TY - JOUR

T1 - Template-based stabilization of relative equilibria in systems with continuous symmetry

AU - Ahuja, S.

AU - Kevrekidis, Yannis

AU - Rowley, Clarence Worth

N1 - Funding Information:
The authors thank Jerry Marsden and Kurt Lust for inspiration and helpful comments on a draft of this manuscript, and the referees for their constructive suggestions. A preliminary version of this work appeared in the Proceedings of the 2006 American Control Conference. This work was supported by the AFOSR, NSF career grant CMS-0347239 of C. W. R. and an NSF/ITR grant of I. G. K.

PY - 2007/4

Y1 - 2007/4

N2 - We present an approach to the design of feedback control laws that stabilize relative equilibria of general nonlinear systems with continuous symmetry. Using a template-based method, we factor out the dynamics associated with the symmetry variables and obtain evolution equations in a reduced frame that evolves in the symmetry direction. The relative equilibria of the original systems are fixed points of these reduced equations. Our controller design methodology is based on the linearization of the reduced equations about such fixed points. We present two different approaches of control design. The first approach assumes that the closed loop system is affine in the control and that the actuation is equivariant. We derive feedback laws for the reduced system that minimize a quadratic cost function. The second approach is more general; here the actuation need not be equivariant, but the actuators can be translated in the symmetry direction. The controller resulting from this approach leaves the dynamics associated with the symmetry variable unchanged. Both approaches are simple to implement, as they use standard tools available from linear control theory. We illustrate the approaches on three examples: a rotationally invariant planar ODE, an inverted pendulum on a cart, and the Kuramoto-Sivashinsky equation with periodic boundary conditions.

AB - We present an approach to the design of feedback control laws that stabilize relative equilibria of general nonlinear systems with continuous symmetry. Using a template-based method, we factor out the dynamics associated with the symmetry variables and obtain evolution equations in a reduced frame that evolves in the symmetry direction. The relative equilibria of the original systems are fixed points of these reduced equations. Our controller design methodology is based on the linearization of the reduced equations about such fixed points. We present two different approaches of control design. The first approach assumes that the closed loop system is affine in the control and that the actuation is equivariant. We derive feedback laws for the reduced system that minimize a quadratic cost function. The second approach is more general; here the actuation need not be equivariant, but the actuators can be translated in the symmetry direction. The controller resulting from this approach leaves the dynamics associated with the symmetry variable unchanged. Both approaches are simple to implement, as they use standard tools available from linear control theory. We illustrate the approaches on three examples: a rotationally invariant planar ODE, an inverted pendulum on a cart, and the Kuramoto-Sivashinsky equation with periodic boundary conditions.

KW - Kuramoto-Sivashinsky equation

KW - differential-algebraic equations

KW - feedback control

KW - geometric methods

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U2 - 10.1007/s00332-005-0801-7

DO - 10.1007/s00332-005-0801-7

M3 - Article

AN - SCOPUS:84867940580

SN - 0938-8974

VL - 17

SP - 109

EP - 143

JO - Journal of Nonlinear Science

JF - Journal of Nonlinear Science

IS - 2

ER -