$$H_\infty $$ dynamic observer design for discrete-time linear systems with time varying delays based on generalized reciprocally convex matrix inequality

Ghali Naami1, Mohamed Ouahi1, Abdelhamid Rabhi2, Mohamed Larbi Elhafyani3
1National School of Applied Sciences, Sidi Mohamed Ben Abdellah University, Fez, Morocco
2Modeling Information, and Systems Laboratory, University of Picardie Jules Verne, Amiens, France
3Higher School of Technology, University Med First Oujda, Oujda, Morocco

Tóm tắt

In this work, we have studied the problem of designing $$H_\infty $$ dynamic observers (HDO) for discrete-time linear systems (DTLS) with time-varying delay (TVD) and disturbance. By designing an augmented Lyapunov-Krasovskii functional (LKF) with double summation terms using the Generalized reciprocally convex matrix inequality (GRCMI), as well as the Jensen-based inequality (JBI) and the Wirtinger-based inequality (WBI) that derive new less conservative time-dependent conditions. The resulting algebraic conditions form a set of linear matrix inequalities (LMIs) which can be solved by the LMI or YALMIP toolboxes. Furthermore, the observer under consideration has more degrees of freedom to be estimated, and is known as a generalized observer, where the proportional and proportional-integral observers are particular cases. Finally, two examples are given to demonstrate the validity and effectiveness of the findings.

Tài liệu tham khảo

Boutat-Baddas L, Osorio-Gordillo GL, Darouach M (2021) \(H_\infty \) dynamic observers for a class of nonlinear systems with unknown inputs. Int J Control 94(3):558–569 Boyd S, El Ghaoui L, Feron E, Balakrishnan V (1994) Linear matrix inequalities in system and control theory. Society for industrial and applied mathematics Busawon KK, Kabore P (2001) Disturbance attenuation using proportional integral observers. Int J Control 74(6):618–627 Chen Q, Liu X, Wang F (2022) Improved results on \( L_2-L_\infty \) state estimation for neural networks with time-varying delay. Circiut Syst Signal Process 41(1):122–146 Cheok KC, Loh NK, Beck RR (1982) General structured observers for discrete-time linear systems. In: American control conference 1982, pp 614–619 Friedland B (1996) Full-order state observer. Department of electrical and computer engineering, New Jersey Institute of Technology, NEWARY, NEW JERSEY, USA Gao N, Darouach M, Voos H, Alma M (2016) New unified \(H_\infty \) dynamic observer design for linear systems with unknown inputs. Automatica 65:43–52 Gao N, Darouach M, Alma M (2017) \(H_\infty \) dynamic observer design for linear discrete-time systems. IFAC-PapersOnLine 50(1):2756–2761 Jiang X, Han QL, Yu X (2005). Stability criteria for linear discrete-time systems with interval-like time-varying delay. In: Proceedings of the 2005, American control conference, 2005, IEEE, pp 2817-2822 Kaczorek T (1979) Proportional-integral observers for linear multivariable time-varying systems/Proportional-Integral-Beobachter für lineare. Zeitvariable Mehrgrößensysteme. at-Automatisierungstechnik 27(1–12):359–363 Lee SY, Park J, Park P (2019) Bessel summation inequalities for stability analysis of discrete-time systems with time-varying delays. Int J Robust Nonlinear Control 29(2):473–491 Leondes CT, Novak LM (1974) Reduced-order observers for linear discrete-time systems. IEEE Trans Autom Control 19(1):42–46 Li X, Wang R, Du S, Li T (2022) An improved exponential stability analysis method for discrete-time systems with a time-varying delay. Int J Robust Nonlinear Control 32(2):669–681 Liu K, Seuret A, Fridman E, Xia Y (2018) Improved stability conditions for discrete-time systems under dynamic network protocols. Int J Robust Nonlinear Control 28(15):4479–4499 Luenberger D (1971) An introduction to observers. IEEE Trans Autom Control 16(6):596602 Naami G, Ouahi M, Sadek BA, Tadeo F, Rabhi A (2022) Delay-dependent \(H_\infty \) dynamic observers for non-linear systems with multiple time-varying delays. Transact Inst Measure Control 44(15):2998–3015 Nam PT, Luu TH (2020) A new delay-variation-dependent stability criterion for delayed discrete-time systems. J Franklin Inst 357(11):6951–6967 Nam PT, Pathirana PN, Trinh H (2015) Discrete Wirtinger-based inequality and its application. J Franklin Inst 352(5):1893–1905 Orjuela R, Marx B, Ragot J, Maquin D, et al (2007) PI observer design for discrete-time decoupled multiple models. In 5th workshop on advanced control and diagnosis, ACD Osorio-Gordillo GL, Darouach M, Astorga-Zaragoza CM, Boutat-Baddas L (2019) Generalised dynamic observer design for Lipschitz non-linear descriptor systems. IET Control Theory Appl 13(14):2270–2280 Park JK, Shin DR, Chung TM (2002) Dynamic observers for linear time-invariant systems. Automatica 38(6):1083–1087 Park P, Ko JW, Jeong C (2011) Reciprocally convex approach to stability of systems with time-varying delays. Automatica 47(1):235–238 Pérez-Estrada AJ, Osorio-Gordillo GL, Alma M, Darouach M, Olivares-Peregrino VH (2018) \(H_\infty \) generalized dynamic unknown inputs observer design for discrete LPV systems. Application to wind turbine. Eur J Control 44:40–49 Qiu SB, Liu XG, Wang FX, Chen Q (2019) Stability and passivity analysis of discrete-time linear systems with time-varying delay. Syst Control Lett 134:104543 Seuret A, Gouaisbaut F, Fridman E (2015) Stability of discrete-time systems with time-varying delays via a novel summation inequality. IEEE Trans Autom Control 60(10):2740–2745 Shafai B, Carroll RL (1985) Design of proportional-integral observer for linear time-varying multivariable systems. In: 1985 24th IEEE conference on decision and control, IEEE, pp 597-599 Zhang CK, Long F, He Y, Yao W, Jiang L, Wu M (2020) A relaxed quadratic function negative-determination lemma and its application to time-delay systems. Automatica 113:108764 Zhang CK, Long F, He Y, Yao W, Jiang L, Wu M (2020) A relaxed quadratic function negative-determination lemma and its application to time-delay systems. Automatica 113:108764