This paper proposes a new design approach for control of periodically time-varying systems. The approach is based on the point-mapping technique to obtain an equivalent linear time-invariant sampled-data system for the linear periodically time-varying system with a piecewise parametrization of the control vector. This allows the known control design techniques for sampled-data systems to be applied. The proposed approach is then extended for analysis of robustness of the control design with respect to plant parametric uncertainties. This is achieved by computation of approximate discrete-time dynamics of the perturbed system by truncated point-mappings. By computing an upper norm bound on the error due to the truncated approximations, the robustness analysis of the system with respect to the parametric uncertainties is then formulated as a discrete-time structured singular value problem. Two numerical examples are considered to illustrate the approach and the extension of the approach for robust stability analysis.
A discrete-time control design approach for periodically time-varying systems is introduced. The method employs a period-to-period (point-mapping) formulation of the system’s dynamics and a parametrization of the control input to obtain an equivalent time-invariant discrete-time representation of the system. The representation is generalized to include sampling within the period and varying sampling rates in different feedback loops. The proposed formulation allows for the design of feedback control laws using established discrete-time control methodologies. In this paper, dead-beat and optimal control laws with state- or output-feedback control are presented. An example of a multivariable control design for double inverted pendulum with periodic forcing is used to illustrate the proposed approach.
An approach for the design of a momentum unloading system for a spacecraft controlled by momentum exchange actuators is proposed. The approach is based on formulating momentum unloading as a control problem of a parametrically perturbed periodic system. An equivalent discrete-time robust control design problem is formulated using a point-mapping-based numerical algorithm. Established robust control analysis can then be used to develop and analyse the robustness of the unloading control law. The method is demonstrated on an Earth-pointing satellite with magnetic torquers serving as unloading actuators. Earth's magnetic field is modelled as a tilted dipole that consists of a periodic term and bounded perturbations. A discrete-time representation of the system is used to develop an unloading control law and the structured singular value theory is employed to analyse its robustness to parametric uncertainties.
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