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Constructive RNNs: An Error-Recurrence Perspective on Time-Variant Zero Finding Problem Solving Under Uncertainty

Mingxuan Sun, Xing Li, Han Wang

TL;DR

An error recurrence system approach is presented by equipping with uncertainty compensation in the pre-specified error dynamics, capable of enhancing robustness properly and effectiveness of the proposed computing schemes for the time-variant QP problem solving.

Abstract

When facing time-variant problems in analog computing, the desirable RNN design requires finite-time convergence and robustness with respect to various types of uncertainties, due to the time-variant nature and difficulties in implementation. It is very worthwhile to explore terminal zeroing neural networks, through examining and applying available attracting laws. In this paper, from a control-theoretic point of view, an error recurrence system approach is presented by equipping with uncertainty compensation in the pre-specified error dynamics, capable of enhancing robustness properly. Novel rectifying actions are designed to make finite-time settling so that the convergence speed and the computing accuracy of time-variant computing can be improved. Double-power and power-exponential rectifying actions are respectively formed to construct specific models, while the particular expressions of settling time function for the former are presented, and for the latter the proximate settling-time estimations are given, with which the fixed-time convergence of the corresponding models is in turn established. Moreover, the uncertainty compensation by the signum/smoothing-signum techniques are adopted for finite-duration stabilization. Theoretical results are presented to demonstrate effectiveness (involving fixed-time convergence and robustness) of the proposed computing schemes for the time-variant QP problem solving.

Constructive RNNs: An Error-Recurrence Perspective on Time-Variant Zero Finding Problem Solving Under Uncertainty

TL;DR

An error recurrence system approach is presented by equipping with uncertainty compensation in the pre-specified error dynamics, capable of enhancing robustness properly and effectiveness of the proposed computing schemes for the time-variant QP problem solving.

Abstract

When facing time-variant problems in analog computing, the desirable RNN design requires finite-time convergence and robustness with respect to various types of uncertainties, due to the time-variant nature and difficulties in implementation. It is very worthwhile to explore terminal zeroing neural networks, through examining and applying available attracting laws. In this paper, from a control-theoretic point of view, an error recurrence system approach is presented by equipping with uncertainty compensation in the pre-specified error dynamics, capable of enhancing robustness properly. Novel rectifying actions are designed to make finite-time settling so that the convergence speed and the computing accuracy of time-variant computing can be improved. Double-power and power-exponential rectifying actions are respectively formed to construct specific models, while the particular expressions of settling time function for the former are presented, and for the latter the proximate settling-time estimations are given, with which the fixed-time convergence of the corresponding models is in turn established. Moreover, the uncertainty compensation by the signum/smoothing-signum techniques are adopted for finite-duration stabilization. Theoretical results are presented to demonstrate effectiveness (involving fixed-time convergence and robustness) of the proposed computing schemes for the time-variant QP problem solving.

Paper Structure

This paper contains 10 sections, 9 theorems, 75 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. TZNNs as systems with error recurrence
  3. ERS design using power laws
  4. Exemplary finite-time stable systems as attracting laws
  5. Single power-rate laws
  6. Double power-rate laws
  7. Double power-rate laws of adding a linear term
  8. ERS design using power-exponential laws
  9. ERS redesign: Disturbance rejection and attenuation techniques
  10. Conclusion

Key Result

Lemma 1

For the positive definite function $V$ satisfying that with the given $\alpha \ge 0$ and $\Delta >0$, and $K > 0$ and $R>0$ being the adjustable gains, then $V$ converges to and remains within the region of the origin, bounded by $\left (\frac{\Delta}{R} \right )^{1/\alpha}$, for which the needed time is at least, i) for $\alpha < 1$, $t_\Delta = \frac{

Figures (1)

  • Figure 1: Error recurrence system

Theorems & Definitions (9)

  • Lemma 1
  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Corollary 1
  • Theorem 5
  • Theorem 6
  • Theorem 7