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Constrained Parameter Update Law for Adaptive Control

Ashwin P. Dani

TL;DR

The paper tackles enforcing hard parameter constraints in adaptive control while maintaining tracking performance. It introduces an inverse-barrier constrained parameter update law derived from a Lagrangian, combined with concurrent learning to relax persistence of excitation requirements. A Lyapunov-based analysis proves uniform ultimate boundedness of tracking and parameter errors, with a sigma-modification option when excitation is limited. Simulations on a multi-parameter, model-parameterized system demonstrate that the method keeps estimates within bounds or norms and offers competitive transient behavior compared with gradient and CL-based methods.

Abstract

In this paper, a constrained parameter update law is derived in the context of adaptive control. The parameter update law is based on constrained optimization technique where a Lagrangian is formulated to incorporate the constraints on the parameters using inverse Barrier function. The constrained parameter update law is used to develop a adaptive tracking controller and the overall stability of the adaptive controller along with the constrained parameter update law is shown using Lyapunov analysis and development in stability of constrained primal-dual dynamics. The performance of the constrained parameter update law is tested in simulation for keeping the parameters within constraints and convergence to true parameters.

Constrained Parameter Update Law for Adaptive Control

TL;DR

The paper tackles enforcing hard parameter constraints in adaptive control while maintaining tracking performance. It introduces an inverse-barrier constrained parameter update law derived from a Lagrangian, combined with concurrent learning to relax persistence of excitation requirements. A Lyapunov-based analysis proves uniform ultimate boundedness of tracking and parameter errors, with a sigma-modification option when excitation is limited. Simulations on a multi-parameter, model-parameterized system demonstrate that the method keeps estimates within bounds or norms and offers competitive transient behavior compared with gradient and CL-based methods.

Abstract

In this paper, a constrained parameter update law is derived in the context of adaptive control. The parameter update law is based on constrained optimization technique where a Lagrangian is formulated to incorporate the constraints on the parameters using inverse Barrier function. The constrained parameter update law is used to develop a adaptive tracking controller and the overall stability of the adaptive controller along with the constrained parameter update law is shown using Lyapunov analysis and development in stability of constrained primal-dual dynamics. The performance of the constrained parameter update law is tested in simulation for keeping the parameters within constraints and convergence to true parameters.
Paper Structure (12 sections, 1 theorem, 38 equations, 8 figures)

This paper contains 12 sections, 1 theorem, 38 equations, 8 figures.

Table of Contents

  1. Introduction
  2. System Model and Control Objective
  3. System Dynamics
  4. Controller Objective
  5. Adaptive Controller Design
  6. Design of constrained adaptive parameter update law
  7. Stability Analysis
  8. Parameter Update Law with Norm Constraints
  9. Simulations
  10. Simulation with Bounds on $\hat{\theta}_i$
  11. Simulation with bounds on $\Vert \hat{\theta} \Vert$
  12. Conclusion

Key Result

Theorem 1

If Assumption ass:FiniteExcitation is satisfied, for the system shown in (eq:SystemModel), the constrained parameter update law (eq:ParameterUpdateLaw) and the adaptive controller (eq:Control) ensues uniformly ultimately bounded (UUB) tracking and parameter estimation errors.

Figures (8)

  • Figure 1: Trajectory tracking with different parameter update laws.
  • Figure 2: Parameter convergence with different parameter update laws for $\theta_1$ and $\theta_2$.
  • Figure 3: Parameter convergence with different parameter update laws for $\theta_3$ and $\theta_4$.
  • Figure 4: Evolution of Lagrange multiplier for upper bound
  • Figure 5: Evolution of Lagrange multiplier for lower bound
  • ...and 3 more figures

Theorems & Definitions (8)

  • Remark 1
  • Remark 2
  • Remark 3
  • Theorem 1
  • proof
  • Remark 4
  • Remark 5
  • Remark 6