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A Spectrum-based Filter Design for Periodic Control of Systems with Time Delay

Can Kutlu Yüksel, Tomáš Vyhlídal, Jaroslav Bušek, Martin Hromčík, Silviu-Iulian Niculescu

TL;DR

This work develops an analytic Internal Model Control (IMC) framework to achieve periodic control for stable time-delayed plants by designing a spectrum-based filter that places zeros at targeted harmonics. It formulates a finite-dimensional sensitivity in the ideal configuration and solves a state-space, quasi-optimal filter design via an LQR-based pole placement, enabling implementation for high-harmonic demands. The approach yields a computable, robust controller realized in state-space that compensates delays through a controller delay and remains stable under model mismatch, as validated both numerically and experimentally on a sixth-order mass-spring-damper system with a $2\,\mathrm{Hz}$ periodic disturbance. Overall, the paper provides a concrete, analytically tractable method to implement spectrum-based periodic control for delayed systems and demonstrates practical feasibility in a vibration-rejection scenario.

Abstract

A fully analytical controller design is proposed to tackle a periodic control problem for stable linear systems with an input delay. Applying the internal model control scheme, the controller design reduces to designing a filter, which is done through the placement of poles and zeros. The zeros are placed to compensate for the harmonics and to achieve the desired degree of properness for the filter. For placing the poles, a quasi-optimal procedure is proposed utilizing the standard LQR method. Given the high-dimensionality of the filter due to targeting a large number of harmonics, the design, as well as controller implementation, is performed over a state-space representation. A thorough experimental case study is included to demonstrate both the practical feasibility and effectiveness of the proposed control design. The experimental validation is performed on a physical system, the goal of which is to reject periodic vibrations acting on a mass-spring-damper setup where the sensor and the actuator are non-collocated.

A Spectrum-based Filter Design for Periodic Control of Systems with Time Delay

TL;DR

This work develops an analytic Internal Model Control (IMC) framework to achieve periodic control for stable time-delayed plants by designing a spectrum-based filter that places zeros at targeted harmonics. It formulates a finite-dimensional sensitivity in the ideal configuration and solves a state-space, quasi-optimal filter design via an LQR-based pole placement, enabling implementation for high-harmonic demands. The approach yields a computable, robust controller realized in state-space that compensates delays through a controller delay and remains stable under model mismatch, as validated both numerically and experimentally on a sixth-order mass-spring-damper system with a periodic disturbance. Overall, the paper provides a concrete, analytically tractable method to implement spectrum-based periodic control for delayed systems and demonstrates practical feasibility in a vibration-rejection scenario.

Abstract

A fully analytical controller design is proposed to tackle a periodic control problem for stable linear systems with an input delay. Applying the internal model control scheme, the controller design reduces to designing a filter, which is done through the placement of poles and zeros. The zeros are placed to compensate for the harmonics and to achieve the desired degree of properness for the filter. For placing the poles, a quasi-optimal procedure is proposed utilizing the standard LQR method. Given the high-dimensionality of the filter due to targeting a large number of harmonics, the design, as well as controller implementation, is performed over a state-space representation. A thorough experimental case study is included to demonstrate both the practical feasibility and effectiveness of the proposed control design. The experimental validation is performed on a physical system, the goal of which is to reject periodic vibrations acting on a mass-spring-damper setup where the sensor and the actuator are non-collocated.

Paper Structure

This paper contains 15 sections, 5 theorems, 31 equations, 10 figures.

Table of Contents

  1. Introduction
  2. Controller design
  3. Internal Model Controller
  4. Filter design through reference sensitivity
  5. Filter structure
  6. Quasi-optimal filter design in state-space form
  7. State-space representation of IMC controller
  8. Robustness against model/plant mismatch
  9. Experimental case study validation
  10. Instrumentation and mechatronic design
  11. Parameter identification and output disturbance analysis
  12. Controller design
  13. Spectral and frequency domain analysis
  14. Experimental Results
  15. Conclusion

Key Result

Proposition 1

Under the ideal configuration, i.e., when the model and the system perfectly match, the controller structure constructed as where $F(s)$ is a low-pass filter with relative degree $n_r$, satisfying $n_r\ge\beta-\alpha$, and with properties satisfies periodic control condition eq:sensitivity_cond.

Figures (10)

  • Figure 1: The Internal Model Control scheme, with reference $r$, system input $u$ generater by the controller, measured system output $y$, and modelled system output $y_m$.
  • Figure 2: The filter structure in state feedback form for $n_r=1$.
  • Figure 3: Scheme of the laboratory setup
  • Figure 4: Mechatronic implementation of the laboratory setup
  • Figure 5: Response of mass $m_2$ to a unit step input from $u$ initiated at $t=0s$. Notice that the effect of the added input-delay $\tau=0.2s$ is visible in the beginning.
  • ...and 5 more figures

Theorems & Definitions (9)

  • Definition 2.1: Ideal Configuration
  • Remark 1
  • Proposition 1
  • Remark 2
  • Proposition 2
  • Remark 3
  • Proposition 3
  • Proposition 4
  • Proposition 5