Finite-time control of linear systems subject to parametric uncertainties and disturbances. (English) Zbl 0983.93060

The authors consider a linear system subject to time-varying parametric uncertainties and to exogenous constant disturbances \[ \dot x(t) = A(p)x(t) + B(p)u(t) + G(p)w, \] where \(A(p)\in\mathbb R^{n\times n}\), \(B(p)\in\mathbb R^{n\times m}\) and \(G(p)\in\mathbb R^{n\times l}\). The concept of finite-time boundedness for the state of a system, when not only given initial conditions but also external constant disturbances are considered, is introduced. The main result provided is a sufficient condition guaranteeing finite-time boundedness via state feedback. It can be applied to problems with both non-zero initial conditions and unknown constant disturbances. This condition is turned into an optimization problem involving LMIs. A detailed example is presented to illustrate the proposed methodology.


93D21 Adaptive or robust stabilization
93C73 Perturbations in control/observation systems
15A39 Linear inequalities of matrices


LMI toolbox
Full Text: DOI


[1] Amato, F.; Garofalo, F.; Glielmo, L.; Pironti, A., Robust and quadratic stability via polytopic set covering, International Journal of Robust and Nonlinear Control, 5, 745-756 (1995) · Zbl 0851.93058
[2] Bhattacharyya, S. P.; Chapellat, H.; Keel, L. H., Robust control: The parametric approach (1995), Prentice Hall PTR: Prentice Hall PTR Upper Saddle River, NJ · Zbl 0838.93008
[3] Boyd, S.; El Ghaoui, L.; Feron, E.; Balakrishnan, V., Linear matrix inequalities in system and control theory (1994), SIAM Press: SIAM Press Philadelphia, PA · Zbl 0816.93004
[4] Corless, M., Robust stability analysis and controller design with quadratic Lyapunov Functions, (Zinober, A. S.I., Variable structure and lyapunov control (1993), Springer: Springer Berlin) · Zbl 0803.93032
[6] Gahinet, P.; Nemirovski, A.; Laub, A. J.; Chilali, M., LMI control toolbox (1995), The Mathworks Inc: The Mathworks Inc Natick, MA
[7] Garofalo, F.; Celentano, G.; Glielmo, L., Stability robustness of interval matrices via quadratic Lyapunov forms, IEEE Transactions on Automatic Control, 38, 281-284 (1993) · Zbl 0774.93061
[9] Geromel, J. C.; Peres, P. L.D.; Bernussou, J., On a convex parameter space method for linear control design of uncertain systems, SIAM Journal on Control and Optimization, 29, 381-402 (1991) · Zbl 0741.93020
[11] Grantham, W. J., Estimating Reachable Sets, ASME Journal of Dynamic Systems and Measurement Control, 103, 420-422 (1981) · Zbl 0467.93032
[12] Weiss, L.; Infante, E. F., Finite time stability under perturbing forces and on product spaces, IEEE Transactions on Automatic Control, 12, 54-59 (1967) · Zbl 0168.33903
[13] Zhou, K.; Doyle, J. C., Essentials of Robust Control (1998), Prentice Hall, Inc: Prentice Hall, Inc Upper Saddle River, NJ
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