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.
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e-mail: julieparish@tamu.edu
e-mail: jehurtado@tamu.edu
e-mail: sinclair@auburn.edu
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July 2009
Research Papers
Direct Linearization of Continuous and Hybrid Dynamical Systems
Julie J. Parish,
Julie J. Parish
Department of Aerospace Engineering,
e-mail: julieparish@tamu.edu
Texas A&M University
, 3141 TAMU, College Station, TX 77843-3141
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John E. Hurtado,
John E. Hurtado
Department of Aerospace Engineering,
e-mail: jehurtado@tamu.edu
Texas A&M University
, 3141 TAMU, College Station, TX 77843-3141
Search for other works by this author on:
Andrew J. Sinclair
Andrew J. Sinclair
Department of Aerospace Engineering,
e-mail: sinclair@auburn.edu
Auburn University
, 211 Davis Hall, Auburn, AL 36849-5338
Search for other works by this author on:
Julie J. Parish
Department of Aerospace Engineering,
Texas A&M University
, 3141 TAMU, College Station, TX 77843-3141e-mail: julieparish@tamu.edu
John E. Hurtado
Department of Aerospace Engineering,
Texas A&M University
, 3141 TAMU, College Station, TX 77843-3141e-mail: jehurtado@tamu.edu
Andrew J. Sinclair
Department of Aerospace Engineering,
Auburn University
, 211 Davis Hall, Auburn, AL 36849-5338e-mail: sinclair@auburn.edu
J. Comput. Nonlinear Dynam. Jul 2009, 4(3): 031002 (11 pages)
Published Online: May 19, 2009
Article history
Received:
January 10, 2008
Revised:
September 26, 2008
Published:
May 19, 2009
Citation
Parish, J. J., Hurtado, J. E., and Sinclair, A. J. (May 19, 2009). "Direct Linearization of Continuous and Hybrid Dynamical Systems." ASME. J. Comput. Nonlinear Dynam. July 2009; 4(3): 031002. https://doi.org/10.1115/1.3124092
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