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Filtering in Projection-based Integrators for Improved Phase Characteristics

Hoang Chu, S. J. A. M van den Eijnden, M. F. Heertjes, W. P. M. H. Heemels

Abstract

Projection-based integrators are effectively employed in high-precision systems with growing industrial success. By utilizing a projection operator, the resulting projection-based integrator keeps its input-output pair within a designated sector set, leading to unique freedom in control design that can be directly translated into performance benefits. This paper aims to enhance projection-based integrators by incorporating well-crafted linear filters into its structure, resulting in a new class of projected integrators that includes the earlier ones, such as the hybrid-integrator gain systems (with and without pre-filtering) as special cases. The extra design freedom in the form of two filters in the input paths to the projection operator and the internal dynamics allows the controller to break away from the inherent limitations of the linear control design. The enhanced performance properties of the proposed structure are formally demonstrated through a (quasi-linear) describing function analysis, the absence of the gain-loss problem, and numerical case studies showcasing improved time-domain properties. The describing function analysis is supported by rigorously showing incremental properties of the new filtered projection-based integrators thereby guaranteeing that the computed steady-state responses are unique and asymptotically stable.

Filtering in Projection-based Integrators for Improved Phase Characteristics

Abstract

Projection-based integrators are effectively employed in high-precision systems with growing industrial success. By utilizing a projection operator, the resulting projection-based integrator keeps its input-output pair within a designated sector set, leading to unique freedom in control design that can be directly translated into performance benefits. This paper aims to enhance projection-based integrators by incorporating well-crafted linear filters into its structure, resulting in a new class of projected integrators that includes the earlier ones, such as the hybrid-integrator gain systems (with and without pre-filtering) as special cases. The extra design freedom in the form of two filters in the input paths to the projection operator and the internal dynamics allows the controller to break away from the inherent limitations of the linear control design. The enhanced performance properties of the proposed structure are formally demonstrated through a (quasi-linear) describing function analysis, the absence of the gain-loss problem, and numerical case studies showcasing improved time-domain properties. The describing function analysis is supported by rigorously showing incremental properties of the new filtered projection-based integrators thereby guaranteeing that the computed steady-state responses are unique and asymptotically stable.
Paper Structure (22 sections, 2 theorems, 39 equations, 10 figures)

This paper contains 22 sections, 2 theorems, 39 equations, 10 figures.

Table of Contents

  1. Introduction
  2. Projected dynamics of FHIGS
  3. System description
  4. Motivation of using the projection operator
  5. ePDS formalization of a control system with FHIGS
  6. PWL dynamics of open-loop FHIGS
  7. Describing function analysis
  8. Sinusoidal response with phase lead ($0\leqslant\phi(\omega)\leqslant \pi$)
  9. Sinusoidal response with phase lag ($-\pi \leqslant \phi <0$)
  10. The $k_1$ gain mode is activated
  11. The $k_1$-gain mode is not activated
  12. Describing function calculation
  13. Comparison with HIGS's DF
  14. Incremental attractivity of open-loop FHIGS
  15. Numerical examples
  16. ...and 7 more sections

Key Result

Theorem 1

Consider the projected dynamics eq: cloop projected with eq: open unprojected fhigs and assume that $C_p B_p = 0$ and that $e$ is a differentiable function. Then, the FHIGS dynamics can be written as with and with $v_2 = C_{v_2} x_{v_2}+ D_{v_2} e$, $\dot v_2 = C_{v_2} (A_{v_2} x_{v_2} + B_{v_2} e)+ D_{v_2} \dot e.$

Figures (10)

  • Figure 1: Projection-based integrator. The LTI filter $F_1$ adheres to the "gain" characteristics of the element, whereas $F_2$ determines the "phase" characteristics. The switching signal, affected by the choice for $F_2$, determines when projection is applied to the integrator state.
  • Figure 2: Filtered HIGS in closed-loop with other linear systems $F_3$ and $G$
  • Figure 3: Example of FHIGS trajectory in a sector set
  • Figure 4: HIGS and FHIGS steady-state sinusoidal responses with their first Fourier approximations. FHIGS (yellow) has approximately the same gain and more phase lead compared with HIGS (purple)
  • Figure 5: Simplified filtered HIGS structure with filter $F$. If $F=1$, FHIGS becomes the HIGS element.
  • ...and 5 more figures

Theorems & Definitions (7)

  • Theorem 1
  • Remark 1
  • Theorem 2
  • proof : Sketch of proof
  • Remark 2
  • Remark 3
  • proof