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Discrete-time Indirect Adaptive Control for Systems with Disturbances via Directional Forgetting: Concurrent Learning Approach

Satoshi Tsuruhara, Kazuhisa Ito

TL;DR

The paper proposes a discrete-time indirect adaptive control method that blends directional forgetting with concurrent learning to achieve stability without persistent excitation in the presence of disturbances. By maintaining an information matrix that forgets in the excitation direction and recording informative data, the approach guarantees Uniformly Ultimately Boundedness with a bound that decreases over time via the forgetting factor. It extends CL to indirect adaptive control and provides a Lyapunov-based proof of UUB, including exponential convergence in disturbance-free conditions. Numerical results show improved tracking, faster parameter convergence, and robust disturbance attenuation compared with conventional CL methods.

Abstract

Recently, adaptive control systems with relaxed persistent excitation (PE) conditions have been proposed to guarantee true parameter convergence and improve the transient response. However, in some cases, sufficient control performance and parameter convergence cannot be easily achieved, with stability demonstrated only under ideal conditions, such as the absence of disturbances and matching conditions required. In this study, we propose a novel adaptive control method for discrete-time systems with disturbances, which is not under an ideal case, that combines directional forgetting and concurrent learning. The proposed method does not require the PE condition, information on disturbances, unknown parameters, or matching conditions, and it guarantees uniformly ultimately bounded (UUB). It was also theoretically demonstrated that the ultimate bound can be designed based on the forgetting factor, which is a design parameter. In addition, the upper bound decreases with time step, which is independent of the system order and/or target trajectory due to forgetting factor. This also implies stronger stability than a normal UUB. Numerical simulation results illustrate the effectiveness of the proposed method.

Discrete-time Indirect Adaptive Control for Systems with Disturbances via Directional Forgetting: Concurrent Learning Approach

TL;DR

The paper proposes a discrete-time indirect adaptive control method that blends directional forgetting with concurrent learning to achieve stability without persistent excitation in the presence of disturbances. By maintaining an information matrix that forgets in the excitation direction and recording informative data, the approach guarantees Uniformly Ultimately Boundedness with a bound that decreases over time via the forgetting factor. It extends CL to indirect adaptive control and provides a Lyapunov-based proof of UUB, including exponential convergence in disturbance-free conditions. Numerical results show improved tracking, faster parameter convergence, and robust disturbance attenuation compared with conventional CL methods.

Abstract

Recently, adaptive control systems with relaxed persistent excitation (PE) conditions have been proposed to guarantee true parameter convergence and improve the transient response. However, in some cases, sufficient control performance and parameter convergence cannot be easily achieved, with stability demonstrated only under ideal conditions, such as the absence of disturbances and matching conditions required. In this study, we propose a novel adaptive control method for discrete-time systems with disturbances, which is not under an ideal case, that combines directional forgetting and concurrent learning. The proposed method does not require the PE condition, information on disturbances, unknown parameters, or matching conditions, and it guarantees uniformly ultimately bounded (UUB). It was also theoretically demonstrated that the ultimate bound can be designed based on the forgetting factor, which is a design parameter. In addition, the upper bound decreases with time step, which is independent of the system order and/or target trajectory due to forgetting factor. This also implies stronger stability than a normal UUB. Numerical simulation results illustrate the effectiveness of the proposed method.
Paper Structure (13 sections, 6 theorems, 77 equations, 10 figures, 1 table, 2 algorithms)

This paper contains 13 sections, 6 theorems, 77 equations, 10 figures, 1 table, 2 algorithms.

Table of Contents

  1. Introduction
  2. Preliminalies and Problem Formulation
  3. Preliminalies
  4. Problem Formulation
  5. Proposed methods
  6. Concurrent learning estimation
  7. Indirect adaptive controller
  8. Stability analysis
  9. Ultimately uniform bounded for CL
  10. The boundedness of the signals in the closed-loop system
  11. Proof of UUB
  12. Nuemerical simulations
  13. Conculusion

Key Result

lemma 1

DF2 We assume that the information matrix $\Omega(k)$ satisfies the rank condition (see Condition 1). For the DF algorithm, the following matrix $U(k)$ is bounded from above to below for some $\mu>0$, $^\forall k\in[k_e,\infty)$:

Figures (10)

  • Figure 1: Overall view
  • Figure 2: Enlarged view with y-axis restricted
  • Figure 4: Comparison of the estimated parameters in the absence of disturbance (Case (A))
  • Figure 5: Comparison of the inverse of the condition number in the absence of disturbance (Case (A))
  • Figure 6: Comparison of control performance with disturbance (Case (B)), ($w(k)=0.3,\ k<500,\ -0.3,\ k\geq 500$)
  • ...and 5 more figures

Theorems & Definitions (17)

  • definition 1
  • definition 2
  • remark 1
  • remark 2
  • lemma 1
  • proof
  • theorem 1
  • proof
  • lemma 2
  • proof
  • ...and 7 more