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Piecewise constant tuning gain based singularity-free MRAC with application to aircraft control systems

Zhipeng Zhang, Yanjun Zhang, Jian Sun

Abstract

This paper introduces an innovative singularity-free output feedback model reference adaptive control (MRAC) method applicable to a wide range of continuous-time linear time-invariant (LTI) systems with general relative degrees. Unlike existing solutions such as Nussbaum and multiple-model-based methods, which manage unknown high-frequency gains through persistent switching and repeated parameter estimation, the proposed method circumvents these issues without prior knowledge of the high-frequency gain or additional design conditions. The key innovation of this method lies in transforming the estimation error equation into a linear regression form via a modified MRAC law with a piecewise constant tuning gain developed in this work. This represents a significant departure from existing MRAC systems, where the estimation error equation is typically in a bilinear regression form. The linear regression form facilitates the direct estimation of all unknown parameters, thereby simplifying the adaptive control process. The proposed method preserves closed-loop stability and ensures asymptotic output tracking, overcoming some of the limitations associated with existing methods like Nussbaum and multiple-model based methods. The practical efficacy of the developed MRAC method is demonstrated through detailed simulation results within an aircraft control system scenario.

Piecewise constant tuning gain based singularity-free MRAC with application to aircraft control systems

Abstract

This paper introduces an innovative singularity-free output feedback model reference adaptive control (MRAC) method applicable to a wide range of continuous-time linear time-invariant (LTI) systems with general relative degrees. Unlike existing solutions such as Nussbaum and multiple-model-based methods, which manage unknown high-frequency gains through persistent switching and repeated parameter estimation, the proposed method circumvents these issues without prior knowledge of the high-frequency gain or additional design conditions. The key innovation of this method lies in transforming the estimation error equation into a linear regression form via a modified MRAC law with a piecewise constant tuning gain developed in this work. This represents a significant departure from existing MRAC systems, where the estimation error equation is typically in a bilinear regression form. The linear regression form facilitates the direct estimation of all unknown parameters, thereby simplifying the adaptive control process. The proposed method preserves closed-loop stability and ensures asymptotic output tracking, overcoming some of the limitations associated with existing methods like Nussbaum and multiple-model based methods. The practical efficacy of the developed MRAC method is demonstrated through detailed simulation results within an aircraft control system scenario.
Paper Structure (14 sections, 5 theorems, 48 equations, 6 figures)

This paper contains 14 sections, 5 theorems, 48 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Problem statement
  3. System model
  4. Control objective and design conditions
  5. Technical issues
  6. Output feedback MRAC design
  7. Construction of the MRAC law
  8. Specification of the new tracking error
  9. Development of the parameter update law
  10. Design of the tuning gain
  11. Stability analysis
  12. Simulation Example
  13. Conclusion
  14. Two useful lemmas

Key Result

Lemma 1

(t03) Constants $\theta_1^* \in \mathbb{R}^{n-1},\theta_2^*\in \mathbb{R}^{n-1},\theta_{3}^* \in \mathbb{R},\theta_4^* = 1/k_p$ exist such that where $b(s)=[1,s,\ldots,s^{n-2}]^T$, and $\Omega(s)$ is an any monic Hurwitz polynomial of degree $n-1$.

Figures (6)

  • Figure 1: Responses of the output $y(t)$ and $y^*(t)$ (Case (i)).
  • Figure 2: Evolutions of the tuning gain $\sigma$ and input $u(t)$ (Case (i)).
  • Figure 3: Evolutions of part of parameters in $\Theta(t)$ (Case (i)).
  • Figure 4: Responses of the output $y(t)$ and $y^*(t)$ (Case (ii)).
  • Figure 5: Evolutions of the tuning gain $\sigma$ and input $u(t)$ (Case (ii)).
  • ...and 1 more figures

Theorems & Definitions (9)

  • Lemma 1
  • Remark 1
  • Remark 2
  • Lemma 2
  • Remark 3
  • Theorem 1
  • Remark 4
  • Lemma 3
  • Lemma 4