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Sampled-Data Control using Hermite-Obreschkoff Methods with an IDA-PBC Example

Le Zhang, Paul Kotyczka

TL;DR

This work addresses the gap between continuous-time nonlinear control designs and discrete-time implementations at relatively slow sampling. It unifies Lobatto IIIA collocation and Hermite interpolation through Hermite-Obreschkoff (HO) methods, enabling higher-order one-step predictions and alternative input-shaping strategies for sampled-data control. An IDA-PBC controller for a magnetic levitation system with a non-constant interconnection is derived and experimentally validated, demonstrating improved stability and reduced model mismatch compared to conventional explicit Euler. The results highlight a practical pathway to high-order discrete-time control with HO-based prediction and interpolation, while revealing trade-offs in noise sensitivity and actuator limitations when adopting high-order input shapes or piecewise-constant inputs. The work opens avenues toward integrating constraints and control-efficiency enhancements, including connections to model predictive control and generalized collocation schemes.

Abstract

The motivation for this paper is the implementation of nonlinear state feedback control, designed based on the continuous-time plant model, in a sampled control loop under relatively slow sampling. In previous work we have shown that using one-step predictions of the target dynamics with higher order integration schemes, together with possibly higher order input shaping, is a simple and effective way to increase the feasible sampling times until performance degradation and instability occur. In this contribution we present a unifying derivation for arbitrary orders of the previously used Lobatto IIIA collocation and Hermite interpolation schemes through the Hermite-Obreschkoff formula. We derive, moreover, an IDA-PBC controller for a magnetic levitation system, which requires a non-constant target interconnection matrix, and show experimental results.

Sampled-Data Control using Hermite-Obreschkoff Methods with an IDA-PBC Example

TL;DR

This work addresses the gap between continuous-time nonlinear control designs and discrete-time implementations at relatively slow sampling. It unifies Lobatto IIIA collocation and Hermite interpolation through Hermite-Obreschkoff (HO) methods, enabling higher-order one-step predictions and alternative input-shaping strategies for sampled-data control. An IDA-PBC controller for a magnetic levitation system with a non-constant interconnection is derived and experimentally validated, demonstrating improved stability and reduced model mismatch compared to conventional explicit Euler. The results highlight a practical pathway to high-order discrete-time control with HO-based prediction and interpolation, while revealing trade-offs in noise sensitivity and actuator limitations when adopting high-order input shapes or piecewise-constant inputs. The work opens avenues toward integrating constraints and control-efficiency enhancements, including connections to model predictive control and generalized collocation schemes.

Abstract

The motivation for this paper is the implementation of nonlinear state feedback control, designed based on the continuous-time plant model, in a sampled control loop under relatively slow sampling. In previous work we have shown that using one-step predictions of the target dynamics with higher order integration schemes, together with possibly higher order input shaping, is a simple and effective way to increase the feasible sampling times until performance degradation and instability occur. In this contribution we present a unifying derivation for arbitrary orders of the previously used Lobatto IIIA collocation and Hermite interpolation schemes through the Hermite-Obreschkoff formula. We derive, moreover, an IDA-PBC controller for a magnetic levitation system, which requires a non-constant target interconnection matrix, and show experimental results.
Paper Structure (14 sections, 42 equations, 4 figures, 2 tables)

This paper contains 14 sections, 42 equations, 4 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Solving Initial Value Problems using Hermite-Obreschkoff Methods
  3. Hermite-Obreschkoff Formula
  4. Lobatto IIIA Collocation
  5. Example.
  6. Hermite Interpolation
  7. Example.
  8. Higher-Order Discrete-Time Control Implementation
  9. Input Shaping Based on Lagrange Interpolation
  10. Conversion to Piecewise Constant Input
  11. Sampled IDA-PBC for the Magnetic Levitation System
  12. IDA-PBC Design
  13. Experimental Results
  14. Conclusions and Outlook

Figures (4)

  • Figure 1: Sketch and photo of the MagLev test bench.
  • Figure 2: Schematic block diagram of the experiment. Continuous-time components are drawn in black, quasi-continuous components in green, and discrete-time components in red. $y(t)=\left[ s(t), i(t) \right]^{T}$
  • Figure 3: Experimental results for the implementation of 3-stage Lobatto IIIA collocation at $h=16\,\textrm{ms}$.
  • Figure 4: Experimental results for the implementation of 5-stage Lobatto IIIA collocation at $h=22\,\textrm{ms}$.