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Research Papers

# Robust Piecewise-Linear State Observers for Flexible Link Mechanisms

[+] Author and Article Information
Roberto Caracciolo, Dario Richiedei

Dipartimento di Tecnica e Gestione dei Sistemi Industriali, Università di Padova, Stradella S. Nicola 3-36100 Vicenza, Italy

Alberto Trevisani1

Dipartimento di Tecnica e Gestione dei Sistemi Industriali, Università di Padova, Stradella S. Nicola 3-36100 Vicenza, Italyalberto.trevisani@unipd.it

1

Corresponding author.

J. Dyn. Sys., Meas., Control 130(3), 031011 (May 12, 2008) (8 pages) doi:10.1115/1.2909600 History: Received October 16, 2006; Revised May 29, 2007; Published May 12, 2008

## Abstract

This paper tackles the problem of designing state observers for flexible link mechanisms: An investigation is made on the possibility of employing observers making use of suitable piecewise-linear truncated dynamics models. A general and novel approach is proposed, which provides an objective way of synthesizing observers preventing the instability that may arise from using reduced-order linearized models. The approach leads to the identification of the regions of the domain of the state variables where the linear approximations of the nonlinear model can be considered acceptable. To this purpose, first of all, the stability of the equilibrium points of the closed-loop system is assessed by applying the eigenvalue analysis to appropriate piecewise-linear models. Admittedly, the dynamics of such a closed-loop system is affected by the perturbation of the poles caused by spillover and by the discrepancies between the linearized models of the plant and the one of the observer. Additionally, when nodal elastic displacements and velocities are not bounded in the infinitesimal neighborhoods of the equilibrium points, the difference between the nonlinear model and the locally linearized one is expressed in terms of unstructured uncertainty and stability is assessed through $H∞$ robust analysis. The method is demonstrated by applying it to a closed-chain flexible link mechanism.

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## Figures

Figure 1

Representation of the open-loop system with uncertainty

Figure 2

Representation of the closed-loop system with uncertainty

Figure 3

Finite element representation of the four-bar linkage

Figure 4

Time history of the output estimation error for the crank angle, obtained using the linear observer

Figure 5

Magnitude of the frequency responses of the output estimation errors

Figure 6

Time histories of the output variables obtained using the piecewise-linear observer. The actual values are plotted as solid lines; the estimated values as dashed lines.

Figure 7

Time histories of the output estimation error for the crank angle, obtained using the piecewise-linear observer

Figure 8

Actual (solid line) and estimated (dotted lines) values of the crank angle about the observer switching instant (vertical line)

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