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Research Papers

A New Approach to Stable Optimal Control of Complex Nonlinear Dynamical Systems

[+] Author and Article Information
Firdaus E. Udwadia

Professor of Aerospace and
Mechanical Engineering
Civil Engineering, Systems Architecture
Engineering, Mathematics, and Information
and Operations Management,
University of Southern California,
430K Olin Hall,
Los Angeles, CA 90089-1453
e-mail: fudwadia@usc.edu

The author is indebted to an anonymous reviewer who brought this to his attention.

Manuscript received October 31, 2011; final manuscript received June 14, 2013; accepted manuscript posted June 21, 2013; published online September 18, 2013. Assoc. Editor: Alexander F. Vakakis.

J. Appl. Mech 81(3), 031001 (Sep 18, 2013) (6 pages) Paper No: JAM-11-1404; doi: 10.1115/1.4024874 History: Received October 31, 2011; Revised June 14, 2013; Accepted June 21, 2013

This paper gives a simple approach to designing a controller that minimizes a user-specified control cost for a mechanical system while ensuring that the control is stable. For a user-given Lyapunov function, the method ensures that its time rate of change is negative and equals a user specified negative definite function. Thus a closed-form, optimal, nonlinear controller is obtained that minimizes a desired control cost at each instant of time and is guaranteed to be Lyapunov stable. The complete nonlinear dynamical system is handled with no approximations/linearizations, and no a priori structure is imposed on the nature of the controller. The methodology is developed here for systems modeled by second-order, nonautonomous, nonlinear, differential equations. The approach relies on some recent fundamental results in analytical dynamics and uses ideas from the theory of constrained motion.

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Figures

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Fig. 1

Displacement and velocity response of the controlled nonlinear system

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Fig. 2

Projections of the phase portrait on the q1-q·1 and the q2-q·2 planes. The squares show the initial values.

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Fig. 3

(a) Time history of the control force QC. (b) Error e(t) in satisfaction of stability requirement showing the extent to which the relation in Eq. (26) is satisfied.

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Fig. 4

Displacement and velocity response of the controlled nonlinear system

Grahic Jump Location
Fig. 5

Projections of the phase portrait on the q1-q·1 and the q2-q·2 planes. The squares show the initial values.

Grahic Jump Location
Fig. 6

(a) Time history of the control force QC. (b) Error e(t) showing the extent to which relation in Eq. (23) is satisfied.

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