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Parameter Update Laws for Adaptive Control with Affine Equality Parameter Constraints

Ashwin P. Dani

TL;DR

The effectiveness of the proposed equality-constrained adaptive control law is demonstrated through simulations, validating its ability to maintain constraints on the parameter estimates, achieving convergence to the true parameters for CL-based update law, and achieving asymptotic and exponential tracking performance for constrained gradient and constrained CL-based update laws, respectively.

Abstract

In this paper, constrained parameter update laws for adaptive control with convex equality constraint on the parameters are developed, one based on a gradient only update and the other incorporating concurrent learning (CL) update. The update laws are derived by solving a constrained optimization problem with affine equality constraints. This constrained problem is reformulated as an equivalent unconstrained problem in a new variable, thereby eliminating the equality constraints. The resulting update law is integrated with an adaptive trajectory tracking controller, enabling online learning of the unknown system parameters. Lyapunov stability of the closed-loop system with the equality-constrained parameter update law is established. The effectiveness of the proposed equality-constrained adaptive control law is demonstrated through simulations, validating its ability to maintain constraints on the parameter estimates, achieving convergence to the true parameters for CL-based update law, and achieving asymptotic and exponential tracking performance for constrained gradient and constrained CL-based update laws, respectively.

Parameter Update Laws for Adaptive Control with Affine Equality Parameter Constraints

TL;DR

The effectiveness of the proposed equality-constrained adaptive control law is demonstrated through simulations, validating its ability to maintain constraints on the parameter estimates, achieving convergence to the true parameters for CL-based update law, and achieving asymptotic and exponential tracking performance for constrained gradient and constrained CL-based update laws, respectively.

Abstract

In this paper, constrained parameter update laws for adaptive control with convex equality constraint on the parameters are developed, one based on a gradient only update and the other incorporating concurrent learning (CL) update. The update laws are derived by solving a constrained optimization problem with affine equality constraints. This constrained problem is reformulated as an equivalent unconstrained problem in a new variable, thereby eliminating the equality constraints. The resulting update law is integrated with an adaptive trajectory tracking controller, enabling online learning of the unknown system parameters. Lyapunov stability of the closed-loop system with the equality-constrained parameter update law is established. The effectiveness of the proposed equality-constrained adaptive control law is demonstrated through simulations, validating its ability to maintain constraints on the parameter estimates, achieving convergence to the true parameters for CL-based update law, and achieving asymptotic and exponential tracking performance for constrained gradient and constrained CL-based update laws, respectively.
Paper Structure (15 sections, 2 theorems, 46 equations, 5 figures)

This paper contains 15 sections, 2 theorems, 46 equations, 5 figures.

Table of Contents

  1. Introduction
  2. System Model and Control Objective
  3. System Dynamics
  4. Controller Objective
  5. Adaptive Controller Design
  6. Equality-Constrained Gradient Update Law
  7. Error dynamics
  8. Stability Analysis
  9. Equality-Constrained Concurrent Learning Update Law
  10. Error dynamics
  11. Stability analysis
  12. Simulations
  13. Simulation for Equality-Constrained Gradient Law
  14. Simulation for Equality-Constrained CL Update Law
  15. Conclusion

Key Result

Theorem 1

If Assumption ass:Assumption1 is satisfied, for the system shown in (eq:SystemModel), the equality-constrained parameter update law (eq:EqualityConstGradUpdateLaw) and the adaptive controller (eq:Control) ensure global asymptotic tracking, i.e., and bounded parameter estimation error with constrained satisfaction on the parameter estimates.

Figures (5)

  • Figure 1: Trajectory tracking with equality-constrained gradient parameter update law.
  • Figure 2: Parameter estimates using equality-constrained gradient parameter update law.
  • Figure 3: Trajectory tracking with equality-constrained CL parameter update law.
  • Figure 4: Parameter estimates using equality-constrained CL parameter update law, where $\hat{\theta}_2$ and $\hat{\theta}_4$ satisfy the equality constraint $\hat{\theta}_2-\hat{\theta}_4=0$.
  • Figure 5: Parameter estimates using equality-constrained gradient update law, where $\hat{\theta}_2$ and $\hat{\theta}_4$ satisfy the equality constraint $\hat{\theta}_2-\hat{\theta}_4=0$.

Theorems & Definitions (10)

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