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Necessary and sufficient conditions for Positive Realness of general transfer function matrices are derived. The conditions are expressed in terms of eigenvalues of matrix functions of the state matrices representation of the LTI system. Illustrative numerical examples are provided. Index Terms--Conditions, passivity, positive real (PR), spectral methods.

In this paper, we present a probabilistic synchronization algorithm whose convergence properties are examined using tools of rowstochastic matrices. The proposed algorithm is particularly well suited for wireless sensor network applications, where connectivity is not guaranteed at all times, and energy efficiency is an important design consideration. The tradeoff between the convergence speed and the energy use is studied. Index Terms--Common Lyapunov function, consensus, scrambling matrix, switched systems, synchronization.

Necessary and sufficient conditions for strict positive realness and positive realness of strictly proper functions are derived. The conditions are expressed in terms of eigenvalues of the state matrices representation of the system. Previous results rendered conditions which were significantly more complex than those for proper (but not strictly proper) functions. The present conditions for strictly proper functions are simpler than the ones for proper functions, which is consistent with intuition in this case. Illustrative numerical examples are provided. Index Terms--Circle criterion, eigenvalues locations, positive real (PR) conditions, state--space representation, time varying systems.

In this note, we introduce the concepts of best case and worst-case convergence rates for switched linear systems. Their connections to the problems of stability and stabilization are briefly discussed. For simultaneously triangularizable switched systems, explicit formulas are presented for the convergence rates in terms of the system eigenvalues. Index Terms--Convergence rate, simultaneous triangularization, switched linear systems, switching path, stability, stabilization.

In this note, the problem of determining necessary and sufficient conditions for the existence of a common quadratic Lyapunov function for a pair of stable linear time-invariant systems whose system matrices [A.sub.1] and [A.sub.2] are in companion form is considered. It is shown that a necessary and sufficient condition for the existence of such a function is that the matrix product [A.sub.1] [A.sub.2] does not have an eigenvalue that is real and negative. Examples are presented to illustrate the result. Index Terms--Quadratic stability, stability theory, switched linear systems.

In this note, we define strong and weak common quadratic Lyapunov functions (CQLFs) for sets of linear time-invariant (LTI) systems. We show that the simultaneous existence of a weak CQLF of a special form, and the nonexistence of a strong CQLF, for a pair of LTI systems, is characterized by easily verifiable algebraic conditions. These conditions are found to play an important role in proving the existence of strong CQLFs for general LTI systems. Index Terms--Quadratic stability, stability theory, switched linear systems.

In this note, we derive necessary and sufficient conditions for a single-input-single-output system to be (strictly) positive real. These conditions take the form of a spectral condition on a matrix product constructed from the state space representation of the linear time-invariant system {A, b, c, d}. Numerical examples are given to illustrate the usefulness of our conditions. Index Terms--Kalman-Yacubovic-Popov, positive real transfer functions, quadratic Lypunov functions, strictly positive real transfer functions.