ {1, …, m}, are upper triangular. We show that, for a special class of such systems, the origin of the switching system x˙ = A(t)x, A(t) ∈ A, is globally attractive. A novel technique is developed to derive this result, and the applicability of this technique to more general systems is discussed towards the end of the paper. [ABSTRACT FROM PUBLISHER]

In this paper we give necessary and sufficient conditions for weak and strong quadratic stability of a class of switched linear systems consisting of two subsystems, associated with symmetric transfer function matrices. These conditions can simply be tested by checking the eigenvalues of the product of two subsystem matrices. This result is an extension of the result by Shorten and Narendra for strong quadratic stability, and the result by Shorten et al. on weak quadratic stability for switched linear systems. Examples are given to illustrate the usefulness of our results. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we derive stability results for large-scale interconnections of “mixed” linear, time-invariant systems using classical Nyquist arguments. We compare our results with Moylan and Hill (1978) . Our results indicate that, if one relaxes assumptions on the subsystems in an interconnection from assumptions of passivity or small gain to assumptions of “mixedness,” then the Moylan and Hill-like conditions on the interconnection matrix become more stringent. Finally, we explore a condition for the stability of large-scale, time-varying interconnections of strictly positive real systems. This condition mirrors the condition obtained in  for time-invariant interconnections and is thus an extension of this work. [Copyright &y& Elsevier]

Abstract: We consider a convex combination of matrices that arise in the study of communication networks and the corresponding convex combination of Kronecker squares of these matrices. We show that the spectrum of the first convex combination is contained in the spectrum of the second set and that the second largest eigenvalues coincide. [Copyright &y& Elsevier]