Feedforward tracking control for non‐uniform Timoshenko beam models: combining differential flatness, modal analysis, and FEM

Abstract

Key words feedforward control, tracking control, Timoshenko beam, differential flatness, modal analysis, finite element method, flexible structure MSC (2000) 35B37 This paper considers the feedforward tracking control problem for flexible structures with spatially varying geometrical and material parameters. The design approach is illustrated for a Timoshenko beam model and is based on the Riesz spectral properties of the system operator, which allow to determine the modal system representation. This serves as the starting point for the derivation of an infinite-dimensional inverse system description in terms of a so-called flat output parameterizing modal states and input. Convergence of the resulting parameterizations can be ensured by appropriate motion planning. The feedforward control design methodology is further complemented to compute appropriate control commands directly from finite element approximations of the considered structure. As a result, feedforward tracking control design for flexible structures with complex geometry, boundary conditions, or material parameters can be performed within a general framework. Simulation scenarios for feedforward boundary control of a clamped-free Timoshenko beam model with variable system parameters illustrate the usefulness of the approach.Here, E is the Young's modulus, G the shear modulus, I the area moment of inertia of the cross section, A the crosssectional area, ρ the mass density, and k the shear correction factor. If not stated otherwise, it is assumed that any material

Flatness-based feedforward control design for flexible structures

“…Convergence of the determined parameterizations can be ensured following the results of [5] for a boundary controlled Timoshenko beam. Thereby, based on results from entire functions and the spectral properties of the system operator A, i.e.…”

“…the growth of ω 2 k , it can be verified that given y(t) holomorphic in a certain domain, |[D n+r u (s)−D n u (s)]{y(t)}| evaluated using s j = d j /dt j , j ∈ N 0 represents a Cauchy sequence, where D n u (s) = n j=1 (1 + s 2 /ω 2 j ), n ∈ N. Hence by the Weierstrass convergence theorem, u(t) = D u (s){y(t)} where s j = d j /dt j , j ∈ N 0 converges to an analytic function. Furthermore, since D k x (s) = D u (s)/(1 + s 2 /ω 2 k ), convergence of the state parameterization (11) can be directly deduced -see [5] for details.…”

“…while d j y(t)/dt j = 0 for all j ∈ N. The parameters y * 0 and y * T of y * (t) can easily be determined from the initial and desired final steady state [5]. Note that the choice of (22) for the trajectory (21) allows a very efficient evaluation of the 2 m derivatives of the basic output needed for the parameterizations.…”

Trajektorienplanung für eine piezo-aktuierte elastische Kirchhoff-Platte

“…beschrieben werden. Fü r elastische Strukturen umfasst dies insbesondere die Analyse von Euler-Bernoulli und Timoshenko Balken (Fliess et al, 1997;Woittennek, Rudolph, 2003;Becker, Meurer, 2007;Meurer et al, 2008) sowie kreisfö rmigen Kirchhoff-Plattenmodellen (Woittennek, Rudolph, 2004). Fü r rechteckige Plattengeometrien und konstante Koeffizienten wurde in (Meurer, Kugi, 2006;Schrö ck et al, 2009) ein projektionsbasierter Ansatz zur Trajektorienplanung vorgestellt.…”

Model-based and model-free control of flexible-link robots: A comparison between representative methods

“…Flatness-based control is a powerful tool for the control of distributed parameter systems which does not follow modal analysis but the description of the flexible robot using the concept of differential flatness [9][10][11]. It has been shown that flexible-link robots and flexible beams are flat systems and thus flatness-based control can be efficiently used for trajectory tracking of flexible-link manipulators [12][13][14][15].…”