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Nonlinear Observer Design in Discrete-time Systems: Incorporating LMI Relaxation Strategies

Shivaraj Mohite

TL;DR

The paper tackles robust observer design for nonlinear discrete-time systems with disturbances by introducing two $\mathcal{H}_\infty$-based LMIs. The LMIs leverage generalized matrix multipliers, reformulated Lipschitz properties, a new variant of Young's inequality, and LPV concepts to increase decision-variable counts and reduce conservatism. Two theorems yield an observer gain $L$ via $L = P^{-1}R^\top$ and a LPV variant that ensures the $\mathcal{H}_\infty$ bound for all admissible nonlinearities within convex sets. The authors validate the approach with numerical examples and a SoC estimation case for Li-ion batteries, showing improved attenuation and RMSE performance compared with state-of-the-art methods. The work advances discrete-time nonlinear observer design by offering less conservative, more flexible LMIs with practical relevance to energy storage and related applications.

Abstract

This manuscript focuses on the $\mathcal{H}_\infty$ observer design for a class of nonlinear discrete systems under the presence of measurement noise or external disturbances. Two new Linear Matrix Inequality (LMI) conditions are developed in this method through the utilization of the reformulated Lipschitz property, a new variant of Young inequality and the well-known Linear Parameter Varying (LPV) approach. One of the key components of the proposed LMIs is the generalized matrix multipliers. The judicious use of these multipliers enables us to introduce more numbers of decision variables inside LMIs than the one illustrated in the literature. It aids in adding some extra degrees of freedom from a feasibility point of view, thus enhancing the LMI conditions. Thus, the established LMIs are less conservative than existing ones. Later on, the effectiveness of the developed LMIs and observer is highlighted through a numerical example and the application of state of charge (SoC) estimation in the Li-ion battery model.

Nonlinear Observer Design in Discrete-time Systems: Incorporating LMI Relaxation Strategies

TL;DR

The paper tackles robust observer design for nonlinear discrete-time systems with disturbances by introducing two -based LMIs. The LMIs leverage generalized matrix multipliers, reformulated Lipschitz properties, a new variant of Young's inequality, and LPV concepts to increase decision-variable counts and reduce conservatism. Two theorems yield an observer gain via and a LPV variant that ensures the bound for all admissible nonlinearities within convex sets. The authors validate the approach with numerical examples and a SoC estimation case for Li-ion batteries, showing improved attenuation and RMSE performance compared with state-of-the-art methods. The work advances discrete-time nonlinear observer design by offering less conservative, more flexible LMIs with practical relevance to energy storage and related applications.

Abstract

This manuscript focuses on the observer design for a class of nonlinear discrete systems under the presence of measurement noise or external disturbances. Two new Linear Matrix Inequality (LMI) conditions are developed in this method through the utilization of the reformulated Lipschitz property, a new variant of Young inequality and the well-known Linear Parameter Varying (LPV) approach. One of the key components of the proposed LMIs is the generalized matrix multipliers. The judicious use of these multipliers enables us to introduce more numbers of decision variables inside LMIs than the one illustrated in the literature. It aids in adding some extra degrees of freedom from a feasibility point of view, thus enhancing the LMI conditions. Thus, the established LMIs are less conservative than existing ones. Later on, the effectiveness of the developed LMIs and observer is highlighted through a numerical example and the application of state of charge (SoC) estimation in the Li-ion battery model.
Paper Structure (13 sections, 2 theorems, 60 equations, 1 figure, 2 tables)

This paper contains 13 sections, 2 theorems, 60 equations, 1 figure, 2 tables.

Table of Contents

  1. Introduction
  2. Nomenclature and Prerequisites
  3. Notations
  4. Preliminaries
  5. Problem statement
  6. Main result
  7. Comments related to the proposed techniques
  8. Case of the nonlinear systems with linear outputs
  9. Case of absence of exogenous input
  10. Illustrative examples
  11. Numerical example 1
  12. Application: SoC estimation of Li-ion batteries
  13. Conclusion

Key Result

Theorem 1

Let us introduce two matrices, $\mathbb{Z}$ and $\mathbb{S}$, illustrated by eq 11 matrix Z and eq 11 matrix S, respectively. If there exist matrices $P>0\in \mathbf{S}^{n}$, $R\in \mathbb{R}^{p \times n}$ and a positive scalar $\mu$ such that the following optimization problem is solvable: where $\Sigma$, $\mathbb{U}$, and $\mathbb{M}$ are described by eq LMI 1 Sigma, eq 13 U and eq 13 M, respec

Figures (1)

  • Figure 1: Graph of estimated and actual SoC

Theorems & Definitions (6)

  • Remark 1
  • Remark 2
  • Theorem 1
  • proof
  • Theorem 2
  • proof