Characterizations of Global Transversal Exponential Stability

Abstract

We study the relationship between the global exponential stability of an invariant manifold and the existence of a positive semi-definite Riemannian metric which is contracted by the flow. In particular, we investigate how the following properties are related to each other (in the global case): i). A manifold is globally "transversally" exponentially stable; ii). The corresponding variational system (c.f. (7) in Section II) admits the same property; iii). There exists a degenerate Riemannian metric which is contracted by the flow and can be used to construct a Lyapunov function. We show that the transverse contraction rate being larger than the expansion of the shadow on the manifold is a sufficient condition for the existence of such a Lyapunov function. An illustration of these tools is given in the context of global full-order observer design.

Distance-based Formation-Motion Control for Unicycle Agents

“…We use Theorem 1, which is based on [22], [20], to prove the proposition by showing that the corresponding transversal linear systems satisfy UES-TL property in (5). In this case, the transversal linear systems are given by ė = ∂F ∂e (0, φ12r ) , φ12r = G ( φ12r ) ,…”

LMI conditions for contraction, integral action, and output feedback stabilization for a class of nonlinear systems

“…In many control applications, it is of primary importance to guarantee some convergence properties of any two trajectories. This is the case, for instance, of observers design [1][2][3], synchronization problems [4][5][6][7], or output regulation [8][9][10]. Such a property has been defined and characterized in the literature in several different manners: incremental stability [3,[11][12][13][14][15][16], convergence [9,17] or quadratic stability [18].…”

“…Differently from the Lyapunov based approaches in [11,12] or the logarithmic norm approach in [14,16], in order to establish the desired contractivity properties we follow the framework based on metric analysis [13,29]. In particular, following [3,13], we look for the existence of a Riemannian metric for which the distance between different trajectories of the closed loop system is monotonically decreasing in forward time. By supposing that such a Riemannian metric is Euclidean, sufficient conditions based on linear matrix inequalities (LMIs) are then derived.…”

Contraction analysis of nonlinear systems and its application

“…As one of the stability analysis methods for time-varying systems, which has gained popularity in recent years, contraction analysis focuses on the relative trajectory of the nonlinear time-varying system rather than a specific equilibrium point. There are many methods to analyze the contractivity of nonlinear timevarying ODE systems in literature, such as, [5,7,27,55,100] among many others. An ODE system is contracting if and only if the associated variational system is uniformly globally exponentially stable (UGES) [7].…”

“…Mode-dependent Lyapunov functions for contracting switched nonlinear systems are presented in [100]. In [5], the focus is on investigating transverse exponential stability (a generalized notion of contraction) by employing a Lyapunov matrix transversal equation. In the context of DAE systems, the contraction analysis thereof has recently been presented in [62].…”