Discrete-Time Control of Linear Time-Periodic Systems

Abstract

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.

Beyond equidistant sampling for performance and cost: A loop‐shaping approach applied to a motion system

“…Most control designs for LPTV systems require a parametric model of the system, including pole placement, linear quadratic regulator (LQR) control, linear quadratic Gaussian (LQG) control, / approaches, internal model principle, and LTI approximations . In addition, designs based on time‐invariant reformulations are often based on parametric models, including Floquet‐Lyapunov transformations,(section 1.2) and lifting approaches,(section 1.6) which enable the use of full state feedback, pole placement, model matching, LQR, LQG, and / control . However, as is argued in the work of Oomen et al, despite the availability of solid control theory, such model‐based designs are demanding because (i) obtaining a parametric LPTV model is difficult and (ii) typical LTI interpretations are not valid, leading to complications for the actual design …”

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“…In this article, a new approach for control design of perturbed LPTV systems [14,15] to develop momentum unloading control laws is presented. In this approach, the perturbed LPTV control design problem is formulated as a discrete-time robust control design problem.…”