Abstract
In this chapter, the Linear Matrix Inequalities (LMIs) are presented. The terms “observability” and “observer” were also discussed. A design of linear observer-based control using the technique of LMIs is described, too.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Chang, S.S.L., Peng, T.K.C.: Adaptive guaranteed cost control systems with uncertain parameters. IEEE Trans. Autom. Control 17, 474–483 (1972)
Wonham, W.M.: Optimal stationary control of linear system with state dependent noise. SIAM J. Control (1967)
Camozzi, P.: Uncertain Modeling Systems: control by Placement. of pole, Thesis, Paul Sabatier University of Toulouse, France (1995)
Bien, Z., Kim, J.H.: A robust stability bound of linear system with structured uncertainty. IEEE Trans. Autom. Control 37(10), 1549–1551 (1992)
Gao, Z., Antsaklis, P.J.: Explicit asymmetric bounds for robust stability of continuous and discret-time systems. IEEE Trans. Autom. Control 38, 332–335 (1993)
Lyapunov, A.M.: General problem of stability of movement. Ann. Fac. Sci. Toulouse 9, 203–474 (1907)
Hahn, W.: Stability of Motion. Springer, Berlin (1967)
Parks, P.C., Hahn, V.: Stabililtäts Theorie. Springer, Berlin (1981)
Khalil, H.K.: Nonlinear System. Prentice Hall, Upper Saddle River, N. J. 07458 (1996)
Narendra, K.S., Annaswamy, A.M.: Stable Adaptive Systems. Prentice Hall Information ans System Sciences Series, N. J. 07632 (1989)
Borne, P., Dauphin-Tanguy, G., Richard, J.P., Rotella, F., Zambettakis, I.: Analysis and Regulation of Industrial Processes. Technip, Paris (1993)
Lien, C.H.: An efficient method to design robust observer-based control of uncertain linear systems. Appl. Math. Comput. 158, 29–44 (2004)
Hua, C., Li, F., Guan, X.: Observer-based adaptive control for uncertain time delay systems. Inf. Sci. (2004). (Article in press)
Chadli, M., Maquin, D., Ragot, J.: On the stability of multiple model systems. In: Proceedings of the European Control Conference, pp. 1894–1899 (2001)
Tanaka, K., Ikeda, T., He, Y.Y.: Fuzzy regulators and fuzzy observers: relaxed stability conditions and LMI-based design. IEEE Trans. Fuzzy Syst. 6(1), 250–256 (1998)
Luenberger, D.G.: Determining the state of the linear system with observers of low dynamic order, Ph.D. dissertation, Departement of Elec. Engineers., Stanford University, California (1963)
Lewis, F.L.: A survey of linear singular systems. Circuits Syst. Signal Process 5, 3–36 (1986)
Chen, J., Zhang, HH.: Robust detection of faulty actuators via unknown input observers. Int. J. Syst. Sci. 22, 1829–1839 (1991)
Soffker, D.: Fault detection using proportional integral observers for application to elastic mechanical structures. In: Proceeding of the Multiconference CESA IMACS, Symposium on Control, Optimization and Supervision. Lille, France, pp. 522–527 (1996)
Jiang, G.P., Wang, S.P., Song, W.Z.: Design of observer with intergrators for linear systems with unknown input disturbances. Electron. Lett. 36(13), 1168–1169 (2000)
Chadli, M., Maquin, D., Ragot, J.: Nonquadratic stability analysis of Takagi Sugeno models. IEEE Conf. Decision Control 2, 2143–2148 (2002)
Karamarkar, N.: A new polynomial time algorithm for linear programming. Combinatorica 4(4), 373–395 (1984)
Nesterov, Y., Balakrishnan, J.: A common Lyapunov function for stable LTI systems with commuting A-matrices. IEEE Trans. Autom. Control 39(12), 2469–2471 (1994)
Hassibi, A., How, J., Boyd, S.: A path-following method for solving BMI problems in control. In: American Control Conference, vol. 2, pp. 1385–1389. San Diego, California (1999)
Boyd, S., El Ghaoui, L., Feron, E., Balakrishnan, V.: Linear Matrix Inequalities in System and Control Theory. SIAM, Philadelphia (1994)
Chilali, M., Gabinet, P.: H∞ Design with pole placement constraints: an LMI approach. IEEE Trans. Autom. Control 41(3), 358–367 (1996)
Fossard, A.J., Normand-Cyrot, D.: Système Non Linéaire, Tome 1 : Modeling-Estimation. Masson, Paris (1993)
O’Reilly, J.: Observer for Linear System, Richard Bellman, Mathematics in Science and Engineering, vol. 140. Academic Press, New York (1983)
Follinger, O.: Regelungstechnik—Einführung in die methoden und ihre anwendungen. Dr. Alfred Huthig Verlag, Heidelberg (1985)
Borne, P., Dauphin-Tanguy, G., Richard, J.P., Rotella, F., Zambettakis, I.: Modeling and Identification of Processes, Tome 1. Editions Technip, Paris (1992)
Kalman, R.E., Betram, J.E.: Control system analysis and design via the “second method” of Lyapunov -I : Continuous-time system. ASME J. Basic Eng. 82, 371–393 (1960)
Borne, P., Dauphin-Tanguy, G., Richard, J.P., Rotella, F., Zambettakis, I.: Control and Optimization of Processes, Collection of Engineering Methods and Techniques. Editions Technip, Paris (1990)
Magni, J.F., Mouyon, P.: A generalized approach to observers for fault diagnosis. In: Proceedings of the 30th Conference on Decision and Control, vol. 3, pp. 2236–2241. Brignton, England (1991)
Dieulessaint, E., Royer, D.: Linear control systems with Brignton sampled signals. Autom. Appl. Masson (1994)
Flugge-Lotz, I., Taylor, C.: Synthesis of nonlinear control systems. IRE Trans. Autom. Control 1(1), 3–9 (1956)
Letov, A.M.: Conditionally stable control systems (on a class of optimal control systems). Autom. Remote Control 7, 649–664 (1957)
Maslennikov, V.A.: High-quality control of neutral plants by means of an astatic controller. Autom. Remote Control 2 (1956)
Ostrovsky, G.M.: Application of nonlinear correction to control systems of the second order. Autom. Remote Control 11 (1956)
Utkin, V.I.: Principles of identification using sliding regimes. Soviet Trans. Autom. Control 26, 271–272 (1981)
Utkin, V.I.: Variable structure systems with sliding modes. IEEE Trans. Autom. Control 22(22), 212–222 (1977)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2022 The Author(s), under exclusive license to Springer Nature Switzerland AG
About this chapter
Cite this chapter
Darwich, I., Etlili, D., Errachdi, A., Naifar, O. (2022). Generalities on Linear Matrix Inequalities and Observer Design of Linear Systems. In: Naifar, O., Ben Makhlouf, A. (eds) Advances in Observer Design and Observation for Nonlinear Systems. Studies in Systems, Decision and Control, vol 410. Springer, Cham. https://doi.org/10.1007/978-3-030-92731-8_2
Download citation
DOI: https://doi.org/10.1007/978-3-030-92731-8_2
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-92730-1
Online ISBN: 978-3-030-92731-8
eBook Packages: EngineeringEngineering (R0)