An intrinsic observer for a class of lagrangian systems

Abstract

Abstract-We propose a new design method of asymptotic observers for a class of nonlinear mechanical systems: Lagrangian systems with configuration (position) measurements. Our main contribution is to introduce a state (position and velocity) observer that is invariant under any changes of the configuration coordinates. The observer dynamics equations, as the Euler-Lagrange equations, are intrinsic. The design method uses the Riemannian structure defined by the kinetic energy on the configuration manifold. The local convergence is proved by showing that the Jacobian of the observer dynamics is negative definite (contraction) for a particular metric defined on the state-space, a metric derived from the kinetic energy and the observer gains. From a practical point of view, such intrinsic observers can be approximated, when the estimated configuration is close to the true one, by an explicit set of differential equations involving the Riemannian curvature tensor. These equations can be automatically generated via symbolic differentiations of the metric and potential up to order two. Numerical simulations for the ball and beam system, an example where the scalar curvature is always negative, show the effectiveness of such approximation when the measured positions are noisy or include high frequency neglected dynamics.

Robust Synchronization via Homogeneous Parameter-Dependent Polynomial Contraction Matrix

“…Next, we will show that the auxiliary system (13) is contracting if there exists a parameter-dependent contraction matrix. A concise proof of exponential convergence of trajectories for contracting system is given in [32] for an uncertainty-free case. Let y 0 and y 1 be two different points and let Υ(y, θ, t) be the associated flow of the auxiliary system (13).…”

Backstepping Design for Incremental Stability

“…By using Theorem 2.5, we conclude that a control system of the form (3.1), equipped with the state feedback control law (3.2), is δ ∃ -GAS. The δ ∃ -GAS condition (2.2), as shown in [AR03], is given by:…”

“…Examples include intrinsic observer design [AR03], consensus problems in complex networks [WS05], output regulation of nonlinear systems [PvdWN05], design of frequency estimators [SK08b], synchronization of coupled identical dynamical systems [RdBS09], construction of symbolic models for nonlinear control systems [PGT08,PT09,GPT09], and the analysis of bio-molecular systems [RdB09]. Our motivation comes from symbolic control where incremental stability was identified as a key property enabling the construction of finite abstractions of nonlinear control systems [PGT08,PT09,GPT09,ZPJT10].…”

Contraction theory on Riemannian manifolds

“…The root piece of literature for this methodology is [1], with a plethora of extensions including graph-theoretic characterizations [2], backstepping design [3], extensions to distributed systems [4], and algorithmic searches for contraction metrics [5]. Rigorous proofs of the main result -a sufficient condition for a vector field to be "contracting" -are varied in style, with some revolving around the use of the matrix measure [6,7] while others utilize the perspective of a contraction metric [8,9,10]. As noted a recent historical review [11], the main ideas of the theory trace to the works [12,13] in the early 1950's, with some similar concepts presented slightly later [14,15].…”

“…Application areas of contraction theory and incremental stability have grown as the topics have proliferated, and now include symbolic models and control [26,9], output regulation [18], synchronization and consensus [27,21,28], bio-molecular systems [7], intrinsic observer design [8,29,30], mechanical system controller design [31], and power system dynamics [3]. Among these works, we also note the useful tutorial paper [32].…”