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Research Papers

Observer Design for Nonlinear Systems With Time-Periodic Coefficients via Normal Form Theory

[+] Author and Article Information
Yandong Zhang

Department of Mechanical Engineering, Nonlinear Systems Research Laboratory, Auburn University, Auburn, AL 36849zhangya@auburn.edu

S. C. Sinha

Department of Mechanical Engineering, Nonlinear Systems Research Laboratory, Auburn University, Auburn, AL 36849ssinha@eng.auburn.edu

J. Comput. Nonlinear Dynam 4(3), 031001 (May 19, 2009) (10 pages) doi:10.1115/1.3124093 History: Received January 28, 2008; Revised December 23, 2008; Published May 19, 2009

For most complex dynamic systems, it is not always possible to measure all system states by a direct measurement technique. Thus for dynamic characterization and controller design purposes, it is often necessary to design an observer in order to get an estimate of those states, which cannot be measured directly. In this work, the problem of designing state observers for free systems (linear as well as nonlinear) with time-periodic coefficients is addressed. It is shown that, for linear periodic systems, the observer design problem is the duality of the controller design problem. The state observer is constructed using a symbolic controller design method developed earlier using a Chebyshev expansion technique where the Floquet multipliers can be placed in the desired locations within the unit circle. For nonlinear time-periodic systems, an observer design methodology is developed using the Lyapunov–Floquet transformation and the Poincaré normal form technique. First, a set of time-periodic near identity coordinate transformations are applied to convert the nonlinear problem to a linear observer design problem. The conditions for existence of such invertible maps and their computations are discussed. Then the local identity observers are designed and implemented using a symbolic computational algorithm. Several illustrative examples are included to show the effectiveness of the proposed methods.

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Copyright © 2009 by American Society of Mechanical Engineers
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Figures

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Figure 1

Dynamics of the system and its observer

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Figure 2

The error dynamics

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Figure 3

Invariant manifold in two coordinate systems

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Figure 5

Dynamics of the system and its observer

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Figure 6

Coupled pendulums under periodic axial loads

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Figure 7

States of the original system

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Figure 8

States of the error dynamics

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