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Adaptive Output Tracking Control with Reference Model System Uncertainties: Extensions

Gang Tao

TL;DR

This work extends previous adaptive output-tracking results to cases where the reference model is uncertain, spanning SISO discrete-time, MIMO discrete-time/continuous-time, and nonlinear feedback-linearizable systems. It introduces a parametrized estimator for the equivalent reference input, yielding completely parametrized error models with known regressors and enabling stable adaptive updates across diverse architectures. The paper develops nominal and adaptive state- and output-feedback controllers, including unified MIMO designs, adaptive laws with Lyapunov guarantees, and K_p decomposition techniques to reduce reliance on exact high-frequency gains. It also treats practical extensions such as unavailable reference states, partial-state information, continuous-time degree-one systems, and leader-follower/multi-agent applications, highlighting broader applicability to uncertain-reference-model tracking. Collectively, the results advance robust adaptive tracking by encoding reference-model uncertainties directly into the adaptive law design, achieving bounded closed-loop signals and asymptotic agreement with unknown reference dynamics.

Abstract

This paper develops some extensions to the work of [1] which studied the continuous-time adaptive output tracking control schemes with the reference output signal generated from an unknown reference model system. The presented extensions include adaptive control schemes with reference model system uncertainties for single-input single-output (SISO) discrete-time systems and multi-input multi-output (MIMO) discrete-time, continuous-time and feedback linearizable systems as well. To deal with such reference model system uncertainties, the adaptive controller structures are expanded to include a parametrized estimator of the equivalent reference input signal, to ensure a completely parametrized error system with a known regressor vector, suitable for stable adaptive controller parameter update law design.

Adaptive Output Tracking Control with Reference Model System Uncertainties: Extensions

TL;DR

This work extends previous adaptive output-tracking results to cases where the reference model is uncertain, spanning SISO discrete-time, MIMO discrete-time/continuous-time, and nonlinear feedback-linearizable systems. It introduces a parametrized estimator for the equivalent reference input, yielding completely parametrized error models with known regressors and enabling stable adaptive updates across diverse architectures. The paper develops nominal and adaptive state- and output-feedback controllers, including unified MIMO designs, adaptive laws with Lyapunov guarantees, and K_p decomposition techniques to reduce reliance on exact high-frequency gains. It also treats practical extensions such as unavailable reference states, partial-state information, continuous-time degree-one systems, and leader-follower/multi-agent applications, highlighting broader applicability to uncertain-reference-model tracking. Collectively, the results advance robust adaptive tracking by encoding reference-model uncertainties directly into the adaptive law design, achieving bounded closed-loop signals and asymptotic agreement with unknown reference dynamics.

Abstract

This paper develops some extensions to the work of [1] which studied the continuous-time adaptive output tracking control schemes with the reference output signal generated from an unknown reference model system. The presented extensions include adaptive control schemes with reference model system uncertainties for single-input single-output (SISO) discrete-time systems and multi-input multi-output (MIMO) discrete-time, continuous-time and feedback linearizable systems as well. To deal with such reference model system uncertainties, the adaptive controller structures are expanded to include a parametrized estimator of the equivalent reference input signal, to ensure a completely parametrized error system with a known regressor vector, suitable for stable adaptive controller parameter update law design.

Paper Structure

This paper contains 29 sections, 3 theorems, 147 equations.

Table of Contents

  1. Introduction
  2. Discrete-Time Designs for SISO Systems
  3. State Feedback Control Designs
  4. Design with $x_m(t)$ Available
  5. Design with $x_m(t)$ Unavailable
  6. Output Feedback Control Designs
  7. Nominal Control Laws
  8. Adaptive Control Schemes
  9. Unified Designs for MIMO Systems
  10. State Feedback Control Laws
  11. Nominal Control Law
  12. Adaptive Control Law
  13. Output Feedback Control Laws
  14. Nominal Control Law
  15. Adaptive Control Law
  16. ...and 14 more sections

Key Result

Lemma 2.1

The adaptive laws (5732A.37d) and (5732A.38d) ensure that $\theta(t) \in L^{\infty}$, $\rho(t) \in L^{\infty}$, $\frac{\epsilon(t)}{m(t)} \in L^{2} \cap L^{\infty}$, $\theta(t+1)-\theta(t) \in L^{2}$, and $\rho(t+1) - \rho(t) \in L^{2}$.

Theorems & Definitions (4)

  • Lemma 2.1
  • Theorem 2.1
  • Remark 3.1
  • Lemma 3.1