Nonlinear Control of Electro-Dynamic Tether System

The electro-dynamic tether (EDT) system uses the Lorentz force as a thruster generated by the interference between the Earth’s magnetic field and the electric current in the tether. This system is expected to provide a new and very efficient thruster system and is considered to have important applications in Earth observation, space observation, communications and satellite constellations. In the present paper, three nonlinear control schemes, which are a decoupling method, a model-following-decoupling method, and a combination of a partial feedback linearization control and a linear quadratic regulator, are investigated with respect to the stabilization of an electro-dynamic tether system that is assumed to consist of a large mother satellite and two sub-satellites connected to each other by conductive tethers. For the combination of a partial feedback linearization and a linear quadratic regulator, the linear quadratic regulator is employed after the state converges to the neighborhood of the equilibrium point. The results of numerical simulations show that the decoupling method can stabilize the librational motion of the electro-dynamic tether system in an elliptic orbit, and the model-following-decoupling method can change it to a periodic motion. The combination of the partial feedback linearization and the linear quadratic regulator can then stabilize the librational motion of the system in a circular orbit. Nomenclature m0, m1, and m2 = masses of the mother satellite, subsatellite 1, and subsatellite 2 ([kg]) l1, l2 = lengths of tether 1 and tether 2 ([km]) 1 θ , 2 θ = angles of tether 1 and tether 2 ([rad]) η = true anomaly ([rad]) c R = orbital radius of the center of mass of the system ([km]) τ = orbital period ([s]) m μ = magnetic moment of the Earth’s dipole ( ) 6 3 8.1 10 [ ] T km × ⋅ B = Earth’s magnetic field vector ([T]) 0 F , , , = Lorentz forces centralized from tether 1 to the mother satellite, from tether 1 to subsatellite 1, from tether 2 to subsatellite 1, and from tether 2 to subsatellite 2 ([N]) 11 F 21 F 2 F u = control input vector (electric current in tether 1 and tether 2 [A]) s = dimension of the output vector ( 1, ) i r i s = = relative degree of the system for a decoupling method ( 1, ) i P i s = = relative degree of a plant model for a model-following-decoupling method ( 1, ) i i s π = = relative degree of the reference model for a model-following-decoupling method 8 x∈ = state vector for a decoupling method and a partial feedback control 8 P x ∈ = state vector of a plant model for a model-following-decoupling method 4 M x ∈ = state vector of a reference model for a model-following-decoupling method 2 y∈ = output vector for a decoupling method * Associate Professor, Department of Aerospace Engineering, hkojima@cc.tmit.ac.jp, Member AIAA. † Graduate student, Department of Aerospace Engineering. American Institute of Aeronautics and Astronautics 1 AIAA/AAS Astrodynamics Specialist Conference and Exhibit 21 24 August 2006, Keystone, Colorado AIAA 2006-6766 Copyright © 2006 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved. 2 p y ∈ = output vector of a plant model for a model-following-decoupling method 2 M y ∈ = output vector of a reference model for a model-following-decoupling method M P e y y = − = difference between the output of the reference model and that of the plant model 1 v , = fictitious control inputs for a decoupling method and a model-following-decoupling method 2 v * 1 v , = fictitious control input for a partial feedback linearization * 2 v 1 1 2 , , , P D P D K K K K 2 = feedback gains for a decoupling method and a model-following-decoupling method 1 p k , = feedback gains for a partial feedback linearization 1 d k 4 Q∈ , = weighting matrix for a linear quadratic regulator 1 R∈ [ ] 1 11 12 13 14 K K K K K = , [ ] 2 21 22 23 24 K K K K K = = feedback gain vectors for a linear quadratic regulator