Finite time stabilization of the four tanks system: extensions to the uncertain systems. (English) Zbl 1406.93236

Summary: We consider the finite time stability and stabilization of linear systems described in continuous time. First, we provide a condition for the stability over time using the state transition matrix standard. Then we give conditions to design a state feedback control that stabilizes the system over time. In some cases where there is uncertainty in the system model, the previous conditions are extended to a certain class of uncertain systems. The considered uncertainties are the polytopic and norm bounded ones. To reveal the proposed approach, an application to the four tanks system was made.


93D05 Lyapunov and other classical stabilities (Lagrange, Poisson, \(L^p, l^p\), etc.) in control theory
93C41 Control/observation systems with incomplete information
93C15 Control/observation systems governed by ordinary differential equations
93C05 Linear systems in control theory
93B52 Feedback control
Full Text: DOI


[1] Tarbouriech, S.; Garcia, G., Control of Uncertain Systems with Bounded Inputs (1997), Berlin, Germany: Springer, Berlin, Germany · Zbl 0868.00028
[2] Bhat, S. P.; Bernstein, D. S., Continuous finite-time stabilization of the translational and rotational double integrators, IEEE Transactions on Automatic Control, 43, 5, 678-682 (1998) · Zbl 0925.93821
[3] Abhilash, P. M.; Mahindrakar, A. D., Stabilization of a circular ball-and-beam system with input and state constraints using linear matrix inequalities, Proceedings of the IEEE International Conference on Systems, Man and Cybernetics (SMC ’08)
[4] Dorato, P., Short time stability in linear time-varying systems, Proceedings of the IRE International Convention Record Part 4
[5] Boyd, S.; El Ghaoui, L.; Feron, E.; Balakrishnan, V., Linear Matrix Inequalities in System and Control Theory (1994), Philadelphia, Pa, USA: SIAM Press, Philadelphia, Pa, USA · Zbl 0816.93004
[6] Bhattacharyya, S. P.; Chapellat, H.; Keel, L. H., Robust Control: The Parametric Approach (1995), Upper Saddle River, NJ, USA: Prentice Hall, Upper Saddle River, NJ, USA · Zbl 0838.93008
[7] Gronwall, T. H., Note on the derivatives with respect to a parameter of the solutions of a system of differential equations, Annals of Mathematics, 20, 4, 292-296 (1919) · JFM 47.0399.02
[8] Weiss, L.; Infante, E. F., Finite time stability under perturbing forces and on product spaces, IEEE Transactions on Automatic Control, 12, 1, 54-59 (1967) · Zbl 0168.33903
[9] Moulay, E., Contribution à l’étude de la stabilité en temps fini et de la stabilisation [Thèse de Doctorat] (2005), Ecole centrale de Lille
[10] Mabrouk, W. B.; Njima, C. B.; Messaoud, H.; Garcia, G., Finite-time stabilization of nonlinear affine systems, Journal Européen des Systèmes Automatisés, 44, 3, 327-343 (2010)
[11] Njima, C. B.; Mabrouk, W. B.; Garcia, G.; Messaoud, H., Robust finite-time stabilization of nonlinear systems, International Review of Automatic Control, 4, 3, 362-369 (2011)
[12] Garcia, G.; Tarbouriech, S.; Bernussou, J., Finite-time stabilization of linear time-varying continuous systems, IEEE Transactions on Automatic Control, 54, 2, 364-369 (2009) · Zbl 1367.93060
[13] Grujic, L. T., On practical stability, International Journal of Control, 17, 4, 881-887 (1973) · Zbl 0254.93039
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