Research Papers

Direct Linearization of Continuous and Hybrid Dynamical Systems

[+] Author and Article Information
Julie J. Parish

Department of Aerospace Engineering, Texas A&M University, 3141 TAMU, College Station, TX 77843-3141julieparish@tamu.edu

John E. Hurtado

Department of Aerospace Engineering, Texas A&M University, 3141 TAMU, College Station, TX 77843-3141jehurtado@tamu.edu

Andrew J. Sinclair

Department of Aerospace Engineering, Auburn University, 211 Davis Hall, Auburn, AL 36849-5338sinclair@auburn.edu

J. Comput. Nonlinear Dynam 4(3), 031002 (May 19, 2009) (11 pages) doi:10.1115/1.3124092 History: Received January 10, 2008; Revised September 26, 2008; Published May 19, 2009

Nonlinear equations of motion are often linearized, especially for stability analysis and control design applications. Traditionally, the full nonlinear equations are formed and then linearized about the desired equilibrium configuration using methods such as Taylor series expansions. However, it has been shown that the quadratic form of the Lagrangian function can be used to directly linearize the equations of motion for discrete dynamical systems. This procedure is extended to directly generate linearized equations of motion for both continuous and hybrid dynamical systems. The results presented require only velocity-level kinematics to form the Lagrangian and find equilibrium configuration(s) for the system. A set of selected partial derivatives of the Lagrangian are then computed and used to directly construct the linearized equations of motion about the equilibrium configuration of interest, without first generating the entire nonlinear equations of motion. Given an equilibrium configuration of interest, the directly constructed linearized equations of motion allow one to bypass first forming the full nonlinear governing equations for the system. Examples are presented to illustrate the method for both continuous and hybrid systems.

Copyright © 2009 by American Society of Mechanical Engineers
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Grahic Jump Location
Figure 1

Rotating one-link structure

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Figure 2

Rotating three-beam T-structure

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Figure 3

Rotating hybrid structure



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