On existence and stability of equilibria of linear time-invariant systems With constant power loads

The problem of existence and stability of equilibria of linear systems with constant power loads is addressed in this paper. First, we correct an unfortunate mistake in our recent paper [10] pertaining to the sufficiency of the condition for existence of equilibria in multiport systems given there. Second, we give two necessary conditions for existence of equilibria. The first one is a simple linear matrix inequality hence it can be easily verified with existing software. Third, we prove that the latter condition is also sufficient if a set defined by the problem data is convex, which is the case for single and two-port systems. Finally, sufficient conditions for stability and instability for a given equilibrium point are given. The results are illustrated with two benchmark examples.Postprint (author's final draft

Binary Operations in the Unit Ball: A Differential Geometry Approach

Within the framework of differential geometry, we study binary operations in the open, unit ball of the Euclidean n-space R n , n ∈ N , and discover the properties that qualify these operations to the title addition despite the fact that, in general, these binary operations are neither commutative nor associative. The binary operation of the Beltrami-Klein ball model of hyperbolic geometry, known as Einstein addition, and the binary operation of the Beltrami-Poincaré ball model of hyperbolic geometry, known as Möbius addition, determine corresponding metric tensors in the unit ball. For a variety of metric tensors, including these two, we show how binary operations can be recovered from metric tensors. We define corresponding scalar multiplications, which give rise to gyrovector spaces, and to norms in these spaces. We introduce a large set of binary operations that are algebraically equivalent to Einstein addition and satisfy a number of nice properties of this addition. For such operations we define sets of gyrolines and co-gyrolines. The sets of co-gyrolines are sets of geodesics of Riemannian manifolds with zero Gaussian curvatures. We also obtain a special binary operation in the ball, which is isomorphic to the Euclidean addition in the Euclidean n-space

On the need of projections in input-error model reference adaptive control

International audienceThe main objective of this note is to contribute, if modestly, toward the understanding of the input‐error model reference adaptive control scheme revealing an instability mechanism that arises if the projection of the plant high‐frequency gain coefficient estimate is omitted. In addition, a self‐contained proof of global convergence of the scheme with the projections for a simple first‐order plant is given

On Global Asymptotic Stability of SPR Adaptive Systems Without Persistent Excitation

International audienceWe present a sufficient condition for global asymptotic stability of linear time-varying systems of the form = Ax + B phi(inverted perpendicular)(t)theta, = -phi(t)C-inverted perpendicular x with strictly positive real transfer function W(s) = C-inverted perpendicular(sI - A)(-1) B and the vector theta(t) not satisfying the well-known persistent excitation condition. It is also shown that the criterion is optimal in some well-defined sense-making the condition "almost" necessary as well. This class of systems arise in many control applications including system identification and adaptive control and, to the best of the authors' knowledge, no necessary and sufficient condition for global asymptotic stability has been reported

Robust stability under relaxed persistent excitation conditions

International audienceFor linear time-varying systems with a persistently excited state matrix it is well-known that input-to-state stability is recovered. In this note a relaxed condition on persistence of excitation is studied together with related robust stability notions (input-to-state stability and integral input-to-state stability). The results are illustrated by simulations for scalar systems

On contraction of time-varying port-Hamiltonian systems

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Robustness of linear time-varying systems with relaxed excitation

International audienceIt is a well-known fact that linear time-varying systems with a persistently excited state matrix are exponentially converging and input-to-state stable with respect to additive perturbations. Recently, several relaxed conditions of persistent excitation have been presented [1], [2], which ensure an asymptotic convergence rate in the system. In the present work it is shown that these conditions are similar, and that under such a relaxed excitation only non-uniform in time input-to-state stability and integral input-to-state stability properties can be obtained. The results are illustrated by simulations for a problem of estimation in the linear regression model

Duality Results for the Joint Spectral Radius and Transient Behavior

For linear inclusions in discrete or continuous time several quantities characterizing the growth behavior of the corresponding semigroup are analyzed. These quantities are the joint spectral radius, the initial growth rate and (for bounded semigroups) the transient bound. It is recalled how these constants relate to one another and how they are characterized by various norms. A complete duality theory is developed in this framework, relating semigroups and dual semigroups and extremal or transient norms with their respective dual norms