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Adaptive Output Tracking Control with Reference Model System Uncertainties

Gang Tao

TL;DR

This work addresses adaptive output tracking when the reference model is unknown or its derivatives are unavailable, by introducing an expanded MRAC framework that parametrizes and estimates the unknown equivalent reference input $r_m(t)$. It develops four adaptive control schemes (state-feedback and output-feedback, using either the reference state $x_m$ or the reference output $y_m$) with explicit parametrizations of $r_m(t)$, and proves stability and asymptotic tracking through Lyapunov analysis, showing $\,\lim_{t\to\infty}(y(t)-y_m(t))=0$. The methodology unifies nominal and adaptive laws across designs, handles cases where the reference model is uncertain, and extends to partial-state, MIMO, discrete-time, and certain nonlinear settings. A simulation with leader-follower aircraft dynamics demonstrates practical tracking under reference-model uncertainties, illustrating the approach’s potential for robust adaptive tracking in aerospace and related domains. Overall, the paper provides a principled way to broaden MRAC applicability when reference dynamics are not fully known.

Abstract

This paper develops adaptive output tracking control schemes with the reference output signal generated from an unknown reference system whose output derivatives are also unknown. To deal with such reference system uncertainties, an expanded adaptive controller structure is developed to include a parametrized estimator of the equivalent reference input signal. Without using the knowledge of the reference system transfer function and equivalent input, both are the critical components of a traditional model reference adaptive control (MRAC) scheme, the developed new MRAC schemes designed for various cases plant and reference model uncertainties, ensure completely parametrized error systems and stable parameter adaptation, leading to the desired closed-loop system stability and asymptotic output tracking.

Adaptive Output Tracking Control with Reference Model System Uncertainties

TL;DR

This work addresses adaptive output tracking when the reference model is unknown or its derivatives are unavailable, by introducing an expanded MRAC framework that parametrizes and estimates the unknown equivalent reference input . It develops four adaptive control schemes (state-feedback and output-feedback, using either the reference state or the reference output ) with explicit parametrizations of , and proves stability and asymptotic tracking through Lyapunov analysis, showing . The methodology unifies nominal and adaptive laws across designs, handles cases where the reference model is uncertain, and extends to partial-state, MIMO, discrete-time, and certain nonlinear settings. A simulation with leader-follower aircraft dynamics demonstrates practical tracking under reference-model uncertainties, illustrating the approach’s potential for robust adaptive tracking in aerospace and related domains. Overall, the paper provides a principled way to broaden MRAC applicability when reference dynamics are not fully known.

Abstract

This paper develops adaptive output tracking control schemes with the reference output signal generated from an unknown reference system whose output derivatives are also unknown. To deal with such reference system uncertainties, an expanded adaptive controller structure is developed to include a parametrized estimator of the equivalent reference input signal. Without using the knowledge of the reference system transfer function and equivalent input, both are the critical components of a traditional model reference adaptive control (MRAC) scheme, the developed new MRAC schemes designed for various cases plant and reference model uncertainties, ensure completely parametrized error systems and stable parameter adaptation, leading to the desired closed-loop system stability and asymptotic output tracking.
Paper Structure (18 sections, 2 theorems, 116 equations, 2 figures)

This paper contains 18 sections, 2 theorems, 116 equations, 2 figures.

Table of Contents

  1. Problem Description
  2. Technical Background
  3. State Feedback Control Designs
  4. Design Using Reference System State $x_m(t)$
  5. Design Using Reference System Output $y_m(t)$
  6. Output Feedback Control Designs
  7. Design Using Reference System State $x_m(t)$
  8. Design Using Reference System Output $y_m(t)$
  9. Adaptive Control System Properties
  10. Adaptive Law Properties
  11. Closed-loop System Properties
  12. Extensions
  13. Partial-State Feedback Control Designs
  14. Other Extensions
  15. Simulation Study
  16. ...and 3 more sections

Key Result

Lemma 5.1

The four sets of adaptive parameter update laws for the four designed adaptive control schemes ensure that $\theta(t) \in L^{\infty}$, $\rho(t) \in L^{\infty}$, $\frac{\epsilon(t)}{m(t)} \in L^{2} \cap L^{\infty}$, $\dot{\theta}(t) \in L^{2} \cap L^{\infty}$, and $\dot{\rho}(t) \in L^{2} \cap L^{\in

Figures (2)

  • Figure 1: System responses for $v_m(t) = 300$.
  • Figure 3: System responses for $v_m(t) = 300 \sin (0.3t) + 250 \sin (0.5t)$.

Theorems & Definitions (4)

  • Lemma 5.1
  • Theorem 5.1
  • Remark 5.1
  • Remark 5.2