Robust Stabilization and Observer-Based Stabilization for a Class of Singularly Perturbed Bilinear Systems

Ding-Horng Chen,Juing-Shian Chiou,Chun-Tang Chao

doi:10.3390/math9192380

Ding-Horng Chen, Juing-Shian Chiou + Show 1 more

Open Access

https://doi.org/10.3390/math9192380

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Abstract

An infinite-bound stabilization of a system modeled as singularly perturbed bilinear systems is examined. First, we present a Lyapunov equation approach for the stabilization of singularly perturbed bilinear systems for all ε∈(0, ∞). The method is based on the Lyapunov stability theorem. The state feedback constant gain can be determined from the admissible region of the convex polygon. Secondly, we extend this technique to study the observer and observer-based controller of singularly perturbed bilinear systems for all ε∈(0, ∞). Concerning this problem, there are two different methods to design the observer and observer-based controller: one is that the estimator gain can be calculated with known bounded input, the other is that the input gain can be calculated with known observer gain. The main advantage of this approach is that we can preserve the characteristic of the composite controller, i.e., the whole dimensional process can be separated into two subsystems. Moreover, the presented stabilization design ensures the stability for all ε∈(0, ∞). A numeral example is given to compare the new ε-bound with that of previous literature.

Highlights

The method is based on the Lyapunov stability theorem
If k1 and k2 are selected inside the region of the convex polygon as shown in Figure 2, the observer-based feedback (10) stabilizes the system (1) in the entire state space for all ε∈(0,∞), where the six intersections of convex polygon with k1 –axis and with k2 –axis are described as Equation (7), and k = min(k1, k2 )
The main advantage of this paper is that the whole n1 + n2 dimensional process can be separated into two parts (n1 and n2 dimensional), which do not require the slow-fast decomposition. i.e., the main advantage of the proposed approach is that the whole dimensional design can be directly separated into two lower dimensions without the decomposition and composition of the slow and fast subsystems

Summary

Introduction

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. Asamoah and Jamshidi [18] extended the results of [19] to singularly perturbed bilinear systems, but some of the results in [17] were proven wrong by [19] These papers [17,18,19] only guarantee the stability for ε∈(0, 1). Presented two kinds of robust controllers for stabilizing singularly perturbed discrete bilinear systems. An infinite ε-bound stabilization of a system modeled as singularly perturbed bilinear systems is examined. The state feedback constant gain can be determined from the admissible region of the convex polygon We extend this technique to study the observer and observer-based controller of singularly perturbed bilinear systems for all ε∈(0, ∞). 1/2 the norm of matrix A, i.e., k A k= Max [λ A T A ]

Problem Formulation and Design Procedure

Nonlinear State Feedback

Observer and Observer-Based State Feedback

Example

Whereas

Conclusions