Fault Detection Filter Design of Polytopic Uncertain Continuous-Time Singular Markovian Jump Systems with Time-Varying Delays

Yunling Shi,Xiuyan Peng

doi:10.1155/2020/3736467

Abstract

This paper investigates the problem of full-order and reduced-order fault detection filter (FDF) design under unified linear matrix inequality (LMI) conditions for a class of continuous-time singular Markovian jump systems (CTSMJSs) with time-varying delays and polytopic uncertain transition rates. By constructing a new Lyapunov function, sufficient conditions are firstly provided for the singular model error augmented system such that the system is stochastically admissible with an H∞ performance level γ. And then, by applying a novel convex polyhedron technique to decoupled linear matrix inequalities, the full-order and reduced-order fault detection filter parameters can be obtained within a convex optimization frame. The reduced-order fault detection filter (FDF) can not only meet the fault detection accuracy requirements of complex systems but also improve the fault detection efficiency. Finally, a DC motor and an illustrative simulation example are given to verify the feasibility and effectiveness of the proposed algorithms.

Highlights

Markovian jump systems (MJSs) are usually defined as a family of complex random jumping parameter systems [1,2,3,4]. e past few decades have witnessed the prosperous research on MJSs because it can simulate the structure and parameter changes of real dynamic systems, such as biological economics systems, network control systems, power electronic systems, and mechanical engineering systems [5,6,7,8,9]
It is worth mentioning that when the structure or parameters of the singular dynamic hybrid system change, the singular Markovian jump systems (SMJSs) can effectively simulate this phenomenon by using the Markov chain transformation, and many fruitful about SMJSs have been shown in recent years [17,18,19]. e transition rates (TRs) are the concrete expression of Markov chains, which can express the random transition state of the actual dynamic systems [20,21,22]
Since the singular Markovian jump systems are more complex than the ordinary Markovian jump systems, the stability, regularity, and nonimpulsiveness need to be considered [32,33,34,35]; especially for polytopic uncertain continuous-time singular Markovian jump systems, few results can be found related to the fault detection filter designed problems under this case

Summary

Introduction

Markovian jump systems (MJSs) are usually defined as a family of complex random jumping parameter systems [1,2,3,4]. e past few decades have witnessed the prosperous research on MJSs because it can simulate the structure and parameter changes of real dynamic systems, such as biological economics systems, network control systems, power electronic systems, and mechanical engineering systems [5,6,7,8,9]. Since the singular Markovian jump systems are more complex than the ordinary Markovian jump systems, the stability, regularity, and nonimpulsiveness need to be considered [32,33,34,35]; especially for polytopic uncertain continuous-time singular Markovian jump systems, few results can be found related to the fault detection filter designed problems under this case. It is worth mentioning that the fault detection of polytopic uncertain CTSMJSs with time-varying delays, which contains completely known, partly unknown, and completely unknown TRs, has not been studied. (II) e innovation reduced-order FDF design method of polytopic uncertain CTSMJSs with time-varying delays is proposed, which satisfies the stochastically admissible sufficient conditions and the H∞ performance standard cmin for the first time. About real symmetric matrix P, P > 0(P ≥ 0) means that P is the real symmetric positive (semipositive) and ∗ represents the symmetric term that has been ellipsis. ‖·‖ denotes the Euclidean norm for vectors. l2[0, ∞) is the space of square summable infinite sequence, and fo􏽱r w { w ( t)} ∈ l2[0, ∞), its norm is given by ‖w‖2 􏽒∞ 0 ‖w‖2dt

Problem Formulation

Performance Analysis of FDFS

Numerical Example

Conclusions