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Stabilization by Controllers Having Integer Coefficients

Joowon Lee, Donggil Lee, Junsoo Kim

TL;DR

There always exists a controller with integer coefficients stabilizing a given discrete-time linear time-invariant plant, and a constructive algorithm to obtain such a controller is provided.

Abstract

The system property of ``having integer coefficients,'' that is, a transfer function has an integer monic polynomial as its denominator, is significant in the field of encrypted control as it is required for a dynamic controller to be realized over encrypted data. This paper shows that there always exists a controller with integer coefficients stabilizing a given discrete-time linear time-invariant plant. A constructive algorithm to obtain such a controller is provided, along with numerical examples. Furthermore, the proposed method is applied to converting a pre-designed controller to have integer coefficients, while the original performance is preserved in the sense that the transfer function of the closed-loop system remains unchanged.

Stabilization by Controllers Having Integer Coefficients

TL;DR

There always exists a controller with integer coefficients stabilizing a given discrete-time linear time-invariant plant, and a constructive algorithm to obtain such a controller is provided.

Abstract

The system property of ``having integer coefficients,'' that is, a transfer function has an integer monic polynomial as its denominator, is significant in the field of encrypted control as it is required for a dynamic controller to be realized over encrypted data. This paper shows that there always exists a controller with integer coefficients stabilizing a given discrete-time linear time-invariant plant. A constructive algorithm to obtain such a controller is provided, along with numerical examples. Furthermore, the proposed method is applied to converting a pre-designed controller to have integer coefficients, while the original performance is preserved in the sense that the transfer function of the closed-loop system remains unchanged.
Paper Structure (12 sections, 6 theorems, 56 equations, 3 figures, 2 algorithms)

This paper contains 12 sections, 6 theorems, 56 equations, 3 figures, 2 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries and Problem Formulation
  3. Preliminaries
  4. Problem formulation
  5. Main Result
  6. Proposed framework
  7. Solution to Problem \ref{['prob:stabilization']}
  8. Overall procedure and main theorem
  9. Numerical example
  10. Application to Conversion Problem
  11. Numerical example
  12. Conclusion

Key Result

Proposition 1

Let $a(z)$ and $r(z)$ be monic polynomials of degree $n$, and $N\ge 0$. Then, a polynomial $a^+(z)$ satisfying eq:w with some polynomial $w(z)$ of degree less than $N+n$ is uniquely determined as $\mathrm{a}^+=\mathrm{a}+\Delta(\mathrm{a})\mathrm{r}$.

Figures (3)

  • Figure 1: Illustration of the state $x_k$, the destination $x^\star$, and the hyperplanes \ref{['eq:hyperplane']} when $n=3$, $n_r=2$, and $n_c=0$.
  • Figure 2: Conversion of a pre-designed controller $C$ to a new controller $C^\prime$ having integer coefficients.
  • Figure 3: Performance of the pre-designed controller $C$ of \ref{['eq:simul pre']} and the converted controller $C^\prime$.

Theorems & Definitions (17)

  • Definition 1
  • Remark 1
  • Proposition 1
  • proof
  • Lemma 1
  • proof
  • Proposition 2
  • proof
  • Proposition 3
  • proof
  • ...and 7 more