Der-Cherng Liaw

Stabilization of tethered satellites during station keeping

After deriving a set of dynamic equations governing the dynamics of a tethered satellite system (TSS), stabilizing tension-control laws are derived in feedback form. The tether is assumed to be rigid and massless, and the equations of motion are derived using the system Lagrangian. It is observed that, to stabilize the system, tools from stability analysis of critical nonlinear systems must be applied. Tools related to the Hopf bifurcation theorem are used in the construction of the stabilizing control laws, which may be taken purely linear. Simulations illustrate the nature of the conclusions and show that nonlinear terms in the feedback can be used to significantly improve the transient response. >

Feedback stabilization of nonlinear systems via center manifold reduction

The center manifold theorem is applied to the local feedback stabilization of nonlinear systems in critical cases. The authors address two particular critical cases, for which the system linearization at the equilibrium point of interest is assumed to possess either a simple zero eigenvalue or a complex conjugate pair of simple, pure imaginary eigenvalues. In either case, the noncritical eigenvalues are taken to be stable. The results on stabilizability and stabilization are given explicitly in terms of the nonlinear model of interest in its original form, i.e. before reduction to the center manifold. Moreover, the formulation given uncovers connections between results obtained using the center manifold reduction and those of an alternative approach.<<ETX>>

Bifurcation Control of Nonlinear Systems

Bifurcation control is discussed in the context of the stabilization of high angle-of-attack flight dynamics. Two classes of stabilization problems for which bifurcation control is useful are discussed. In the first class, which is emphasized in this presentation, a nonlinear control system operates at an equilibrium point which persists only under very small perturbations of a parameter. Such a system will tend to exhibit a jump, or divergence, instability in the absence of appropriate control action. In the second class of systems, an instance of which arises in a tethered satellite system model [14], eigenvalues of the system linearization appear on (or near) the imaginary axis in the complex plane, regardless of the values of system parameters or admissible linear feedback gains.

Stabilization of linear parameter-dependent systems using eigenvalue expansion with application to two time-scale systems

An eigenvalue expansion method is applied to the stability analysis and stabilization of linear parameter-dependent systems in the neighborhood of a specified system parameter. The system, at the parameter of interest, is assumed to possess distinct eigenvalues on the imaginary axis. The asymptotic stability conditions, which rely on Taylor series expansions of the continuous extensions of the critical eigenvalues with respect to system parameters, are algebraic and are given in terms of the system dynamics and the critical eigenvectors. Asymptotically stabilizing control laws are designed for the case in which the system at the specified parameter has uncontrollable eigenvalues on the imaginary axis. Application of these results yields computational algorithms for the stability analysis and stabilization design of two-time scale linear systems.<<ETX>>

Nonlinear stabilization of tethered satellites

A set of dynamic equations governing the dynamics of a tethered satellite system and stabilizing tension control laws in feedback form are derived. The tether is assumed rigid and massless, and the equations of motion are derived using the system Lagrangian. It is observed that to stabilize the system, tools from stability analysis of critical nonlinear systems must be applied. Tools related to the Hopf bifurcation theorem are used in the construction of the stabilizing control laws, which may be taken as purely linear. Simulations illustrate the nature of the conclusions, and show that nonlinear terms in the feedback can be used to improve significantly the transient response.<<ETX>>