Journal of Scientific and Industrial Research

Open Access Open Access  Restricted Access Subscription Access

A Novel Approach to Ensure Robust Stability using Unsymmetric Lyapunov Matrix for 2-D Discrete Model


Affiliations
1 School of Electrical & Electronics Engineering, VIT Bhopal University, Sehore 466 114, Madhya Pradesh, India
 

This paper addresses the issue of ensuring the asymptotic stability of the two-dimensional discrete Roesser model. Most of the emphasis is given now a days on the stability analysis of two-dimensional discrete models because of their wide variety of real time applications. When it comes of ensuring the stability of any system, the most generalized method is to use symmetric Lyapunov function. There have been a lot of published articles in which the stability of the system has been ensured using the symmetrical Lyapunov function, but use of unsymmetrical Lyapunov function has not been adopted due to the computational complexity. One of the very popular two dimensional discrete model is the Roesser model, which is structurally different from other two dimensional discrete models and has its wide applications in the field of image processing. In this article the stability of a 2-D discrete Roesser model has been ensured using the unsymmetrical Lyapunov function, which is a more generalized way of ensuring the stability of any system. Accordingly, new stability conditions have been developed, which is an extension of the previously reported methods in which the stability is made certain using the symmetrical Lyapunov matrix. In some cases, it has been shown numerically that it is difficult to ensure stability using the symmetric Lyapunov matrix but still, the stability for such cases may be ensured using the unsymmetric Lyapunov matrix. In addition, symmetrical Lyapunov matrix stability conditions have also been derived using the unsymmetric Lyapunov matrix. The stability criteria have been checked and ensured based on newly developed stability conditions by considering two different examples. An effort has been put in reducing the conservatism with the new stability conditions.

Keywords

Asymptotic stability analysis, Roessermodel, Two-dimensional discrete system, Unsymmetric lyapunov function.
User
Notifications
Font Size

  • Kanellakis A, Tawfik A & Agathoklis P, Stability analysis of 2‐D discrete and continuous state‐space systems, IET Control Theory A, 16(3) (2022) 273–282, doi: org/10.1049/ cth2.12224.

  • Wang F, Wang Z, Liang J & Yang J, A survey on filtering issues for two-dimensional systems: Advances and challenges, Int J Control Autom, 18(3) (2020) 629–642, doi: 10.1007/s12555-019-1000-x.

  • Wang J, Guo T & Xiang P, Stability analysis and control synthesis of uncertain Roesser-type discrete-time two-dimensional systems, Chinese Phys B, 22(3) (2013) 030206, doi: 10.1088/1674-1056/22/3/030206.

  • Tandon A, Dhawan A & Tiwari M, Optimal guaranteed cost control of uncertain 2-D discrete state-delayed systems described by the Roesser model via memory state feedback, T I Meas Control, 41(1) (2019) 285–294, doi:10.1177/ 0142331218754623.

  • Mohsenipour R & Agathoklis P, Algebraic necessary and sufficient conditions for testing stability of 2-D linear systems, IEEE T Automat Contr, 66(4) (2020) 1825–1831, doi: 10.1109/tac.2020.2999020.

  • Ettore F & Giovanni M, State-space realization theory of two-dimensional filters, IEEE T Automat Contr, 21(4) (1976) 484–492, doi: 10.1109/tac.1976.1101305.

  • Mohamed B, LP conditions for stability stabilization of positive 2D discrete state-delayed Roesser models, Int J Control Autom, 16(6) (2018) 2814–2821, doi: org/10.1007/s12555-017-0464-9.

  • Roesser R A, Discretestate-space model for linear image processing, IEEE T Automat Contr, 20(1)(1975) 1–10, doi: 10.1109/tac.1975.1100844.

  • Dhawan A & Kar H, An LMI approach to robust optimal guaranteed cost control of 2-D discrete systems described by the Roesser model, Signal Process, 90(9) (2010) 2648–2654, doi: 10.1016/j.sigpro.2010.03.008.

  • Gopal M, Modern Control System Theory, 4th edn, (New Age International, New Delhi) 1993.

  • Bachelier O, Paszke W, Yeganefar N, Mehdi D & Cherifi A, LMI stability conditions for 2D Roesser models, IEEE T Automat Contr, 61(3) (2015) 766–770, doi: 10.1109/tac.2015. 2444051.

  • Kandanvli V K R & Kar H, Global asymptotic stability of 2-D digital filters with a saturation operator on the state-space, IEEE T Circuits-II, 67(11) (2022) 2742–2746, doi: 10.1109/tcsii.2022.3229072.

  • Chairez I & Poznyak A, Sliding-mode control of full-state constraint nonlinear systems: A barrier Lyapunov function approach, IEEE T Syst Man Cyb, 52(10) (2020) 6593–6606, doi: 10.1109/tsmc.2022.3148695.

  • Pandey S, Tadepalli K, Leite S, Nigam R & Bhusnur S, Stability of 2-D discrete systems in the presence of saturation function and delays, Int J Control Autom, 21(3) (2023) 788–799, doi: 10.1007/s12555-021-1061-5.

  • Soni B P, Saxena A, Gupta V & Surana S, Transient stability‐oriented assessment and application of preventive control action for power system, J Eng, (18) (2019) 5345–5350, doi: org/10.1049/joe.2018.9353.

  • Gupta E & Saxena A, Robust generation control strategy based on grey wolf optimizer, J Elect Syst, 11(2) (2015) 174–188.

  • Thakur S & Barai R K, Addressing model uncertainties and external disturbances in optimal robust control for vibration reduction in a Flexible Link Manipulator, J Sci Ind Res, 82(7) (2023) 700–710, doi: 10.56042/jsir.v82i07.1844.

  • Peng D & Zhang Y, Stability analysis for 2‐D switched Takagi‐Sugeno fuzzy systems with stable and unstable subsystems, Asian J Control, 25(3) (2023) 2204–2213, doi: org/10.1002/asjc.2901.

  • Emelianova J P, Stabilization of nonlinear Fornasini–Marchesini systems, Automat Rem Contr, 79(10) (2018) 1903–1911, doi: 10.1134/s0005117918100132.

  • Massera C M, Terra M H & Wolf D F, Optimal guaranteed cost control of discrete-time linear systems subject to structured uncertainties, Int J Contr, 94(4) (2021) 1132–1142, doi: 10.1080/00207179.2019.1634838.

  • Saxena A, Sharma A & Shekhawat S, Parameter extraction of solar cell using intelligent grey wolf optimizer, Ecol Intel, 15(1) (2022) 167–183, doi: org/10.1007/s12065-020-00499-1.

  • Singh V, Stability analysis of 2-D linear discrete systems based on the Fornasini-Marchesini second model: Stability with asymmetric Lyapunov matrix, Digit Signal Process, 26(1) (2014) 183–186, doi: 10.1016/j.dsp.2013.11.008.

  • Anderson B D O, Panajotis A, Jury E I & Mansour M, Stability and the matrix Lyapunov equation for discrete 2-D systems, IEEE T Circuits Syst, 33(3) (1986) 261–267, doi: 10.1109/tcs.1986.1085912.

  • Singh V, Stability of discrete-time systems joined with a saturation operator on the statespace: generalized form of Liu-Michel’s criterion, Automatica, 47(3) (2011) 634–637, doi: 10.1016/j.automatica.2011.01.041.


Abstract Views: 106

PDF Views: 57




  • A Novel Approach to Ensure Robust Stability using Unsymmetric Lyapunov Matrix for 2-D Discrete Model

Abstract Views: 106  |  PDF Views: 57

Authors

Govind Prasad Pandiya
School of Electrical & Electronics Engineering, VIT Bhopal University, Sehore 466 114, Madhya Pradesh, India
Abhay Vidyarthi
School of Electrical & Electronics Engineering, VIT Bhopal University, Sehore 466 114, Madhya Pradesh, India

Abstract


This paper addresses the issue of ensuring the asymptotic stability of the two-dimensional discrete Roesser model. Most of the emphasis is given now a days on the stability analysis of two-dimensional discrete models because of their wide variety of real time applications. When it comes of ensuring the stability of any system, the most generalized method is to use symmetric Lyapunov function. There have been a lot of published articles in which the stability of the system has been ensured using the symmetrical Lyapunov function, but use of unsymmetrical Lyapunov function has not been adopted due to the computational complexity. One of the very popular two dimensional discrete model is the Roesser model, which is structurally different from other two dimensional discrete models and has its wide applications in the field of image processing. In this article the stability of a 2-D discrete Roesser model has been ensured using the unsymmetrical Lyapunov function, which is a more generalized way of ensuring the stability of any system. Accordingly, new stability conditions have been developed, which is an extension of the previously reported methods in which the stability is made certain using the symmetrical Lyapunov matrix. In some cases, it has been shown numerically that it is difficult to ensure stability using the symmetric Lyapunov matrix but still, the stability for such cases may be ensured using the unsymmetric Lyapunov matrix. In addition, symmetrical Lyapunov matrix stability conditions have also been derived using the unsymmetric Lyapunov matrix. The stability criteria have been checked and ensured based on newly developed stability conditions by considering two different examples. An effort has been put in reducing the conservatism with the new stability conditions.

Keywords


Asymptotic stability analysis, Roessermodel, Two-dimensional discrete system, Unsymmetric lyapunov function.

References