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Indirect Adaptive Control Using a Static Update Law

Tom Kaufmann, Johann Reger

TL;DR

This work advances indirect adaptive control by integrating error-feedthrough into the PI update law and analyzing its stability under σ-modification, showing stability over a broader gain range. It then derives a static update law by setting the feedthrough gain to $K = \Sigma^{-1}$, eliminating integration and yielding a weight update $\hat{W}(t)=K\beta(e)B^T P e$ with an adjustable ultimate bound that can be made arbitrarily small by increasing the gain. The authors establish a KYP-based criterion to guarantee the required Lyapunov conditions and present an illustrative second-order plant to compare PI and static updates, revealing that the static law reduces weight oscillations and can act as a disturbance observer without sacrificing low-frequency disturbance rejection. They also show that, under a linear-growth bound on the structured uncertainty, the static update law can drive the error to zero in the absence of unstructured disturbances. Overall, the paper provides theoretical guarantees, a practical criterion for parameter selection, and insights into the disturbance-observer interpretation of adaptation in a simplified, static-update setting.

Abstract

The update law in the indirect adaptive control scheme can be extended to include feedthrough of an error term. This reduces undesired oscillations of the calculated weights. When the $σ$-modification is used for achieving robustness against unstructured uncertainties, the gain of the feedthrough in the update law cannot be chosen arbitrarily. Compared to our previous result, we show stability of the closed loop for a larger parameter-range for the gain of the feedthrough in the update law. This parameter-range includes a configuration for which the influence of the integration in the update law diminishes over time, i.e. for which the adaptation for large times is governed solely by the feedthrough in the update law. By initializing at zero, this allows for removing the integration from the update law, resulting in a static update law. For the purely linear case, the adaptation acts like a disturbance observer. Frequency-domain analysis of the closed loop with a second order plant shows that removing the integration from the update law with $σ$-modification and feedthrough affects how precisely disturbances in the low-frequency band are observed. If the damping injected into the adaptation process by the $σ$-modification exceeds certain bounds, then the precision is increased by using the static update law.

Indirect Adaptive Control Using a Static Update Law

TL;DR

This work advances indirect adaptive control by integrating error-feedthrough into the PI update law and analyzing its stability under σ-modification, showing stability over a broader gain range. It then derives a static update law by setting the feedthrough gain to , eliminating integration and yielding a weight update with an adjustable ultimate bound that can be made arbitrarily small by increasing the gain. The authors establish a KYP-based criterion to guarantee the required Lyapunov conditions and present an illustrative second-order plant to compare PI and static updates, revealing that the static law reduces weight oscillations and can act as a disturbance observer without sacrificing low-frequency disturbance rejection. They also show that, under a linear-growth bound on the structured uncertainty, the static update law can drive the error to zero in the absence of unstructured disturbances. Overall, the paper provides theoretical guarantees, a practical criterion for parameter selection, and insights into the disturbance-observer interpretation of adaptation in a simplified, static-update setting.

Abstract

The update law in the indirect adaptive control scheme can be extended to include feedthrough of an error term. This reduces undesired oscillations of the calculated weights. When the -modification is used for achieving robustness against unstructured uncertainties, the gain of the feedthrough in the update law cannot be chosen arbitrarily. Compared to our previous result, we show stability of the closed loop for a larger parameter-range for the gain of the feedthrough in the update law. This parameter-range includes a configuration for which the influence of the integration in the update law diminishes over time, i.e. for which the adaptation for large times is governed solely by the feedthrough in the update law. By initializing at zero, this allows for removing the integration from the update law, resulting in a static update law. For the purely linear case, the adaptation acts like a disturbance observer. Frequency-domain analysis of the closed loop with a second order plant shows that removing the integration from the update law with -modification and feedthrough affects how precisely disturbances in the low-frequency band are observed. If the damping injected into the adaptation process by the -modification exceeds certain bounds, then the precision is increased by using the static update law.

Paper Structure

This paper contains 13 sections, 9 theorems, 49 equations, 3 figures.

Table of Contents

  1. Introduction
  2. System and Assumptions
  3. Results
  4. Proportional Integral Update Law
  5. Static Update Law
  6. Ultimate Bound
  7. Transient Behavior
  8. Structured Uncertainty with Zero Residual Error for Adaptation with the Static Update Law
  9. Criterion based on the KYP Lemma
  10. Illustrative Example: Second Order Plant without Transfer Zeros
  11. Simulation
  12. Adaptation acting as a Disturbance Observer
  13. Conclusion

Key Result

Theorem 4

Consider the error dynamics $(ErrorDyn)$. Let Assumption assumption:WellPosedness apply and suppose, $0<P,Q\in\mathrm{S}^n$ satisfy the Lyapunov equation $(NomStab)$. The parameters $\Gamma,\Sigma,K\in\mathrm{S}^{n_\mathrm{\beta}}$ admit $0<\Gamma,\Sigma$ and $0\leq K<4\Sigma^{-1}$. Then, the adapti guarantees UUB of $e$ and $\hat{W}$ if the unstructured uncertainty is bounded with $|\eta(t)|\leq

Figures (3)

  • Figure 1: We use $v\!=\!(1\ \ 2/2^{1/2})^\top$ such that $H(s)$ is SPR. The slack variables are set to $\varrho\!=\!0.75$ and $\kappa\!=\!0.9\,\sup\mathcal{K}$. The increase in the minimal eigenvalue $g(\phi,Q,PB)$ with $\phi\!\geq\! 0$ shown in the lower figure is a consequence of Corollary \ref{['corollary:GraphicalCriterion']} since the $\psi$-free criterion~(\ref{['psiFreeConditionEigenvalueHub']}) is fulfilled, compare the red graph. The solution $0\!<\!P\!\in\!\mathrm{S}^2$ of the associated LMI (\ref{['strictLMI']}) can be found in the Appendix.
  • Figure 2: Position error $e_1$ after an initial error in the velocity with different gains of the static update law (dashed) and the feedthrough in the PI update law (solid).
  • Figure 3: Sensitivity of $\hat{W}(t)$ against the input disturbance $d(t)=W+\eta(t)$ for the purely linear case $\beta\equiv 1$ with different gains of the static update law (dashed) and the feedthrough in the PI update law (solid).

Theorems & Definitions (16)

  • Definition 1
  • Remark 3
  • Theorem 4
  • Remark 5
  • Remark 6
  • Lemma 7
  • Corollary 8
  • Remark 9
  • Theorem 10
  • Remark 11
  • ...and 6 more