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Adaptive Control with Rate-Limited Integral Action for Systems with Matched, Time-Varying Uncertainties

Ying-Chun Chen, Craig Woolsey

TL;DR

The paper tackles control of piecewise continuously differentiable systems under matched time-varying uncertainties by combining conventional model-reference adaptive control (MRAC) for the parameterized, time-invariant portion with disturbance observer-based control (DOBC) for the time-varying unstructured part. A lumped-disturbance observer estimates the disturbance and a magnitude- and rate-limited integral action in the disturbance-rejection term counteracts it, providing a balance between estimation performance and control-input smoothness. Stability is maintained through a Lyapunov-based MRAC framework with projection, and a formal performance bound on tracking error is derived when the disturbance-rejection term operates in the unsaturated regime; a robust stabilization mode is available when saturation occurs. The approach is illustrated on a nonlinear mass-spring-damper system, showing that integrating DOBAC (I-DOBAC) yields more accurate disturbance estimates and reduced lumped-disturbance peaks compared with conventional MRAC and direct DOBAC, with tunable parameters such as $k_\eta$ governing speed and bound on the disturbance rejection.

Abstract

This paper considers the problem of controlling a piecewise continuously differentiable system subject to time-varying uncertainties. The uncertainties are decomposed into a time-invariant, linearly-parameterized portion and a time-varying unstructured portion. The former is addressed using conventional model reference adaptive control. The latter is handled using disturbance observer-based control. The objective is to ensure good performance through observer-based disturbance rejection when possible, while preserving the robustness guarantees of adaptive control. A key feature of the observer-based disturbance compensation is a magnitude and rate limit on the integral action that prevents fast fluctuations in the control command due to the observer dynamics.

Adaptive Control with Rate-Limited Integral Action for Systems with Matched, Time-Varying Uncertainties

TL;DR

The paper tackles control of piecewise continuously differentiable systems under matched time-varying uncertainties by combining conventional model-reference adaptive control (MRAC) for the parameterized, time-invariant portion with disturbance observer-based control (DOBC) for the time-varying unstructured part. A lumped-disturbance observer estimates the disturbance and a magnitude- and rate-limited integral action in the disturbance-rejection term counteracts it, providing a balance between estimation performance and control-input smoothness. Stability is maintained through a Lyapunov-based MRAC framework with projection, and a formal performance bound on tracking error is derived when the disturbance-rejection term operates in the unsaturated regime; a robust stabilization mode is available when saturation occurs. The approach is illustrated on a nonlinear mass-spring-damper system, showing that integrating DOBAC (I-DOBAC) yields more accurate disturbance estimates and reduced lumped-disturbance peaks compared with conventional MRAC and direct DOBAC, with tunable parameters such as governing speed and bound on the disturbance rejection.

Abstract

This paper considers the problem of controlling a piecewise continuously differentiable system subject to time-varying uncertainties. The uncertainties are decomposed into a time-invariant, linearly-parameterized portion and a time-varying unstructured portion. The former is addressed using conventional model reference adaptive control. The latter is handled using disturbance observer-based control. The objective is to ensure good performance through observer-based disturbance rejection when possible, while preserving the robustness guarantees of adaptive control. A key feature of the observer-based disturbance compensation is a magnitude and rate limit on the integral action that prevents fast fluctuations in the control command due to the observer dynamics.

Paper Structure

This paper contains 9 sections, 3 theorems, 68 equations, 9 figures.

Table of Contents

  1. Introduction
  2. System description and the control framework
  3. The disturbance observer
  4. A Disturbance Estimation Tradeoff
  5. Design of the disturbance-rejection term
  6. Performance analysis
  7. Example: Position control of a mass-spring-damper system with a nonlinear spring
  8. Remarks on the role of $k_{\eta}$
  9. Conclusions

Key Result

Lemma 1

Under the control law eq:DesiredAdpativeControl_LinearSystem_Dist, the system state $\bm{x}$ remains bounded.

Figures (9)

  • Figure 1: $x_1$ tracking with $u_{\rm drj}=0$. Dash: the desired trajectory $x_{1{\rm r}}$; solid: $x_1$.
  • Figure 2: $x_1$ tracking with the DOBAC using different $u_{\rm drj}$. Dash: the desired trajectory $x_{1{\rm r}}$; solid: $x_1$.
  • Figure 3: The tracking error history and the upper bound guaranteed by the integrating DOBAC
  • Figure 4: $\bm{d}_u$ with the DOBACs using different $u_{\rm drj}$
  • Figure 5: Estimate of $d$ with the DOBAC using different $u_{\rm drj}$. Dash: $d$; solid: $\hat{d}$.
  • ...and 4 more figures

Theorems & Definitions (6)

  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Theorem 1
  • proof