Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

The Waterbed Effect on Quasiperiodic Disturbance Observer: Avoidance of Sensitivity Tradeoff with Time Delays

Hisayoshi Muramatsu

TL;DR

This work analyzes a quasiperiodic disturbance observer (QDOB) that uses time delays to suppress harmonics without amplifying aperiodic disturbances, addressing the waterbed sensitivity tradeoff. It introduces Bode-like sensitivity integrals for the QDOB in both continuous-time and discrete-time, showing that the integrals equal $-\frac{\pi}{4}\omega_b\omega_c L$ and $2\pi\ln(2+2\omega_b T) - 2\pi\ln(2+2\omega_b T+\omega_b\omega_c LT)$ respectively, and that increasing the separation frequency $\rho$ (and thus the harmonic suppression bandwidth) reduces these integrals. The analysis demonstrates that, under a phase condition placing the open-loop phase within $\pm90^{\circ}$, the QDOB avoids the traditional waterbed tradeoff, while $\omega_a$ does not affect the integrals and may cause amplification if not properly ordered relative to $\omega_b$. Numerical results corroborate convergence to the theoretical values and highlight design guidelines: keep $\omega_a\ll\omega_b$, exploit $\rho$ to widen bandwidth, and maintain appropriate phase behavior to ensure non-amplification. Overall, the work provides a rigorous basis for employing time-delay-based disturbance observers to achieve robust, wideband harmonic rejection without compromising stability or introducing misalignment of harmonic suppression frequencies.

Abstract

In linear time-invariant systems, the sensitivity function to disturbances is designed under a sensitivity tradeoff known as the waterbed effect. To compensate for a quasiperiodic disturbance, a quasiperiodic disturbance observer using time delays was proposed. Its sensitivity function avoids the sensitivity tradeoff, achieving wideband harmonic suppression without amplifying aperiodic disturbances or shifting harmonic suppression frequencies. However, its open-loop transfer function is not rational and does not satisfy the assumptions of existing Bode sensitivity integrals due to its time delays. This paper provides Bode-like sensitivity integrals for the quasiperiodic disturbance observer in both continuous-time and discrete-time representations and clarifies the avoided sensitivity tradeoff with time delays.

The Waterbed Effect on Quasiperiodic Disturbance Observer: Avoidance of Sensitivity Tradeoff with Time Delays

TL;DR

This work analyzes a quasiperiodic disturbance observer (QDOB) that uses time delays to suppress harmonics without amplifying aperiodic disturbances, addressing the waterbed sensitivity tradeoff. It introduces Bode-like sensitivity integrals for the QDOB in both continuous-time and discrete-time, showing that the integrals equal and respectively, and that increasing the separation frequency (and thus the harmonic suppression bandwidth) reduces these integrals. The analysis demonstrates that, under a phase condition placing the open-loop phase within , the QDOB avoids the traditional waterbed tradeoff, while does not affect the integrals and may cause amplification if not properly ordered relative to . Numerical results corroborate convergence to the theoretical values and highlight design guidelines: keep , exploit to widen bandwidth, and maintain appropriate phase behavior to ensure non-amplification. Overall, the work provides a rigorous basis for employing time-delay-based disturbance observers to achieve robust, wideband harmonic rejection without compromising stability or introducing misalignment of harmonic suppression frequencies.

Abstract

In linear time-invariant systems, the sensitivity function to disturbances is designed under a sensitivity tradeoff known as the waterbed effect. To compensate for a quasiperiodic disturbance, a quasiperiodic disturbance observer using time delays was proposed. Its sensitivity function avoids the sensitivity tradeoff, achieving wideband harmonic suppression without amplifying aperiodic disturbances or shifting harmonic suppression frequencies. However, its open-loop transfer function is not rational and does not satisfy the assumptions of existing Bode sensitivity integrals due to its time delays. This paper provides Bode-like sensitivity integrals for the quasiperiodic disturbance observer in both continuous-time and discrete-time representations and clarifies the avoided sensitivity tradeoff with time delays.

Paper Structure

This paper contains 10 sections, 58 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Waterbed Effect on QDOB
  4. Main Results
  5. Numerical Results
  6. Conclusion
  7. Proof of Lemma lem:Phi
  8. Proof of Lemma lem:OLTF
  9. Proof of Theorem thm:Bode:Integral:QDOB:CT
  10. Proof of Theorem thm:Bode:Integral:QDOB:DT

Figures (3)

  • Figure 1: Bode plots of the QDOB in the discrete-time representation with $l=3$, $N_\mathrm{max}=256$, $\omega_0=1$ rad/s, $\omega_\mathrm{a}=10$ rad/s, $\omega_\mathrm{b}=100$ rad/s, and $T=1$ ms.
  • Figure 2: Illustrative diagram of the gain of the sensitivity function for the QDOB.
  • Figure 3: Convergence of the Bode-like sensitivity integrals toward the theoretical values with $l=3$, $N_\mathrm{max}=256$, $\omega_\mathrm{a}=10$ rad/s, $T=1$ ms, and Table \ref{['tab:para']}.