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Adaptive Tracking Control with Binary-Valued Output Observations

Lantian Zhang, Lei Guo

Abstract

This paper considers real-time control and learning problems for finite-dimensional linear systems under binary-valued and randomly disturbed output observations. This has long been regarded as an open problem because the exact values of the traditional regression vectors used in the construction of adaptive algorithms are unavailable, as one only has binary-valued output information. To overcome this difficulty, we consider the adaptive estimation problem of the corresponding infinite-impulse-response (IIR) dynamical systems, and apply the double array martingale theory that has not been previously used in adaptive control. This enables us to establish global convergence results for both the adaptive prediction regret and the parameter estimation error, without resorting to such stringent data conditions as persistent excitation and bounded system signals that have been used in almost all existing related literature. Based on this, an adaptive control law will be designed that can effectively combine adaptive learning and feedback control. Finally, we are able to show that the closed-loop adaptive control system is optimal in the sense that the long-run average tracking error is minimized almost surely for any given bounded reference signals. To the best of the authors' knowledge, this appears to be the first adaptive control result for general linear systems with general binary sensors and arbitrarily given bounded reference signals.

Adaptive Tracking Control with Binary-Valued Output Observations

Abstract

This paper considers real-time control and learning problems for finite-dimensional linear systems under binary-valued and randomly disturbed output observations. This has long been regarded as an open problem because the exact values of the traditional regression vectors used in the construction of adaptive algorithms are unavailable, as one only has binary-valued output information. To overcome this difficulty, we consider the adaptive estimation problem of the corresponding infinite-impulse-response (IIR) dynamical systems, and apply the double array martingale theory that has not been previously used in adaptive control. This enables us to establish global convergence results for both the adaptive prediction regret and the parameter estimation error, without resorting to such stringent data conditions as persistent excitation and bounded system signals that have been used in almost all existing related literature. Based on this, an adaptive control law will be designed that can effectively combine adaptive learning and feedback control. Finally, we are able to show that the closed-loop adaptive control system is optimal in the sense that the long-run average tracking error is minimized almost surely for any given bounded reference signals. To the best of the authors' knowledge, this appears to be the first adaptive control result for general linear systems with general binary sensors and arbitrarily given bounded reference signals.

Paper Structure

This paper contains 13 sections, 8 theorems, 123 equations, 3 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Problem formulation
  3. Main results
  4. Adaptive parameter estimation
  5. Adaptive tracking control
  6. Main ideas
  7. Adaptive controller
  8. Proof of main results
  9. Proof of Theorem \ref{['le1']}
  10. Proof of Theorem \ref{['thm2']}
  11. Numerical simulation
  12. Concluding remarks
  13. Proof of Theorem \ref{['le1']}

Key Result

Theorem 1

Under Assumptions ass3-ass1, suppose for some $\delta>0$, where $d_{n}$ is defined in $(e21)$. Then the following properties hold:

Figures (3)

  • Figure 1: $r=1.5$: Adaptive tracking control for fixed reference signals. a) Specially designed binary observation thresholds b) General binary observation thresholds.
  • Figure 2: $r=3.44$: Adaptive tracking control for periodic reference signals.
  • Figure 3: $r=3.8$: Adaptive tracking control for chaotic reference signals.

Theorems & Definitions (13)

  • Remark 1
  • Remark 2
  • Theorem 1
  • Remark 3
  • Theorem 2
  • Lemma 1
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • ...and 3 more