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Discrete-time Integral Resonant Control of Negative Imaginary Systems: Application to a High-speed Nanopositioner

Kanghong Shi, Erfan Khodabakhshi, Prosanto Biswas, Ian R. Petersen, S. O. Reza Moheimani

Abstract

We propose a discrete-time integral resonant control (IRC) approach for negative imaginary (NI) systems, which overcomes several limitations of continuous-time IRC. We show that a discrete-time IRC has a step-advanced negative imaginary property. A zero-order hold-sampled NI system can be asymptotically stabilized using a discrete-time IRC with suitable parameters. A hardware experiment is conducted where a high-speed flexure-guided nanopositioner is efficiently damped using the proposed discrete-time IRC with the discrete-time controller being implemented in FPGA hardware at the sampling rate of 1.25 MHz.

Discrete-time Integral Resonant Control of Negative Imaginary Systems: Application to a High-speed Nanopositioner

Abstract

We propose a discrete-time integral resonant control (IRC) approach for negative imaginary (NI) systems, which overcomes several limitations of continuous-time IRC. We show that a discrete-time IRC has a step-advanced negative imaginary property. A zero-order hold-sampled NI system can be asymptotically stabilized using a discrete-time IRC with suitable parameters. A hardware experiment is conducted where a high-speed flexure-guided nanopositioner is efficiently damped using the proposed discrete-time IRC with the discrete-time controller being implemented in FPGA hardware at the sampling rate of 1.25 MHz.

Paper Structure

This paper contains 10 sections, 4 theorems, 45 equations, 10 figures.

Table of Contents

  1. INTRODUCTION
  2. PRELIMINARIES
  3. Continuous-time integral resonant controller
  4. Discrete-time NI systems
  5. DISCRETE-TIME INTEGRAL RESONANT CONTROLLER
  6. HIGH-SPPED FLEXURE-GUIDED NANOPOSITIONER
  7. Experimental Setup
  8. FPGA Implementation
  9. Discrete-time IRC Design
  10. CONCLUSION

Key Result

Lemma 1

shi2023discrete Suppose the linear system (eq:G(z)) satisfies $\det(I-A)\neq 0$. Then the system (eq:G(z)) is NI with a quadratic positive definite storage function satisfying (eq:NNI ineq) if and only if there exists a real matrix $P=P^T>0$ such that

Figures (10)

  • Figure 1: Closed-loop interconnection of an integrator $C(s)=\frac{\Gamma}{s}$ and $G(s)+D$.
  • Figure 2: Closed-loop interconnection of an IRC and a plant. This is equivalent to the closed-loop system in Fig. \ref{['fig:CT_IRC']}.
  • Figure 3: Positive feedback interconnection of the plant $G(z)$ of the form (\ref{['eq:plant']}) and a discrete-time IRC $F(z)$ of the form (\ref{['eq:DT IRC model']}).
  • Figure 4: Nanopositioner and interferometer sensor
  • Figure 5: Frequency response of the nanopositioner.
  • ...and 5 more figures

Theorems & Definitions (8)

  • Definition 1: discrete-time negative imaginary systems
  • Lemma 1
  • Definition 2
  • Remark 1
  • Lemma 2
  • Lemma 3
  • Theorem 1
  • Remark 2