Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

A nonparametric learning framework for nonlinear robust output regulation

Shimin Wang, Martin Guay, Zhiyong Chen, Richard D. Braatz

TL;DR

This work introduces a nonparametric learning framework for global nonlinear robust output regulation, relaxing the common assumption that the steady-state generator is linear in the exogenous signal to a polynomial form and removing the need for explicit regressor construction. It develops an explicit nonlinear mapping χ for polynomial generators and a nonparametric learning approach to online estimate the generator coefficients a(σ), enabling robust, non-adaptive stabilization of the augmented system with iISS dynamics. The framework also extends to parameter estimation of multi-tone sinusoids with unknown frequencies, yielding exponential convergence of the estimated parameters and the reconstructed exosignals, and it provides a feedforward control design to solve linear output regulation. The approach is validated on nonlinear examples such as the Lorenz system, highlighting practical implications for simplifying implementation and extending applicability to systems with uncertain exosystems and unknown harmonic content.

Abstract

A nonparametric learning solution framework is proposed for the global nonlinear robust output regulation problem. We first extend the assumption that the steady-state generator is linear in the exogenous signal to the more relaxed assumption that it is polynomial in the exogenous signal. Additionally, a nonparametric learning framework is proposed to eliminate the construction of an explicit regressor, as required in the adaptive method, which can potentially simplify the implementation and reduce the computational complexity of existing methods. With the help of the proposed framework, the robust nonlinear output regulation problem can be converted into a robust non-adaptive stabilization problem for the augmented system with integral input-to-state stable (iISS) inverse dynamics. Moreover, a dynamic gain approach can adaptively raise the gain to a sufficiently large constant to achieve stabilization without requiring any a priori knowledge of the uncertainties appearing in the dynamics of the exosystem and the system. Furthermore, we apply the nonparametric learning framework to globally reconstruct and estimate multiple sinusoidal signals with unknown frequencies without the need for adaptive parametric techniques. An explicit nonlinear mapping can directly provide the estimated parameters, which will exponentially converge to the unknown frequencies. Finally, a feedforward control design is proposed to solve the linear output regulation problem using the nonparametric learning framework. Two simulation examples are provided to illustrate the effectiveness of the theoretical results.

A nonparametric learning framework for nonlinear robust output regulation

TL;DR

This work introduces a nonparametric learning framework for global nonlinear robust output regulation, relaxing the common assumption that the steady-state generator is linear in the exogenous signal to a polynomial form and removing the need for explicit regressor construction. It develops an explicit nonlinear mapping χ for polynomial generators and a nonparametric learning approach to online estimate the generator coefficients a(σ), enabling robust, non-adaptive stabilization of the augmented system with iISS dynamics. The framework also extends to parameter estimation of multi-tone sinusoids with unknown frequencies, yielding exponential convergence of the estimated parameters and the reconstructed exosignals, and it provides a feedforward control design to solve linear output regulation. The approach is validated on nonlinear examples such as the Lorenz system, highlighting practical implications for simplifying implementation and extending applicability to systems with uncertain exosystems and unknown harmonic content.

Abstract

A nonparametric learning solution framework is proposed for the global nonlinear robust output regulation problem. We first extend the assumption that the steady-state generator is linear in the exogenous signal to the more relaxed assumption that it is polynomial in the exogenous signal. Additionally, a nonparametric learning framework is proposed to eliminate the construction of an explicit regressor, as required in the adaptive method, which can potentially simplify the implementation and reduce the computational complexity of existing methods. With the help of the proposed framework, the robust nonlinear output regulation problem can be converted into a robust non-adaptive stabilization problem for the augmented system with integral input-to-state stable (iISS) inverse dynamics. Moreover, a dynamic gain approach can adaptively raise the gain to a sufficiently large constant to achieve stabilization without requiring any a priori knowledge of the uncertainties appearing in the dynamics of the exosystem and the system. Furthermore, we apply the nonparametric learning framework to globally reconstruct and estimate multiple sinusoidal signals with unknown frequencies without the need for adaptive parametric techniques. An explicit nonlinear mapping can directly provide the estimated parameters, which will exponentially converge to the unknown frequencies. Finally, a feedforward control design is proposed to solve the linear output regulation problem using the nonparametric learning framework. Two simulation examples are provided to illustrate the effectiveness of the theoretical results.
Paper Structure (15 sections, 12 theorems, 104 equations, 3 figures)

This paper contains 15 sections, 12 theorems, 104 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Problem Formulation and Objectives
  3. System models and problem formulation
  4. Standard assumptions
  5. Linear generic internal model design and error system dynamics
  6. Objectives
  7. A Nonparametric Learning Framework for Nonlinear Robust Output Regulation
  8. An explicit nonlinear mapping for the polynomial steady-state generator
  9. A nonparametric learning framework for the nonlinear robust output regulation problem
  10. A Nonparametric Learning Framework for Parameter Estimation and Feedforward Control
  11. A single-input single-output (SISO) linear system
  12. Parameter estimation problem of the multi-tone sinusoidal signals with unknown frequencies
  13. Feedforward control design for output regulation
  14. Numerical and Practical Examples
  15. Example 1: Controlled Lorenz systems

Key Result

Theorem 1

For the composite system Main-sys1 and Exosys1 under Assumptions H1 and H2, there is a positive smooth function $\rho(\cdot)$, positive number $k^*$ and the controller, solves Problem ldlesp. Furthermore, the closed-loop system composed of augmen-1 and ESC-1 has the property that there exists a continuous positive definite function $V{\color{black} \equiv}V(\bar{Z}, e )$ such that, for all $\mu\i

Figures (3)

  • Figure 1: A nonparametric learning solution to the time-varying linear equation.
  • Figure 2: Nonparametric learning framework to parameter/frequency estimation.
  • Figure :

Theorems & Definitions (24)

  • Theorem 1
  • Corollary 1
  • Remark 1
  • Remark 2
  • Lemma 1
  • Lemma 2
  • proof
  • Remark 3
  • Lemma 3
  • proof
  • ...and 14 more