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Adaptive Neural Control with Desired Approximation: An Integral Lyapunov Function Approach

Mingxuan Sun, Shengxiang Zou

TL;DR

This work addresses adaptive neural control for nonlinear systems in which the practical state need not be bounded a priori. It introduces a desired approximation (DA) framework that separates the unknown nonlinearity into a part depending on a known desired trajectory, thereby relaxing the compact-set prerequisite on the actual state. An integral Lyapunov-function-based controller is developed to accommodate state-dependent input gains, and two adaptation schemes—integral and incremental—are analyzed, yielding bounded closed-loop behavior and mean-square performance characterized by computable bounds. The results show that tracking errors converge to an adjustable neighborhood without requiring knowledge of the region where the state evolves or initial weight estimates, and they provide explicit transient-performance guarantees. The incremental adaptation extension further improves practicality by avoiding online numerical integration while maintaining boundedness and performance promises.

Abstract

The inherent approximation ability of neural networks plays an essential role in adaptive neural control, where the prerequisite for existence of the compact set is crucial in the control designs. Instead of using practical system state, in this paper, the desired approximation approach is characterized to tackle such a problem, where the desired state signal is required only as the input to the network. An integral Lyapunov function-based adaptive controller is designed, in the sense of the error tracking, where the treatment of the state-dependent input gain is adopted. Theoretical results for the performance analysis of the integral and incremental adaptation algorithms are presented in details. In particular, the boundedness of the variables in the closed-loop is characterized, while the transient performance of the output error is analytically quantified. It is shown that the proposed control schemes assure that the tracking error converges to an adjustable set without any requirement on the knowledge of the region that the practical variables evolve, and remove the requirement for the setting of initial conditions including system states and weight estimates.

Adaptive Neural Control with Desired Approximation: An Integral Lyapunov Function Approach

TL;DR

This work addresses adaptive neural control for nonlinear systems in which the practical state need not be bounded a priori. It introduces a desired approximation (DA) framework that separates the unknown nonlinearity into a part depending on a known desired trajectory, thereby relaxing the compact-set prerequisite on the actual state. An integral Lyapunov-function-based controller is developed to accommodate state-dependent input gains, and two adaptation schemes—integral and incremental—are analyzed, yielding bounded closed-loop behavior and mean-square performance characterized by computable bounds. The results show that tracking errors converge to an adjustable neighborhood without requiring knowledge of the region where the state evolves or initial weight estimates, and they provide explicit transient-performance guarantees. The incremental adaptation extension further improves practicality by avoiding online numerical integration while maintaining boundedness and performance promises.

Abstract

The inherent approximation ability of neural networks plays an essential role in adaptive neural control, where the prerequisite for existence of the compact set is crucial in the control designs. Instead of using practical system state, in this paper, the desired approximation approach is characterized to tackle such a problem, where the desired state signal is required only as the input to the network. An integral Lyapunov function-based adaptive controller is designed, in the sense of the error tracking, where the treatment of the state-dependent input gain is adopted. Theoretical results for the performance analysis of the integral and incremental adaptation algorithms are presented in details. In particular, the boundedness of the variables in the closed-loop is characterized, while the transient performance of the output error is analytically quantified. It is shown that the proposed control schemes assure that the tracking error converges to an adjustable set without any requirement on the knowledge of the region that the practical variables evolve, and remove the requirement for the setting of initial conditions including system states and weight estimates.
Paper Structure (7 sections, 8 theorems, 68 equations)

This paper contains 7 sections, 8 theorems, 68 equations.

Table of Contents

  1. Introduction
  2. Preliminaries: Conventional Approximation Versus Desired Approximation
  3. Adaptive Neural Control Using Desired Approximation
  4. Desired error trajectories
  5. DA-based adaptive neural control with integral adaptation
  6. DA-based Adaptive Neural Control with Incremental Adaptation
  7. Conclusion

Key Result

Lemma 1

For the positive function $f(t)$, being continuously differentiable for all $t > 0$ and satisfying $f(0) = 0$, and the positive function $g(t)$ is continuous for all $t \ge 0$, then $\int^t_0 g(f(s))f'(s)ds > 0$ for all $t>0$.

Theorems & Definitions (15)

  • Lemma 1
  • Lemma 2
  • Remark 1
  • Remark 2
  • Theorem 1
  • Remark 3
  • Corollary 1
  • Remark 4
  • Remark 5
  • Theorem 2
  • ...and 5 more