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Arbitrarily Fast Multivariable Least-squares MRAC

Liu Hsu, Ramon R. Costa, Fernando Lizarralde, Alessandro Jacoud Peixoto

TL;DR

This paper introduces a direct LS-MRAC method for MIMO systems with uniform relative degree $1$, combining SDU-based control parametrization and a Monopoli-inspired multiplier to yield an error model with zero relative degree and enable output-feedback LS adaptation. A complete Lyapunov-based stability analysis demonstrates that, for suitably chosen adaptation gains and sufficient PE, the controller achieves arbitrarily fast tracking while maintaining bounded parameter convergence, and the LS updates rely on regressor structures $oldsymbol{Ω}$, $oldsymbol{Ξ}$ and a dynamic gain matrix $R$. The Fast-Converging LS-MRAC and M-MRAC specializations illustrate trade-offs between tracking speed and parameter convergence, with simulations showing substantial improvements over existing MIMO MRAC methods in both transient performance and convergence. The work also discusses parametric convergence behavior under PE and highlights future directions, including covariance reset, forgetting factors, LS learning variants, and extensions to general relative degree.

Abstract

A novel least-squares model-reference direct adaptive control (LS-MRAC) algorithm for multivariable (MIMO) plants is presented. The controller parameters are directly updated based on the output tracking error. The control law is crucially modified to reduce the relative degree of the error model to zero. A complete Lyapunov-based stability analysis as well as a tracking error convergence characterization is provided demonstrating that the LS-MRAC can achieve arbitrarily fast tracking while maintaining satisfactory parameter convergence for appropriate adaptation gains. Simulation results show a significant improvement in tracking performance compared to previous methods.

Arbitrarily Fast Multivariable Least-squares MRAC

TL;DR

This paper introduces a direct LS-MRAC method for MIMO systems with uniform relative degree , combining SDU-based control parametrization and a Monopoli-inspired multiplier to yield an error model with zero relative degree and enable output-feedback LS adaptation. A complete Lyapunov-based stability analysis demonstrates that, for suitably chosen adaptation gains and sufficient PE, the controller achieves arbitrarily fast tracking while maintaining bounded parameter convergence, and the LS updates rely on regressor structures , and a dynamic gain matrix . The Fast-Converging LS-MRAC and M-MRAC specializations illustrate trade-offs between tracking speed and parameter convergence, with simulations showing substantial improvements over existing MIMO MRAC methods in both transient performance and convergence. The work also discusses parametric convergence behavior under PE and highlights future directions, including covariance reset, forgetting factors, LS learning variants, and extensions to general relative degree.

Abstract

A novel least-squares model-reference direct adaptive control (LS-MRAC) algorithm for multivariable (MIMO) plants is presented. The controller parameters are directly updated based on the output tracking error. The control law is crucially modified to reduce the relative degree of the error model to zero. A complete Lyapunov-based stability analysis as well as a tracking error convergence characterization is provided demonstrating that the LS-MRAC can achieve arbitrarily fast tracking while maintaining satisfactory parameter convergence for appropriate adaptation gains. Simulation results show a significant improvement in tracking performance compared to previous methods.
Paper Structure (16 sections, 4 theorems, 81 equations, 8 figures, 1 table)

This paper contains 16 sections, 4 theorems, 81 equations, 8 figures, 1 table.

Table of Contents

  1. Introduction
  2. Multivariable MRAC
  3. $SDU$ factorization
  4. Control parametrization
  5. Design procedure
  6. Fast converging LS-MRAC
  7. Summary of the MIMO LS-MRAC
  8. The M-MRAC
  9. Convergence analysis
  10. Simulation results
  11. Simulation 1.
  12. Simulation 2.
  13. Simulation 3.
  14. Simulation 4.
  15. On Parametric Convergence
  16. ...and 1 more sections

Key Result

Lemma 1

Every $m \times m$ real matrix $K_p$ with nonzero leading principal minors $\{\Delta_i\}_{i=1}^m$ can be factored as where $S$ is symmetric positive definite, $D$ is diagonal, and $U$ is unity upper triangular.

Figures (8)

  • Figure 1: Simulation result of the first order plant (\ref{['P1']}) with the MIMO MRAC algorithm.
  • Figure 2: Simulation result of the first order plant (\ref{['P1']}) with the MIMO M-MRAC algorithm.
  • Figure 3: Simulation result of the first order plant (\ref{['P1']}) with the MIMO LS-MRAC algorithm.
  • Figure 4: Simulation result of the third order plant (\ref{['P2']}) with the MIMO LS-MRAC algorithm, $x_p(0) = [0.65\ \ 1\ \ -0.37]^T$, $\gamma=10$ and $R(0)=I$.
  • Figure 5: As in Fig. \ref{['Fig4']}, with larger R(0)=10. Note that the tracking transient is faster and with smaller errors, as expected. The parameters converge nicely, as shown for $\Theta_1$.
  • ...and 3 more figures

Theorems & Definitions (14)

  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Remark 1
  • proof
  • Remark 2
  • Lemma 3
  • proof
  • Lemma 4
  • ...and 4 more