$\ell_{1}/\ell_{-}$ Filter-Based Fault Detection for Positive Markov Jump Linear Systems With Time Delay

The purpose of this paper is to design a positive fault detection filter for positive Markov jump linear systems (MJLSs) using <inline-formula> <tex-math notation="LaTeX">$\ell _{1}$ </tex-math></inline-formula> and <inline-formula> <tex-math notation="LaTeX">$\ell _{-}$ </tex-math></inline-formula> performance indices, which represent respectively, robustness and fault sensitivity. First, by converting the MJLS to an associated deterministic system, sufficient conditions are derived for the positivity, stochastic stability and <inline-formula> <tex-math notation="LaTeX">$\ell _{1}/\ell _{-}$ </tex-math></inline-formula> performance of an augmented system consisting of the original system and the fault detection filter. Then an iterative linear programming (LP) algorithm is provided to solve these bilinear conditions, with two numerical examples presented to show the superiority of the developed approach over the existing ones.

[1]  Hamid Reza Karimi,et al.  Mixed L - / L 1 fault detection filter design for fuzzy positive linear systems with time-varying delays , 2014 .

[2]  Minyue Fu,et al.  H∞ state estimation via asynchronous filtering for descriptor Markov jump systems with packet losses , 2019, Signal Process..

[3]  James Lam,et al.  On ℓ∞ and L∞ gains for positive systems with bounded time-varying delays , 2015, Int. J. Syst. Sci..

[4]  Oswaldo Luiz do Valle Costa,et al.  H∞-filtering for Markov jump linear systems with partial information on the jump parameter , 2017, IFAC J. Syst. Control..

[5]  Bo Wang,et al.  Delay-dependent stochastic finite-time ℓ1-gain filtering for discrete-time positive Markov jump linear systems with time-delay , 2017, J. Frankl. Inst..

[6]  James Lam,et al.  Stability and Performance Analysis for Positive Fractional-Order Systems With Time-Varying Delays , 2016, IEEE Transactions on Automatic Control.

[7]  Patrizio Colaneri,et al.  Stochastic stability of Positive Markov Jump Linear Systems , 2014, Autom..

[8]  Jun Shen,et al.  Stability and L∞‐gain analysis for a class of nonlinear positive systems with mixed delays , 2017 .

[9]  Corentin Briat.  Robust stability and stabilization of uncertain linear positive systems via Integral Linear Constraints : L 1-and L ∞-gains characterization , 2014 .

[10]  Shengyuan Xu,et al.  Robust non-fragile H∞ fault detection filter design for delayed singular Markovian jump systems with linear fractional parametric uncertainties , 2019, Nonlinear Analysis: Hybrid Systems.

[11]  W. Haddad,et al.  Nonnegative and Compartmental Dynamical Systems , 2010 .

[12]  F. Tadeo,et al.  Controller Synthesis for Positive Linear Systems With Bounded Controls , 2007, IEEE Transactions on Circuits and Systems II: Express Briefs.

[13]  Qing-Long Han,et al.  I1-gain performance analysis and positive filter design for positive discrete-time Markov jump linear systems: A linear programming approach , 2014, Autom..

[14]  Qing-Long Han,et al.  Investigating the effects of time-delays on stochastic stability and designing l1-gain controllers for positive discrete-time Markov jump linear systems with time-delay , 2016, Inf. Sci..

[15]  Steven X. Ding,et al.  Model-based Fault Diagnosis Techniques: Design Schemes, Algorithms, and Tools , 2008 .

[16]  Ho Jae Lee,et al.  H-/H∞ fault detection and isolation for nonlinear systems with state delay in T-S form , 2016, J. Frankl. Inst..

[17]  Shengyuan Xu,et al.  Control for stability and positivity: equivalent conditions and computation , 2005, IEEE Transactions on Circuits and Systems II: Express Briefs.

[18]  James Lam,et al.  Static output-feedback stabilization with optimal L1-gain for positive linear systems , 2016, Autom..

[19]  Feiqi Deng,et al.  Delay-dependent H∞ filtering for discrete singular Markov jump systems with Wiener process and partly unknown transition probabilities , 2018, J. Frankl. Inst..

[20]  Heng Wang,et al.  H-/H∞ fault detection observer design for two-dimensional Roesser systems , 2015, Syst. Control. Lett..

[21]  Peng Shi,et al.  H- / H∞ fault detection observer in finite frequency domain for linear parameter-varying descriptor systems , 2017, Autom..

[22]  Peng Shi,et al.  Dissipativity-based asynchronous filtering for periodic Markov jump systems , 2017, Inf. Sci..

[23]  Qing-Guo Wang,et al.  Event-triggered H∞ filtering of Markov jump systems with general transition probabilities , 2017, Inf. Sci..

[24]  Wassim M. Haddad,et al.  Stability theory for nonnegative and compartmental dynamical systems with time delay , 2004, Proceedings of the 2004 American Control Conference.

[25]  Zidong Wang,et al.  Fault detection for switched positive systems under successive packet dropouts with application to the Leslie matrix model , 2016 .

[26]  Corentin Briat,et al.  Robust stability and stabilization of uncertain linear positive systems via integral linear constraints: L1‐gain and L∞‐gain characterization , 2012, ArXiv.

[27]  Z. Xiang,et al.  Robust fault detection for switched positive linear systems with time-varying delays. , 2014, ISA transactions.

[28]  Xiaojie Su,et al.  Fault detection filtering for nonlinear switched systems via event-triggered communication approach , 2019, Autom..

[29]  Peng Shi,et al.  Multi-Objective H-/H∞ Fault Detection Filter Design For T-S Fuzzy Systems With Finite Frequency Specifications , 2018 .

[30]  Renquan Lu,et al.  Event-triggered H∞ filter design for Markovian jump systems with quantization , 2018 .

[31]  James Lam,et al.  $\ell_{\infty}/L_{\infty} $-Gain Analysis for Positive Linear Systems With Unbounded Time-Varying Delays , 2015, IEEE Transactions on Automatic Control.

[32]  Peng Shi,et al.  Robust Fault Detection for Switched Linear Systems With State Delays , 2009, IEEE Transactions on Systems, Man, and Cybernetics, Part B (Cybernetics).

[33]  P. Shi,et al.  Robust fault detection for continuous-time switched delay systems: an linear matrix inequality approach , 2010 .