l₁-Gain Controller Design for 2-D Markov Jump Positive Systems With Directional Delays

Zhaoxia Duan, Choon Ki Ahn, Zhengrong Xiang, Imran Ghous

Research output: Contribution to journalArticlepeer-review

9 Citations (Scopus)


This article is concerned with the stochastic stability and l₁-gain control of two-dimensional (2-D) positive Markov jump systems (PMJSs) with directional delays based on the Roesser model. First, necessary and sufficient conditions (NSCs) for the stochastic stability of the addressed system are established by constructing a deterministic ``equivalent'' system and applying a stochastic copositive Lyapunov function. This reveals that the stochastic stability of 2-D PMJSs with delays is affected by the size of directional delays, the transition matrix, and system matrices. Second, the exact l₁-gain index is calculated and NSCs in the form of linear programming (LP) are established for the addressed system. Systematic methods for the l₁-gain controller design are proposed so that the closed-loop system (CLS) is positive and stochastically stable and has an optimal l₁-gain performance, which is achieved using an iterative algorithm and an analytical calculation method for a single-input case. Finally, the potency and accuracy of the theoretical results are verified using two examples.

Original languageEnglish
JournalIEEE Transactions on Systems, Man, and Cybernetics: Systems
Publication statusAccepted/In press - 2022

Bibliographical note

Publisher Copyright:


  • Asymptotic stability
  • Controller design
  • Delays
  • Markov processes
  • Process control
  • Stability criteria
  • Switches
  • Two dimensional displays
  • directional delays
  • l₁-gain performance
  • stochastic stability
  • two-dimensional (2-D) Markov systems

ASJC Scopus subject areas

  • Software
  • Control and Systems Engineering
  • Human-Computer Interaction
  • Computer Science Applications
  • Electrical and Electronic Engineering


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