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Adaptive Tracking Control of Euler-Lagrange Systems with Time-Varying State and Input Constraints

Poulomee Ghosh, Shubhendu Bhasin

Abstract

This paper presents an adaptive control framework for Euler-Lagrange (E-L) systems that enforces user-defined time-varying state and input constraints in the presence of parametric uncertainties and bounded disturbances. The proposed design integrates a time-varying barrier Lyapunov Function (TVBLF) with a saturated control law to guarantee constraint satisfaction without resorting to real-time optimization. A key contribution is the development of an offline, verifiable feasibility condition that certifies the existence of a feasible control policy for any prescribed pair of time-varying state and input envelopes. Additionally, we prove boundedness of all closed-loop signals. Real-time experiments conducted on a 2-DoF helicopter model validate the efficacy and practical viability of the proposed method.

Adaptive Tracking Control of Euler-Lagrange Systems with Time-Varying State and Input Constraints

Abstract

This paper presents an adaptive control framework for Euler-Lagrange (E-L) systems that enforces user-defined time-varying state and input constraints in the presence of parametric uncertainties and bounded disturbances. The proposed design integrates a time-varying barrier Lyapunov Function (TVBLF) with a saturated control law to guarantee constraint satisfaction without resorting to real-time optimization. A key contribution is the development of an offline, verifiable feasibility condition that certifies the existence of a feasible control policy for any prescribed pair of time-varying state and input envelopes. Additionally, we prove boundedness of all closed-loop signals. Real-time experiments conducted on a 2-DoF helicopter model validate the efficacy and practical viability of the proposed method.
Paper Structure (16 sections, 4 theorems, 61 equations, 5 figures, 1 table, 1 algorithm)

This paper contains 16 sections, 4 theorems, 61 equations, 5 figures, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Motivation
  3. State of the Art: Constrained Control with Static Bounds
  4. What if the admissible safe regions are time-varying?
  5. State of the Art: Constrained Control with Dynamic Bounds
  6. Contribution
  7. Notation
  8. Problem Formulation
  9. Proposed Methodology
  10. Constraint Transformation
  11. Error Dynamics
  12. Saturated Control Design for Time-Varying Input Constraints
  13. TVBLF-based Adaptive Law Design for Time-Varying State Constraints
  14. Designing Time-varying Constraints as Prescribed Performance Functions (PPF)
  15. Experimental Results
  16. ...and 1 more sections

Key Result

Lemma 1

[Constraint conversion from $(e,\dot e)$ to $r$] Let $\alpha$ satisfy the gain condition and suppose there exists a continuously differentiable function $\phi_r:[0,\infty)\rightarrow(0,\infty)$, which satisfies the following inequality Provided Assumption error_assumption holds and the filtered tracking error remains bounded, i.e., the trajectory tracking errors remain within the time-varying e

Figures (5)

  • Figure 1: BLF \ref{['tvblf_def']} with fixed and time-varying bounds.
  • Figure 2: (a) Experimental setup and (b) simplified block diagram of the Quanser 2-DoF helicopter.
  • Figure 3: Effect of varying (a) $\kappa$ and (b) $\nu$ on the performance function $\phi_{x}(t)$, where $\phi_{x}^{0}=2$, $\phi_{x}^{\infty}=0.1$ and $\epsilon=0.2$. $\nu=1$ and $\kappa=1$ are fixed in (a) and (b), respectively.
  • Figure 4: Required control input for the 2-DoF helicopter using the proposed controller.
  • Figure 5: Experimental results using the proposed controller.

Theorems & Definitions (17)

  • Remark 1
  • Remark 2
  • Lemma 1
  • proof
  • Remark 3
  • Definition 1: TVBLF
  • Theorem 1
  • proof
  • Remark 4
  • Remark 5
  • ...and 7 more