Observer Design for Unilaterally Constrained Lagrangian Systems: A Passivity-Based Approach

Abstract

Abstract-This paper addresses the problem of state estimation in nonlinear Lagrangian dynamical systems with frictionless unilateral constraints using the position measurement as output. The discontinuous velocity variable in such systems is modeled as a function of bounded variation (so that Zeno phenomenon is not ruled out). Since the derivative of such functions is represented with the Lebesgue-Stieltjes measure, the framework of measure differential inclusions (MDIs) is used to describe the dynamics. A class of estimators is proposed, which also uses the framework of MDIs, and is shown to generate asymptotically converging state estimates. The existence and uniqueness of solutions for the proposed estimators is rigorously proven. The global stability of error dynamics is analyzed using the generalized Lyapunov methods for functions of bounded variation. As particular cases of our estimators, we provide an explicit construction of a full-order observer, and a reduced-order observer.Index Terms-Complementarity systems, functions of bounded variation, convex analysis, systems with impacts, Lagrangian systems, Lyapunov methods, Moreau's sweeping process, observers, state estimation.

Well-Posedness and Output Regulation for Implicit Time-Varying Evolution Variational Inequalities

“…We can write e + − e − = G η, so that e − P e − = e + P e + − 2e + P G η + η G P G η ≥ e + P e + where the middle term is nonnegative due to the same arguments as used in the proof of Theorem 2. The formal arguments to deduce asymptotic stability for systems with BV solutions can be found in [32,33,34].…”

Motion Signals With Velocity Jumps: Velocity Estimation Employing Only Quantized Position Data

“…with |σ k+1 | ≤ r/2 the measurement error at t = t k+1 . Using (10), (11), and triangular inequality, it follows that the prediction error satisfies the bound…”

“…Techniques exist to retain part of the spectrum at the higher frequency range, as for example the mixed-bandwidth filter presented in [9]. Other approaches make use of additional information other than position data (e.g., the acceleration provided by an accelerometer) or an observer that encapsulates a hybrid model of the impact process (including the knowledge of the coefficient of restitution) done, e.g., in [10], [11]. All these approaches however require additional sensors or a hybrid model of the system (and possibly its force/acceleration input) and its interaction with the environment.…”

The Minimum Time Function for the Controlled Moreau's Sweeping Process

“…x(0) = x 0 ∈ C(0) are well known for a Lipschitz continuous f , even in the case in which C is mildly non-convex (actually, uniformly prox-regular, see [43,Theorem 4.4]). There has been an increasing interest in a number of variants of the sweeping process as a model, e.g., for some electric circuits [1], for crowd motion [36], for hysteresis [29], and as a tool for identification of parameters [8] for some mechanical systems subject to unilateral constraints. On the other hand, a natural modification of such given models concerns the case in which the behavior of the whole system is modified in order to satisfy some expected performance requirements.…”